Big Ideas Math Answers Grade 8 Chapter 7 Functions

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Big Ideas Math Book 8th Grade Answer Key Chapter 7 Functions

Big Ideas Math Grade 8 Ch 7 Functions Solutions provided here cover the questions from Lessons 7.1 to 7.5, Practice Tests, Review Tests, Cumulative Practice, Assessment Tests, etc. You can have deeper insights into all the Topics of Functions such as the representation of functions, linear functions, analyzing and sketching graphs, etc. Use the below available quick links for Big Ideas Math 8th Grade Chapter 7 Functions Answer Key and clear all your ambiguities regarding the concerned topics.

STEAM Video/Performance Task

Getting Ready for Chapter 7

Lesson 1 Relations and Functions

Lesson 2 Representations of Functions

Lesson 3 Linear Functions

Lesson 4 Comparing Linear and Non Linear Functions

Lesson 5 Analyzing and Sketching Graphs

Functions Connecting Concepts

Functions STEAM Video/Performance Task

STEAM Video

Apparent Temperature
Sometimes it feels hotter or colder outside than the actual apparent temperature. How hot or cold it feels is called the temperature. What weather factors might contribute to the apparent temperature?
Watch the STEAM Video “Apparent Temperature.” Then answer the following questions.

1. Robert says that the Wet-Bulb Globe Temperature (WBGT)index is used as a measure of apparent temperature.

In the formula, TW is the natural wet-bulb temperature, TG is the black-globe temperature, TD and is the dry-bulb temperature. Find WBGT when TW = 75ºF, TG = 100ºF, and TD = 84ºF.
2. Different categories of Wet-Bulb Globe Temperatures are shown in the chart. Each category can be represented by a different-colored ﬂag. Which flag color is displayed when WGBT = 87.5ºF?

Performance Task

Heat Index
After completing this chapter, you will be able to use the STEAM concepts you learned to answer the questions in the Video Performance Task. You will be given information about heat index.

You will be asked to create a graph of the temperatures and heat indices. Why is it useful to know the heat index?

Functions Getting Ready for Chapter 7

Chapter Exploration

Work with a partner. Copy and complete the diagram.

1.  Answer: ( 1 , 2 ) , ( 2 , 4 ) , ( 3 , 6 ) , ( 4 , 8 ) .

Explanation:
Given , Width w = 2 , Length l = x  where x = 1 , 2 , 3 , 4 .
To find Area A of a rectangle we have , A = w × l
for x = 1 , A = 2 × 1 = 2 ,
for x = 2 , A = 2 × 2 = 4 ,
for x = 3 , A = 2 × 3 = 6 ,
for x = 4 , A = 2 × 4 = 8 ,
So, for every value of Input x = 1 , 2 , 3 , 4 we have Output A = 2 , 4 , 6 , 8 , respectively .
That is ( 1 , 2 ) , ( 2 , 4 ) , ( 3 , 6 ) , ( 4 , 8 ) .

2. Answer: ( 1 , 6 ) , ( 2 , 8 ) , ( 3 , 10 ) , ( 4 , 12 ).

Explanation:
Given , Width w = 2 , Length l = x  where x = 1 , 2 , 3 , 4 .
To find Perimeter of a rectangle we have , P = 2( l + w )
for x = 1 ,P = 2( 1 + 2 ) = 2 × 3 = 6 ,
for x = 2 , P = 2( 2 + 2 ) = 2 × 4 = 8  ,
for x = 3 , P = 2( 3 + 2 ) = 2 × 5 = 10  ,
for x = 4 , P = 2( 4 + 2 ) = 2 × 6 = 12  ,
So, for every value of Input x = 1 , 2 , 3 , 4 we have Output P = 6 , 8 , 10 , 12 , respectively .
That is ( 1 , 6 ) , ( 2 , 8 ) , ( 3 , 10 ) , ( 4 , 12 ) .

3. Answer : ( 1 , 6 ) , ( 2 , 12 ) , ( 3 , 18 ) , ( 4 , 24 ) .

Explanation:
Given , Radius of a circle , where as  r = 1 , 2 , 3 , 4
To find the circumference of a circle , we have C = 2Òr , Ò = 3.14 , or we can write it as 3 .
for r = 1 , C = 2 × 3 × 1 = 6 ,
for r = 2 , C = 2 × 3 × 2 = 12 ,
for r = 3 , C = 2 × 3 × 3 = 18 ,
for r = 4 , C = 2 × 3 × 4 = 24 ,
So, for every value of Input r = 1 , 2 , 3 , 4 we have Output C = 6 , 12 , 18 , 24 , respectively .
That is ( 1 , 6 ) , ( 2 , 12 ) , ( 3 , 18 ) , ( 4 , 24 ) .

4. Answer: ( 1 , 9 ) , ( 2 , 18 ) , ( 3 , 27 ) , ( 4 , 36 )

Explanation:
Given , Two Edges of a cube = 3 , h = 1 , 2 , 3 , 4
To find the Volume of the cube we have , V = a³
for h = 1 , V = 3 × 3 × 1 = 9 ,
for h = 2 , V = 3 × 3 × 2 = 12 ,
for h = 3 , V = 3 × 3 × 3 = 27 ,
for h = 4 , V = 3 × 3 × 4 = 36 ,
So, for every value of Input h = 1 , 2 , 3 , 4 we have Output V = 9 , 18 , 27 , 36 , respectively .
That is ( 1 , 9 ) , ( 2 , 18 ) , ( 3 , 27 ) , ( 4 , 36 ) .

Vocabulary
The following vocabulary terms are defined in this chapter. Think about what each term might mean and record your thoughts.
input
mapping diagram
nonlinear function
output
linear function

Answer : Input : The ordered pairs can be used to show inputs and outputs of a function ,The input is the number you feed into the expression, and the output is what you get after the calculations are finished.

mapping diagram : A relation pairs inputs with outputs , A relation can be represented by ordered pairs or a mapping diagram .

nonlinear function : nonlinear functions are functions which are not linear. Quadratic functions are one type of nonlinear function. It is a relation between two variables , function that does not form a line when graphed.

output ; The ordered pairs can be used to show inputs and outputs of a function ,The input is the number you feed into the expression, and the output is what you get after the calculations are finished.

linear function : A linear function is a relation between two variables that produces a straight line when graphed. And it has one dependent variable and one independent variable .

Lesson 7.1 Relations and Functions

EXPLORATION 1

Interpreting Diagrams
Work with a partner. Describe the relationship between the inputs and outputs in each diagram. Then complete each diagram. Is there more than one possible answer? Explain your reasoning.

Answer: a. The relation between the inputs and outputs is outputs are the result of twice as many times the inputs.
b. The relation between the inputs and outputs is outputs are the result of colors of inputs . In this case we can notice that , for any one input we can have more than one output .

Explanation:
a. As shown in the diagrams , The relation between the inputs and outputs is outputs are the result of twice as many times the inputs , so for input 1 = 1 × 1 = 1 as output ,
for input 2 = 2 × 2 = 4  ,
for input 3 = 3 × 3 = 9  ,
for input 5 = 5 × 5 = 25 ,
for input 8 = 8 × 8 = 64 ,
for input 9 = 9 × 9 = 81 ,

So, for every value of Input = 1 , 2 , 3 , 5 , 8 , 9 , we have Output = 1 , 4 , 9 , 25 , 64 , 81 , respectively .
That is ( 1 , 1 ) , ( 2 , 4 ) , ( 3 , 9 ) , ( 5 , 25 ) , ( 8 , 64 ) , ( 9 , 81 ) .

b. The relation between the inputs and outputs is outputs are the result of colors of inputs .
for input Blueberry = color is blue as output
for  input lemon = color is yellow as output
for input Apple = color is yellow , red and green as output
for input Grape = color is green as output.

In this case we can notice that , for any one input we can have more than one output .

EXPLORATION 2

Describing Relationships Between Quantities
Work with a partner. The diagrams show the numbers of tickets bought by customers for two different plays and the total costs (in dollars).

a. For each diagram, how many outputs does each input have?
b. Describe the prices of tickets for each play.
c. A person buys 4 tickets for each play. Can you determine the total cost of all 8 tickets? Explain.
Answer:
a. For Play A ,The number of inputs are equal to number of outputs ,
For Play B , The number of inputs are not equal to number of outputs .
So, input 1 = 2 outputs , input 2 = 3 outputs , input 3 = 4 outputs .

b. For Play A , The price of the each ticket is $8 . For Play B , The price of each ticket is$4 or $8 . c. For Play A , each ticket is$8 , Then for 4 tickets = 4 × $8 =$32 .
For Play B , each ticket is $4 or$8 , Then for 4 tickets = 4 × $8 =$32 . or 4 × $4 =$16 .

Explanation:
a. For Play A ,
The number of inputs are equal to number of outputs , 4 inputs = 4 outputs
That is ( 1 , 8 ) , ( 2 , 16 ) , ( 3 , 24 ) , ( 4 , 32 ) .
For Play B ,
The number of inputs are not equal to number of outputs , 3 inputs are not equal to 7 outputs
That is , for input 1 = 4 , 8 as outputs ,
for input 2 = 8 , 12 , 16 as outputs ,
for input 3 = 12 , 16 , 20 , 24 as outputs ,
So, input 1 = 2 outputs , input 2 = 3 outputs , input 3 = 4 outputs .

b. For Play A ,
The price of the each ticket is $8 . For Play B , The price of each ticket is$4 or $8 . c. Given , A person buys 4 tickets for each play. For Play A , each ticket is$8 , Then for 4 tickets = 4 × $8 =$32 .
And for 8 tickets = 8 × $8 =$64 .
For Play B , each ticket is $4 or$8 , Then for 4 tickets = 4 × $8 =$32 . or 4 × $4 =$16 .
And for 8 tickets =8 × $4 =$32  or 8 × $8 =$64 .

Try It

List the ordered pairs shown in the mapping diagram.
Question 1.

Answer: Ordered pairs are ( 0 , 12 ) , ( 2 , 10 ) , ( 4 , 8 ) , ( 6 , 6 ) .

Explanation:
As shown , Ordered pairs are the combinations of input and output
So , Ordered pairs are ( 0 , 12 ) , ( 2 , 10 ) , ( 4 , 8 ) , ( 6 , 6 ) .

Question 2.

Answer: Ordered pairs are ( 1 , -1 ) , ( 1 , -2 ) , ( 2 , -3 ) , ( 2 , -4 ) .

Explanation:
As shown , Ordered pairs are the combinations of input and output
So , Ordered pairs are ( 1 , -1 ) , ( 1 , -2 ) , ( 2 , -3 ) , ( 2 , -4 ) .

Determine whether the relation is a function.
Question 3.

Answer: The relation is not a function

Explanation:
The each input has  more than two outputs , Even one of those inputs are unclear of outputs
So , The relation is not a function .

Question 4.

Answer: The relation is a function .

Explanation:
Each input has exactly one output ,
So , The relation is a function .

Self-Assessment for Concepts & Skills
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 5.
PRECISION
Describe how relations and functions are different.
Answer: Relations are nothing but the ordered pairs with Inputs and Outputs . On the other hand , Functions are The relation that pairs with one input with exactly one output  are called Functions.

IDENTIFYING FUNCTIONS List the ordered pairs shown in the mapping diagram. Then determine whether the relation is a function.
Question 6.

Answer: The ordered pairs are ( 10 , 1 ) , ( 15 , 1 ) , ( 20 , 13 ) , ( 25 , 7 ) and The relation is a function .

Explanation:
As shown , The ordered pairs are ( 10 , 1 ) , ( 15 , 1 ) , ( 20 , 13 ) , ( 25 , 7 ) .
Each input has exactly one output ,
So, The relation is a function .

Question 7.

Answer: The ordered pairs are ( 0 , -5 ) , ( 0 , -4 ) , ( 1 , -4 ) , ( 2 , -3 ) , ( 3 , -2 ) and relation is not a function .

Explanation:
As shown , The ordered pairs are ( 0 , -5 ) , ( 0 , -4 ) , ( 1 , -4 ) , ( 2 , -3 ) , ( 3 , -2 ) .
The input 0 has more than one output ,
So, The relation is not a function .

Question 8.
OPEN-ENDED
Copy and complete the mapping diagram at the left to represent a relation that is a function. Then describe how you can not modify the mapping diagram so that the relation is a function.

Answer: ordered pairs are ( -8 , -4 ) , ( 0 , -2 ) , ( 8 , 0 ) , ( 16 , 2 ) . To have the relation as a function we must have only one output for one input.

Explanation:
The ordered pairs of the diagram are ( -8 , -4 ) , ( 0 , -2 ) , ( 8 , 0 ) , ( 16 , 2 ) .
Each Input must have only one output in order to be the relation is a function ,
If ,The mapping diagram has the right to left representation or each input has more than one output , then the relation is not a function .
So , To have the relation as a function we must have only one output for one input.

Self-Assessment for Problem Solving
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 9.
The mapping diagram represents the costs of reserving a hotel room for different numbers of nights.

a. Is the cost a function of the number of nights reserved?
b. Describe the relationship between the cost and the number of nights reserved.
Answer: a. Yes , The cost is a function of the number of nights reserved .
b. The relationship between the cost and the number of nights reserved is , For every night reservation of the room is increasing by $85 with increase in the next reservation , Explanation: a. From the diagram we have , Ordered pairs are ( 1 , -$85 ) , ( 2 , $170 ) , ( 3 ,$255 ) , ( 4 , $340 ) . each input has exactly one output , So , the relation is a function and , Yes , The cost is a function of the number of nights reserved . b. The relationship between the cost and the number of nights reserved is , For every night reservation of the room is increasing by$85 with increase in the next reservation,
that is , input 1 = $85 as output Input 2 =$85 + $85 =$170 as output
Input 3 = $170 +$85 = $255 as output Input 2 =$255 + $85 =$340 as output

So, The relationship between the cost and the number of nights reserved is ,
For every night reservation of the room is increasing by $85 with increase in the next reservation. Question 10. DIG DEEPER! The graph represents the number of contestants in each round of a talent competition. a. Is the number of contestants a function of the round number? b. Predict the number of contestants in the talent competition during Round 7. Explain your reasoning. Answer: a. The number of contestants is a function of the round number. b. The number of contestants in the talent competition during Round 7 are 2. Explanation: a. From the given graph , The ordered pairs are ( 1 , 128 ) , ( 2 , 64 ) , ( 3 , 32 ) , ( 4 , 16 ) . Each input has only one output , The relation is a function . So , the number of contestants is a function of the round number. b. Firstly , The relation between the input and output is, With every increase in the round number the number of contestants are decreasing by half the number of the previous round , That is, for input 1 = 128 as output For input 2 = 128 – 64 = 64 as output For input 3 = 64 – 32 = 32 as output For input 4 = 32 – 16 = 16 as output For input 5 = 16 – 8 = 8 as output For input 6 = 8 – 4 = 4 as output For input 7 = 4 – 2 = 2 as output, So, The number of contestants in the talent competition during Round 7 are 2 . Relations and Functions Homework & Practice 7.1 Review & Refresh Choose an appropriate data display for the situation. Explain your reasoning. Question 1. the number of runners in each age group at a marathon Answer: In a marathon ,the people of all age group are participating for a promotion on healthy lifestyle, The number of runners in each group has kids, adults and old people to spread the awareness of leading a healthy life by running daily in the morning . Running or jogging in the morning can help us to maintain our body mass index at an optimal level which is good for heart. The Marathon is conducted by the government of health ministry to be example for the future generations. Question 2. the high temperature and the attendance at a water park each day Answer: Generally, The water park is normally crowded depending on the season and the temperature, In summer the attendance in the waterpark is at the utmost point because of the high temperature and the seasonal vacation. Going to the water park in summer is super fun due to the number of water slides , water rides will be a nice place to the whole family trip and as well as friends . In order to be there at a less crowded time spring is also a nice time to visit the water park . Graph the linear equation. Question 3. y = 2x – 3 Answer: Explanation: Given , y = 2x – 3 , we know y = mx + c , where m = slope , c = constant To obtain the graph , we should have ordered pairs , So , if x = 1 , then y = 2(1) – 3 = 2 – 3 = -1 . co-ordinates are (1 , -1) if x = 2 , then y = 2(2) – 3 = 4 – 3 = 1 , co-ordinates are (2 , 1) The co-ordinates (1 , -1) , (2 , 1) form a straight line . So, y = 2x – 3 is a linear equation. Question 4. y = – 0.5x Answer: Explanation: Given , y = -0.5x , we know y = mx + c , where m = slope , c = constant To obtain the graph , we should have ordered pairs , So , if x = 0 , then y = -0.5(0) = 0 . co-ordinates are (0 , 0) if x = 2 , then y = -0.5(2) = -1 , co-ordinates are (2 , -1) The co-ordinates (0 , 0) , (2 , -1) form a straight line . So, y = -0.5x is a linear equation. Question 5. y = – 3x + 4 Answer: Explanation: Given , y = – 3x + 4 , we know y = mx + c , where m = slope , c = constant To obtain the graph , we should have ordered pairs , So , if x = 0 , then y = – 3(0) + 4 = 4 . co-ordinates are (0 , 4) if x = 1 , then y = – 3(1) + 4 = -3 + 4 = 1 , co-ordinates are (1 , 1) if x = 2 , then y = – 3(2) + 4 = -6 + 4 = -2 , co-ordinates are (2 , -2) The co-ordinates (0 , 4) , (1 , 1) , (2 , -2) form a straight line . So, y = – 3x + 4 is a linear equation. Question 6. Which word best describes two figures that have the same size and the same shape? A. congruent B. adjacent C. parallel D. similar Answer: A. congruent Explanation: Two figures which have the same size and shape are congruent. Concepts, Skills, &Problem Solving INTERPRETING DIAGRAMS Describe the relationship between the inputs and outputs in the diagram. Then complete the diagram. Is there more than one possible answer? Explain your reasoning. (See Exploration 1, p. 275.) Question 7. Answer: The relationship between the inputs and outputs in the diagram is , For every increase in number of input is having the output of adding -4 to the previous output. Explanation: The relationship between the inputs and outputs in the diagram is , For every increase in number of input is having the output of adding -4 to the previous output , for input 1 = -1 as output for input 2 = -1 + (-4) = -5 as output for input 3 = -5 + (-4) = -9 as output for input 4 = -9 + (-4) = -13 as output for input 5 = -13 + (-4) = -17 as output for input 6 = -17 + (-4) = -21 as output. So, The relationship between the inputs and outputs in the diagram is , For every increase in number of input is having the output of adding -4 to the previous output . In this case , we are witnessing only one output for one input. Question 8. Answer: The relationship between the inputs and outputs in the diagram is, Each input has the sports name and the output has the Starting letter of sports name. Explanation: The relationship between the inputs and outputs in the diagram is, Each input has the sports name and the output has the Starting letter of sports name. For input basketball = b as output For input baseball = b as output For input football = f as output For input soccer = s as output For input swimming = s as output, So, The relationship between the inputs and outputs in the diagram is, Each input has the sports name and the output has the Starting letter of sports name. In this case we have more than one output for input. LISTING ORDERED PAIRS List the ordered pairs shown in the mapping diagram. Question 9. Answer: Ordered pairs are ( 0 , 4 ) , ( 3 , 5 ) , ( 6 , 6 ) , ( 9 , 7 ) . Explanation: As shown , Ordered pairs are the combinations of input and output So , Ordered pairs are ( 0 , 4 ) , ( 3 , 5 ) , ( 6 , 6 ) , ( 9 , 7 ) . Question 10. Answer: Ordered pairs are ( 1 , 8 ) , ( 3 , 8 ) , ( 3 , 4 ) , ( 5 , 6 ) , ( 7 , 2 ). Explanation: As shown , Ordered pairs are the combinations of input and output So , Ordered pairs are ( 1 , 8 ) , ( 3 , 8 ) , ( 3 , 4 ) , ( 5 , 6 ) , ( 7 , 2 ). Question 11. Answer: Ordered pairs are ( 6 , -5 ) , ( 7 , -5 ) , ( 8 , -10 ) , ( 9 , -10 ). Explanation: As shown , Ordered pairs are the combinations of input and output So , Ordered pairs are ( 6 , -5 ) , ( 7 , -5 ) , ( 8 , -10 ) , ( 9 , -10 ). IDENTIFYING FUNCTIONS Determine whether the relation is a function. Question 12. Answer: The relation is not a function . Explanation: The each input has more than two outputs , That is one input has multiple number of outputs. Here , input 0 has two outputs which are 10 and 20 . So , The relation is not a function . Question 13. Answer: The relation is a function . Explanation: Each input has exactly one output , So , The relation is a function . Question 14. Answer: The relation is a function . Explanation: Each input has exactly one output , So , The relation is a function . Question 15. YOU BE THE TEACHER Your friend determines whether the relation shown in the mapping diagram is a function. Is your friend correct? Explain your reasoning. Answer: The relation is not a function . Explanation: The each input has more than two outputs , That is one input has multiple number of outputs. Here , input 4 has four outputs which are 5, 6 , 7 and 8. So , The relation is not a function . REASONING Draw a mapping diagram that represents the relation. Then determine whether the relation is a function. Explain. Question 16. Answer: The mapping diagram representing the relation is Explanation: From the given graph , co-ordinates of the ordering pairs are( 1 , 1 ), ( 3 , 3 ), ( -1 , -1 ), ( -3 , -3 ). Each input has exactly one output , So , The relation is a function . Question 17. Answer: The mapping diagram representing the relation is Explanation: From the given graph , co-ordinates of the ordering pairs are( 0 , 8 ),( 2 , 8 ),( 4 , 8 ),( 6 , 8 ),( 8 , 8 ),( -2 , 8 ),( -4 , 8 ). Each input has exactly one output , So , The relation is a function. Question 18. Answer: The mapping diagram representing the relation is Explanation: From the given graph , co-ordinates of the ordering pairs are( -2 , 1 ),( -2 , 2 ),( -2 , 3 ),( -2 , 4 ),( -2 , 5 ),( -2 , 6 ). Each input has more than one output , So , The relation is not a function. Question 19. MODELING REAL LIFE The normal pressure at sea level is 1 atmosphere of pressure(1 ATM). As you dive below sea level, the pressure changes. The mapping diagram represents the pressures at different depths. a. Complete the mapping diagram. b. Is pressure a function of depth? c. Describe the relationship between pressure and depth. d. List the ordered pairs. Then plot the ordered pairs in a coordinate plane. What do you notice about the points? e. RESEARCH What are common depths for beginner scuba divers? What are common depths for experienced scuba divers? Answer: The detailed explanation of all the answers are given below . Explanation: a. The mapping diagram is b. Yes , the pressure is a function of depth, Because depth is related to pressure in the given mapping diagram. c. The relationship between pressure and depth is, for every 10m increase in Depth of input there is an increase in 1 ATM pressure . d. The ordered pairs are ( 0 , 1 ) , ( 10 , 2 ) , (20 , 3 ) , ( 30 , 4 ) , ( 40 , 5 ), ( 50 , 6 ). The plot of the ordered pairs in a coordinate plane is From the graph, we have seen that, if the depth of the diving of scuba drivers increases then the water pressure increases with increase in depth. So, the graph have straight line . e. The common depths for beginner scuba divers is 30 feet to 60 feet or 9 to 18 meters , The common depths for experienced scuba divers is more than 60 feet or more than 18 meters . Question 20. DIG DEEPER! The table shows the cost of purchasing 1, 2, 3, or 4 T-shirts from a souvenir shop. a. Is the cost a function of the number of T-shirts purchased? b. Describe the relationship between the cost and the number cost per T-shirt of T-shirts purchased. How does the change as you purchase more T-shirts? Answer: The detailed explanation of all the answers are given below . Explanation: a. Yes , The cost is a function of the number of T-shirts purchased, Because the cost of the purchased T-shirts is varying with the number of T-shirts purchased. b. The relationship between the cost and the number cost per T-shirt of T-shirts purchased is, Input is the cost of 1 T-shirt is$10 as output , Then for 2 T-shirts cost will be $20 If 2 T-shirts will be purchased at same time, cost will be decreased by$2 so it will be $10 + 8 =$18 for 2 T-shirts.
As per the single T-shirt cost , For 3 T-shirts will be $30, So in the table given that 3 T-shirts will cost$24 , because it cost $18 + 6 =$24 for 3 T-shirts.
It goes same for 4 T-shirts , For 4 T-shirts will be $40, because it cost$24 + 4 = $28 for 4 T-shirts. The change as you purchase more T-shirts is For every increase in purchase of the number of T-shirts is decrease in the cost of total T-shirts purchased. Question 21. REPEATED REASONING The table shows the outputs for several inputs. Use two methods to predict the output for an input of 200. Answer: The output for an input of 200 is 1025. Explanation: Method 1. The relation between inputs and outputs is as follows, y = 25 + 5x As input increases by 1 , output increases by 5 units, To find output of 200 as input , put x = 200 in the equation, y = 25 + 5(200) = 25 + 1000 = 1025. So , y = 1025. Method 2. As the table shown, for every increase in input there is an increase in 5 numbers in output, So , For 1 input = 25 + 5 = 30 as output For 2 input = 30 + 5 = 35 as out put For 3 input = 35 + 5 = 40 as out put For 4 input = 40 + 5 = 45 as out put By doing this for number 200 as input we have , 1025 as output. Lesson 7.2 Representations of Functions EXPLORATION 1 Using a Table to Describe Relationships Work with a partner. Make a table that shows the relationship between the figure number x and the area A of each figure. Then use an equation to find which figure has an area of 81 square units when the pattern continues. Answer: a. The equation is y = 2x – 1, For figure has an area of 81 square units is 41. b. The equation is y = x², For figure has an area of 81 square units is 9. Explanation: a. figure shows the 1 square unit of each box for and it has a pattern of 2x – 1 figure 1 = 1 square unit figure 2 =3 square units figure 3 = 5 square unit and so on So, the equation is y = 2x – 1 , it is in the form of y = mx + c, Given to which figure has an area of 81 square units substitute y = 81, we have y = 2x – 1 81 = 2x – 1 2x = 82 x = 41 So, For figure has an area of 81 square units is 41. b. As shown above , we know that , figure 1 = 1 square unit figure 2 =4 square units figure 3 = 9 square unit and so on Here we have a pattern of power of its own number, So, the Equation will be y = x² Given to which figure has an area of 81 square units substitute y = 81, we have x = 9 So, For figure has an area of 81 square units is 9. EXPLORATION 2 Using a Graph Work with a partner. Use a graph to test the truth of each statement. If the statement is true, write an equation that shows how to obtain one measurement from the other. a. “You can find the horsepower of a race-car car engine if you know its volume in cubic inches” b. “You can find the volume of a race-car engine in cubic centimeters if you know its volume in cubic inches.” Answer: a. Given ordered pairs are (200 , 375) , (350 , 650) , (350 , 250) , (500 , 600) We can not find the horsepower of a race-car car engine if you know its volume in cubic inches b. Given ordered pairs are (100 , 1640) , (200 , 3280) , (300 , 4920) , Yes, You can find the volume of a race-car engine in cubic centimeters if you know its volume in cubic inches Explanation: a. Given ordered pairs are (200 , 375) , (350 , 650) , (350 , 250) , (500 , 600) We can not find the horsepower of a race-car car engine if you know its volume in cubic inches b. Given ordered pairs are (100 , 1640) , (200 , 3280) , (300 , 4920) , Yes, You can find the volume of a race-car engine in cubic centimeters if you know its volume in cubic inches Try It Question 1. Write a function rule for “The output is one-fourth of the input.” Answer: y = $$\frac{x}{4}$$ Explanation: Let us say x is input and y is output , then The output is one-fourth of the input, will be , y = $$\frac{x}{4}$$. Find the value of y when x = 5. Question 2. y = 4x – 1 Answer: y = 19. Explanation: Given, y = 4x – 1 substitute x = 5 , we get y = 4(5) – 1 y = 20 – 1 = 19 So, y = 19. Question 3. y = 10x Answer: y = 50 Explanation: Given, y =10x substitute x = 5 , we get y = 10(5) y = 50 So, y = 50. Question 4. y = 7 – 3x Answer: y = -8. Explanation: Given, y = 7 – 3x substitute x = 5 , we get y = 7 – 3(5) y = 7 – 15 = -8 So, y = -8. Graph the function. Question 5. y = x + 1 Answer: Explanation: Given , y = x + 1 , we know y = mx + c , where m = slope , c = constant To obtain the graph , we should have ordered pairs , So , if x = 0 , then y = 0 + 1 = 1 . co-ordinates are (0 , 1) if x = 1 , then y = 1 + 1 = 2 . co-ordinates are (1 , 2) if x = 2 , then y = 2 + 1 = 3 , co-ordinates are (2 , 3) The co-ordinates (0 , 1) , (1 , 2) , (2 , 3) form a straight line . Question 6. y = – 3x Answer: Explanation: Given , y = – 3x , we know y = mx + c , where m = slope , c = constant To obtain the graph , we should have ordered pairs , So , if x = 0 , then y = -3(0) = 0 . co-ordinates are (0 , 0) if x = 1 , then y = -3(1) = -3 . co-ordinates are (1 , -3) if x = 2 , then y = -3(2) = -6 , co-ordinates are (2 , -6) if x = 3 , then y = -3(3) = -9 , co-ordinates are (3 , -9) The co-ordinates (0 , 0) , (1 , -3) , (2 , -6) ,(3 , -9) form a straight line . Question 7. y = 3x + 2 Answer: Explanation: Given , y = 3x + 2 , we know y = mx + c , where m = slope , c = constant To obtain the graph , we should have ordered pairs , So , if x = 0 , then y =3(0) + 2 = 2 . co-ordinates are (0 , 2) if x = 1 , then y = 3(1) + 2= 5 . co-ordinates are (1 , 5) if x = 2 , then y =3(2) + 2 = 7 , co-ordinates are (2 , 7) The co-ordinates (0 , 2) , (1 , 5) , (2 , 7) form a straight line . Self-Assessment for Concepts & Skills Solve each exercise. Then rate your understanding of the success criteria in your journal. WRITING FUNCTION RULES Write a function rule for the statement. Question 8. The output is three times the input. Answer: y = 3x Explanation: Let us say x is input and y is output , then The output is three times the input. will be , So , y = 3x . Question 9. The output is eight more than one-seventh of the input. Answer: y = 8 + $$\frac{x}{7}$$ . Explanation: Let us say x is input and y is output , then The output is eight more than one-seventh of the input., will be , So, y = 8 + $$\frac{x}{7}$$ . EVALUATING A FUNCTION Find the value of y when x = 5. Question 10. y = 6x Answer: y = 30 Explanation: Given, y = 6x substitute x = 5 , we get y = 6(5) =30 So, y = 30 Question 11. y = 11 – x Answer: y = 6 Explanation: Given, y = 11 – x substitute x = 5 , we get y = 11 – 5 = 6 So, y = 6. Question 12. y = $$\frac{1}{5}$$x + 1 Answer: y = 2. Explanation: Given, y = $$\frac{1}{5}$$x + 1 substitute x = 5 , we get y = $$\frac{x}{5}$$ + 1 y= $$\frac{5}{5}$$ + 1 y = 1 + 1 = 2 So, y = 2 . GRAPHING A FUNCTION Graph the function. Question 13. y = – 2x Answer: Explanation: Given , y = – 2x , we know y = mx + c , where m = slope , c = constant To obtain the graph , we should have ordered pairs , So , if x = 0 , then y = – 2(0) = 0 . co-ordinates are (0 , 0) if x = 1 , then y = – 2(1)= -2 . co-ordinates are (1 , -2) if x = 2 , then y =- 2(2) = -4 , co-ordinates are (2 , -4) if x = 3 , then y =- 2(3) = -6 , co-ordinates are (3 , -6) The co-ordinates (0 , 0) , (1 , -2) , (2 , -4) , (3 , -6) form a straight line . Question 14. y = x – 3 Answer: Explanation: Given , y = x – 3 , we know y = mx + c , where m = slope , c = constant To obtain the graph , we should have ordered pairs , So , if x = 0 , then y = 0 – 3 = -3 . co-ordinates are (0 , -3) if x = 1 , then y = 1 – 3= -2 . co-ordinates are (1 , -2) if x = 2 , then y = 2 – 3 = -1 , co-ordinates are (2 , -1) if x = 3 , then y = 3 – 3 = 0 , co-ordinates are (3 , 0) The co-ordinates (0 , -3) , (1 , -2) , (2 , -1) , (3 , 0) form a straight line . Question 15. y = 9 – 3x Answer: Explanation: Given , y = 9 – 3x , we know y = mx + c , where m = slope , c = constant To obtain the graph , we should have ordered pairs , So , if x = 0 , then y = 9 – 3(0) = 9 . co-ordinates are (0 , 9) if x = 1 , then y = 9 – 3(1) = 6 . co-ordinates are (1 , 6) if x = 2 , then y = 9 – 3(2) = 3 , co-ordinates are (2 , 3) if x = 3 , then y = 9 – 3(3) = 0 , co-ordinates are (3 , 0) The co-ordinates (0 , 9) , (1 , 6) , (2 , 3) , (3 , 0) form a straight line . Question 16. DIFFERENT WORDS, SAME QUESTION Which is different? Find “both” answers. Answer: As mentioned in the explanation below a & d , b & c are different . Explanation: Given , a. what output is 4 more than twice the input 3? Let us say that , y is output and x is input and given as 3 , then, we have y = 4 + 2(3) = 10. b. What output is twice the sum of the input 3 and 4? Let us say that , y is output and x is input and given as 3 , then, we have y = 2( 3 + 4 ) = 14. c. what output is the sum of 2 times the input 3 and 4? Let us say that , y is output and x is input and given as 3 , then, we have y = 2( 3 + 4 ) = 14. d. what output is 4 increased by twice the input 3? Let us say that , y is output and x is input and given as 3 , then, we have y = 4 + 2(3) = 10. So, a & d , b & c are different . Self-Assessment for Problem Solving Solve each exercise. Then rate your understanding of the success criteria in your journal. Question 17. The World Health Organization(WHO) suggests having 23 health-care workers for every 10,000 people. How many health-care workers are needed to meet the WHO suggestion for a population of 250,000 people? Justify your answer using a graph. Answer: So, 575 health-care workers are needed to meet the WHO suggestion for a population of 250,000 people Explanation: Given, The World Health Organization(WHO) suggests having 23 health-care workers for every 10,000 people. we need to find how many health-care workers are needed to meet the WHO suggestion for a population of 250,000 people, For every 10,000 people we have 23 care takers Then for 250,000 people we have $$\frac{23 × 250,000}{10,000}$$ = 23 × 25 = 575 So, 575 health-care workers are needed to meet the WHO suggestion for a population of 250,000 people Question 18. DIG DEEPER! A truck produces 22 pounds of carbon dioxide for every gallon of diesel fuel burned. The fuel economy of the truck is 18 miles per gallon. Write and graph a function that describes the relationship between carbon dioxide produced and distance traveled. Answer: y = 22x + 18 is the linear equation Explanation: Given, A truck produces 22 pounds of carbon dioxide for every gallon of diesel fuel burned. The fuel economy of the truck is 18 miles per gallon. So, we have y = 22x + 18 is in the form of y = mx +c To obtain the graph , we should have ordered pairs , So , if x = 0 , then y = 22(0) + 18 = 18 . co-ordinates are (0 , 18) if x = 1 , then y =22(1) + 18= 40 . co-ordinates are (1 , 40) if x = 2 , then y =22(2) + 18 = 62 , co-ordinates are (2 , 62) if x = 3 , then y =22(3) + 18 = 84 , co-ordinates are (3 , 84) The co-ordinates (0 , 18) , (1 , 40) , (2 , 62) , (3 , 84) form a straight line . The graph is Representations of Functions Homework & Practice 7.2 Review & Refresh Determine whether the relation is a function. Question 1. Answer: The relation is a function . Explanation: Each input has exactly one output , So , The relation is a function . Question 2. Answer: The relation is a function . Explanation: Each input has exactly one output , So , The relation is a function . Question 3. Answer: The relation is not a function . Explanation: The each input has more than two outputs , That is one input has multiple number of outputs. Here , input 2 has two outputs which are 0 and -4 . So , The relation is not a function . Find the slope of the line. Question 4. Answer: slope = 1. Explanation: By using the slope equation , we know that Slope = $$\frac{change in y}{change in x}$$ or slope = $$\frac{▲y}{▲x}$$ From the graph we know that change in y or ▲y is change from -2 to -4 =2 change in x or ▲x is change from 1 to 3 = 2 , So, slope = $$\frac{▲y}{▲x}$$ slope = $$\frac{2}{2}$$ slope = 1. Question 5. Answer: slope = $$\frac{5}{2}$$ . Explanation: By using the slope equation , we know that Slope = $$\frac{change in y}{change in x}$$ or slope = $$\frac{▲y}{▲x}$$ From the graph we know that change in y or ▲y is change from -4 to 1 = 5 change in x or ▲x is change from -1 to -3 = 2 , So, slope = $$\frac{▲y}{▲x}$$ slope = $$\frac{5}{2}$$ . Question 6. Answer: slope = $$\frac{1}{3}$$ . Explanation: By using the slope equation , we know that Slope = $$\frac{change in y}{change in x}$$ or slope = $$\frac{▲y}{▲x}$$ From the graph we know that change in y or ▲y is change from -4 to -3 = 1 change in x or ▲x is change from 1 to 4 = 3 , So, slope = $$\frac{▲y}{▲x}$$ slope = $$\frac{1}{3}$$ . Concepts, Skills, & Problem Solving USING A GRAPH Use a graph to test the truth of the statement. If the statement is true, write an equation that shows how to obtain one measurement from the other measurement. (See Exploration 2, p. 281.) Question 7. “You can find the weight of a cell phone in ounces if you know its screen size in inches.” Answer: we can does not find the weight of a cell phone in ounces if you know its screen size in inches. From the given table , Ordered pairs are (4 , 4) , (4.7 , 4.8) , (5 , 4.8) , (5.5 , 6.4) First find the slope m of the line containing the two given points (4, 4) and (4.7, 4.8) m = (y2-y1) / (x2-x1) m= (4.8 – 4) / (4.7 – 4) m = 0.8/0.7 . So, we can does not find the weight of a cell phone in ounces if you know its screen size in inches. Question 8. “You can find the age of a child in years if you know the age of the child in months.” Answer: YES, y = 0.08x + 0.04 is a linear equations Explanation: From the given table , Ordered pairs are (9 , 0.75) , (12 , 1) , (15 , 1.25) , (24 , 2) First find the slope m of the line containing the two given points (12 ,1) and (24, 2) m = (y2-y1) / (x2-x1) m= (2 – 1) / (24 – 12) m = 1/12 m = 0.08. substitute the slope in the (12 ,1) to get point slope to form a line. y-y1 = m (x-x1) y – 1 = 0.08(x – 12) y –1 = 0.08x – 0.96 y = 0.08x –0.96 + 1 y =0.08 x + 0.04 So, y = 0.08x + 0.04 is a linear equation WRITING FUNCTION RULES Write a function rule for the statement. Question 9. The output is half of the input. Answer: y = $$\frac{x}{2}$$. Explanation: Let us say x is input and y is output , then The output is half of the input, will be , y = $$\frac{x}{2}$$. Question 10. The output is eleven more than the input. Answer: y = x + 11 Explanation: Let us say x is input and y is output , then The output is eleven more than the input, will be , y = x + 11 Question 11. The output is three less than the input. Answer: y = x – 3 Explanation: Let us say x is input and y is output , then The output is three less than the input, will be , y = x – 3 Question 12. The output is the cube of the input. Answer: y = x³ Explanation: Let us say x is input and y is output , then The output is the cube of the input, will be , y = x³ Question 13. The output is six times the input. Answer: y = 6x Explanation: Let us say x is input and y is output , then The output is six times the input, will be , y = 6x Question 14. The output is one more than twice the input. Answer: y = 2x + 1 Explanation: Let us say x is input and y is output , then The output is one more than twice the input, will be , y = 2x + 1 EVALUATING A FUNCTION Find the value of y for the given value of x. Question 15. y = x + 5; x = 3 Answer: y = 8 Explanation: Given, y = x + 5 substitute x = 3 , we get y = 3 + 5 So, y = 8. Question 16. y = 7x; x = – 5 Answer: y = -35. Explanation: Given, y = 7x substitute x = -5 , we get y = 7(-5) So, y = -35. Question 17. y = 1 – 2x; x = 9 Answer: y = -17 Explanation: Given, y = 1 – 2x substitute x = 9 , we get y = 1 – 2(9) y = 1 – 18 So, y = -17. Question 18. y = 3x + 2; x = 0.5 Answer: y = 5.5 Explanation: Given, y = 3x + 2 substitute x = 0.5 , we get y = 3(0.5) + 2 y = 3.5 + 2 So, y = 5.5 . Question 19. y = 2x3; x = 3 Answer: y = 54 Explanation: Given, y = 2x3 substitute x = 3 , we get y = 2(3)³ y = 2 × 27 = 54 So, y = 54. Question 20. y = $$\frac{x}{2}$$ + 9; x = – 12 Answer: y = 3 Explanation: Given, y = $$\frac{x}{2}$$ + 9 substitute x = -12 , we get y = $$\frac{-12}{2}$$ + 9 y = -6 + 9 So, y = 3 . GRAPHING A FUNCTION Graph the function. Question 21. y = x + 4 Answer: Explanation: Given , y = x + 4 , we know y = mx + c , where m = slope , c = constant To obtain the graph , we should have ordered pairs , So , if x = 0 , then y = 0 + 4 = 4 . co-ordinates are (0 , 4) if x = 1 , then y = 1 + 4 = 5 . co-ordinates are (1 , 5) if x = 2 , then y = 2 + 4 = 6 , co-ordinates are (2 , 6) if x = 3 , then y = 3 + 4 = 7 , co-ordinates are (3 , 7) The co-ordinates (0 , 4) , (1 , 5) , (2 , 6) , (3 , 7) form a straight line . Question 22. y = 2x Answer: Explanation: Given , y = 2x , we know y = mx + c , where m = slope , c = constant To obtain the graph , we should have ordered pairs , So , if x = 0 , then y = 2(0) = 0 . co-ordinates are (0 , 0) if x = 1 , then y = 2(1) = 2 . co-ordinates are (1 , 2) if x = 2 , then y = 2(2) = 4 , co-ordinates are (2 , 4) if x = 3 , then y = 2(3) = 6 , co-ordinates are (3 , 6) The co-ordinates (0 , 0) , (1 , 2) , (2 , 4) , (3 , 6) form a straight line . Question 23. y = – 5x + 3 Answer: Explanation: Given , y = – 5x + 3 , we know y = mx + c , where m = slope , c = constant To obtain the graph , we should have ordered pairs , So , if x = 0 , then y =- 5(0) + 3 = 3 . co-ordinates are (0 , 3) if x = 1 , then y = – 5(1) + 3 = -2 . co-ordinates are (1 , -2) if x = 2 , then y = – 5(2) + 3 = -7 , co-ordinates are (2 , -7) if x = 3 , then y = – 5(3) + 3 = -12 , co-ordinates are (3 , -12) The co-ordinates (0 , 3) , (1 , -2) , (2 , -7) , (3 , -12) form a straight line . Question 24. y = $$\frac{x}{4}$$ Answer: Explanation: Given , y = $$\frac{x}{4}$$ , we know y = mx + c , where m = slope , c = constant To obtain the graph , we should have ordered pairs , So , if x = 0 , then y = $$\frac{0}{4}$$ = 0 . co-ordinates are (0 , 0) if x = 1 , then y = $$\frac{1}{4}$$ = 0.25 . co-ordinates are (1 , 0.25) if x = 2 , then y = $$\frac{2}{4}$$ = 0.5 , co-ordinates are (2 , 0.5) if x = 3 , then y = $$\frac{3}{4}$$ = 0.75 , co-ordinates are (3 , 0.75) The co-ordinates (0 , 0) , (1 , 0.25) , (2 , 0.5) , (3 , 0.75) form a straight line . Question 25. y = $$\frac{3}{2}$$x + 1 Answer: Explanation: Given , y = $$\frac{3}{2}$$x + 1 , we know y = mx + c , where m = slope , c = constant To obtain the graph , we should have ordered pairs , So , if x = 0 , then y =$$\frac{3}{2}$$(0) + 1 = 1 . co-ordinates are (0 , 1) if x = 1 , then y = $$\frac{3}{2}$$(1) + 1= 2.5 . co-ordinates are (1 , 2.5) if x = 2 , then y = $$\frac{3}{2}$$(2) + 1 = 4 , co-ordinates are (2 , 4) if x = 3 , then y = $$\frac{3}{2}$$(3) + 1 = 5.5 , co-ordinates are (3 , 5.5) The co-ordinates (0 , 1) , (1 , 2.5) , (2 , 4) , (3 , 5.5) form a straight line . Question 26. y = 1 + 0.5x Answer: Explanation: Given , y = 1 + 0.5x , we know y = mx + c , where m = slope , c = constant To obtain the graph , we should have ordered pairs , So , if x = 0 , then y =1 + 0.5(0) = 1 . co-ordinates are (0 , 1) if x = 1 , then y = 1 + 0.5(1) = 1.5 . co-ordinates are (1 , 1.5) if x = 2 , then y = 1 + 0.5(2) = 2 , co-ordinates are (2 , 2) if x = 3 , then y = 1 + 0.5(3) = 2.5 , co-ordinates are (3 , 2.5) The co-ordinates (0 , 1) , (1 , 1.5) , (2 , 2) , (3 , 2.5) form a straight line . MATCHING Match the graph with the function it represents. A. y = $$\frac{x}{3}$$ B. y = x + 1 C. y = – 2x + 6 Question 27. Answer: B. y = x + 1. Explanation: Given , y = x + 1 , we know y = mx + c , where m = slope , c = constant To obtain the graph , we should have ordered pairs , So , if x = 0 , then y = 0 + 1 = 1 . co-ordinates are (0 , 1) if x = 1 , then y = 1 + 1 = 2 . co-ordinates are (1 , 2) if x = 2 , then y = 2 + 1 = 3 , co-ordinates are (2 , 3) The co-ordinates (0 , 1) , (1 , 2) , (2 , 3) form a straight line . Question 28. Answer: c. y = – 2x + 6 Explanation: Given , y = – 2x + 6 , we know y = mx + c , where m = slope , c = constant To obtain the graph , we should have ordered pairs , So , if x = 0 , then y = – 2(0) + 6 = 6 . co-ordinates are (0 , 6) if x = 1 , then y = – 2(1) + 6 = 4 . co-ordinates are (1 , 4) if x = 2 , then y = – 2(2) + 6 = 2 , co-ordinates are (2 , 2) if x = 3 , then y = – 2(3) + 6 = 0 , co-ordinates are (3 , 0) The co-ordinates (0 , 6) , (1 , 4) , (2 , 2) , (3 , 0) form a straight line . Question 29. Answer: A. y = $$\frac{x}{3}$$ Explanation: Given , y = $$\frac{x}{3}$$ , we know y = mx + c , where m = slope , c = constant To obtain the graph , we should have ordered pairs , So , if x = 0 , then y =$$\frac{0}{3}$$ = 0 . co-ordinates are (0 , 0) if x = 1 , then y = $$\frac{1}{3}$$= 0.3 . co-ordinates are (1 , 0.3) if x = 2 , then y = $$\frac{2}{3}$$= 0.6 , co-ordinates are (2 , 0.6) The co-ordinates (0 , 0) , (1 , 0.3) , (2 , 0.6) form a straight line . Question 30. YOU BE THE TEACHER Your friend graphs the function represented by the input-output table. Is your friend correct? Explain your reasoning. Answer: Yes , He is correct Explanation: Ordered pairs are (-1 , -4) , (1 , -2) , (3 ,0) , (5 , 2) these points form a straight line when graphed. Yes , He is correct Question 31. MODELING REAL LIFE A dolphin eats 30 pounds of fish per day. a. Write and graph a function that relates the number p of pounds of fish that a dolphin eats in d days. b. How many total pounds of fish does a dolphin eat in 30 days? Answer: Explanation: a. Given , A dolphin eats 30 pounds of fish per day. by each passing day eating fish is increased by the day passes . So, y = 30x is the function, The graph represents the function as b. Given , A dolphin eats 30 pounds of fish per day. then for 30 days , 30 × 30 = 900 pounds So, A dolphin eats 900 pounds of fish in 30 days Question 32. MODELING REAL LIFE You fill a fish tank with 55 gallons of water on Saturday. The water evaporates at a rate of 1.5 gallons per day. You plan to add water when the tank reaches 49 gallons. When will you add water? Justify your answer. Answer: As the action starts on Saturday , the tank will reach 49 gallons after 4 days , That is on Wednesday. Explanation: Given data ,, implies that slope of the function m = -1.5 The y intercept b= 55, Then the equation will be y = 55 – 1.5x Given , You plan to add water when the tank reaches 49 gallons. determine x for y = 49 , So, 49 = 55 – 1.5x , 1.5x = 55 – 49 1.5x = 6 x = $$\frac{6}{1.5}$$ x = 4. As the action starts on Saturday , the tank will reach 49 gallons after 4 days , That is on Wednesday. USING AN EQUATION Find the value of x for the given value of y. Question 33. y = 5x – 7; y = – 22 Answer: x = -3 Explanation: Given, y = 5x – 7 x = $$\frac{y + 7}{5}$$ substitute y = -22 , we get x = $$\frac{-22 + 7}{5}$$ x = $$\frac{- 15}{5}$$ x = -3 So, x = -3 . Question 34. y = 9 – 7x; y = 37 Answer: x = -4 Explanation: Given, y = 9 – 7x x = $$\frac{9 – y}{7}$$ substitute y = 37 , we get x = $$\frac{9 – 37}{7}$$ x = $$\frac{- 28}{7}$$ x = -4 So, x = -4 . Question 35. y = $$\frac{x}{4}$$ – 7; y = 2 Answer: x = 36 Explanation: Given, y = $$\frac{x}{4}$$ – 7 x = 4( y + 7) substitute y = 2 , we get x = 4( 2 + 7) x = 4(9) x = 36 So, x = 36 . Question 36. PROBLEM SOLVING You decide to make and sell bracelets. The cost of your materials is$84.00. You charge $3.50 for each bracelet. a. P Write a function that represents the profit for selling b bracelets. b. Which variable is independent? dependent? Explain. c. You will break even when the cost of your materials equals your income. How many bracelets must you sell to break even? Answer: a. A function that represents the profit for selling b bracelets is p = 3.5b – 84. b. Here , the profit depends on the number of bracelets sold , b is the independent variable and p is the dependent variable. c. To break even you must sell 24 bracelets. Explanation: a. Given , The cost of your materials is$84.00. You charge $3.50 for each bracelet, Let p be the profit , b be the number of bracelets sold, So, profit = income – cost . p = 3.5b – 84. Thus , A function that represents the profit for selling b bracelets is p = 3.5b – 84. b. Here , the profit depends on the number of bracelets sold , b is the independent variable and p is the dependent variable. c. set the income expression from part a equal to the cost of 84 and solve for b , So, income = cost . 3.5b = 84 , b = $$\frac{84}{3.5}$$ b = 24. To break even you must sell 24 bracelets. Question 37. MODELING REAL LIFE A furniture store is having a sale where everything is 40% off. a. Write and graph a function that represents the amount of discount on an item at regular price. b. You buy a bookshelf that has a regular price of$85. What is the sale price of the bookshelf?
Answer: a. The function is y = 0.4x and the graph is given below.
b. The sale price of the bookshelf s $51. Explanation: a. A function that represents the amount of discount on an item at regular price is , Given , 40% = 0.4 , To find the percent of the number , we should multiply the number by the percent in the decimal form , so, the equation is d = 0.4p , let us convert it in to a function form , y = 0.4x we know y = mx + c , where m = slope , c = constant To obtain the graph , we should have ordered pairs , So , if x = 0 , then y = 0.4(0) = 0 . co-ordinates are (0 , 0) if x = 1 , then y = 0.4(1)= 0.4 . co-ordinates are (1 , 0.4) if x = 2 , then y =0.4(2) = 0.8 , co-ordinates are (2 , 0.8) if x = 3 , then y = 0.4(3) = 1.2 , co-ordinates are (3 , 1.2) The co-ordinates (0 , 0) , (1 , 0.4) , (2 , 0.8) , (3 , 1.2) form a straight line . The graph is b. Given , You buy a bookshelf that has a regular price of$85.
The sale price of the bookshelf is ,
substituting the given price in p = 85 ,
it will be the discount d = 0.4 (85) = 34
Then the sale price is $85 –$34 = $51. So, The sale price of the bookshelf s$51.

Question 38.
REASONING
You want to take a two-hour air boat tour. Which is a better deal, Snake Tours or Gator Tours? Use functions to justify your answer.

Answer: By using functions , $50 >$40 , So, Gator tours are cheaper than the snake tours .

Explanation:
Given , You want to take a two-hour air boat tour.
Let x be the hours of  air boat tour and y be the cost of air boat tour ,
Snake tours , y = 25x
putt x = 2 ,
So , y = 25 (2) = 50 .
y = 50.

Gator tour , y = 35 + $$\frac{5}{2}$$x
Put x = 2 ,
So, y = 35 + $$\frac{5}{2}$$ x
y = 35 + 2.5x
y = 35 + 2.5 (2)
y = 35 + 5
y = 40 .

Finally $50 >$40 , So, Gator tours are cheaper than the snake tours

Question 39.
REASONING
The graph of a function is a line that passes through the points (3, 2), (5, 8), and (8, y). What is the value of y?
Answer: The value of y is 17 , so, The third given point is (8, 17)

Explanation:
First find the slope m of the line containing the two given points (3,2) and (5,8)
m = (y2-y1) / (x2-x1)
m= (8 – 2) / (5 – 3)
m = 6 / 2
m = 3
Then use the slope and one of the given points (3,2) to find the y-intercept
y = mx +
2 = 3(3) + b
2 = 9 + b
-7 = b
The equation is   y = 3x -7
Then find the third point (8, y) by replacing x by 8
y = 3x -7
y = 3(8) -7
y = 24 -7
y = 17

so the third given point is (8, 17)

Question 40.
CRITICAL THINKING
Make a table where the independent variable is the side length of a square and the dependent variable is the perimeter. Make a second table where the independent variable is the side length of a square and the dependent variable is the area. Graph both functions in the same coordinate plane. Compare the functions.
Answer: The graph for the perimeter is linear , The graph for the Area is Quadratic .

Explanation:
Let us say , s be the side length of the square ,
Then the perimeter is P = 4s ,
The function will be y= 4x,
we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = 4(0) = 0 . co-ordinates are (0 , 0)
if x = 1 , then y = 4(1) = 4 . co-ordinates are (1 , 4)
if x = 2 , then y = 4(2) =8 , co-ordinates are (2 , 8)
if x = 3 , then y = 4(3) = 0 , co-ordinates are (3 , 12)
The co-ordinates (0 , 0) , (1 , 4) , (2 ,8) , (3 , 12) form a straight line .

Table will be ,

Let us say , s be the side length of the square ,
Then the Area is A = s² ,
The function will be y=x²,
we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = 0² = 0 . co-ordinates are (0 , 0)
if x = 1 , then y = 1² = 1 . co-ordinates are (1 , 1)
if x = 2 , then y = 2² =4 , co-ordinates are (2 , 4)
if x = 3 , then y = 3² = 9 , co-ordinates are (3 , 9)
The co-ordinates (0 , 0) , (1 , 1) , (2 ,4) , (3 , 9) form a straight line .

Second table is
Then the graph is
The graph for the perimeter is linear , The graph for the Area is Quadratic .

Question 41.
PUZZLE
The blocks that form the diagonals of each square are shaded. Each block has an area of one square unit. Find the “green area” of Square 20. Find the “green area” of Square 21. Explain your reasoning.

Answer:  The green area of the Square 20 is 46 square units and The green area of the Square 21 is 48 square units.

Explanation:
Given , Each block has an area of one square unit,
Square 1 has  the diagonals of each square are shaded. the “green area” is 3 + 3 = 6 square units ,
Square 2 has  the diagonals of each square are shaded. the “green area” is 4 + 4 = 8 square units ,
Square 3 has  the diagonals of each square are shaded. the “green area” is 5 + 5 = 10 square units ,
Square 4 has  the diagonals of each square are shaded. the “green area” is 6 + 6 = 12 square units,
Square 5 has  the diagonals of each square are shaded. the “green area” is 7 + 7 = 14 square units ,
Here , The number of squares are increasing by one block with the square numbers.
So for the , Square 20 has  the diagonals of each square are shaded. the “green area” is 23 + 23 = 46 square units,
And Square 21 has  the diagonals of each square are shaded. the “green area” is 24 + 24 = 48 square units.

Lesson 7.3 Linear Functions

EXPLORATION 1

Writing and Graphing Functions
Work with a partner. Each table shows a familiar pattern from geometry.

• Determine what the variables x and y represent. Then write a function rule that relates y to x.
• Is the function a linear function? Explain your reasoning.

Answer: All of them are explained below

Explanation:
The variables x and y represents a rectangle
a. From the given table , Ordered pairs are (1 , 10) , (2 , 12) , (3 , 14) , (4 , 16)
First find the slope m of the line containing the two given points (1 ,-1) and (2, -2)
m = (y2-y1) / (x2-x1)
m= (-2 – (-1)) / (2 – 0)
m = -1 / 2 .
substitute the slope in the (1 , 10) to get point slope to form a line.
y-y1 = m (x-x1)
y – 10 = -1/2 ( x –1)
2(y – 10) = -x  + 1
2y – 20 = -x+ 1
2y = -x  + 21
y = $$\frac{-1}{2}$$ (x – 21)
So ,  y = $$\frac{-1}{2}$$ (x – 21) is linear function.

b. The variables x and y represent a circle
Ordered pairs are (1 , 3.14 ) , (2 , 6.28) , (3 , 9.42) , (4 , 12.5 )
Plot the points in the table , Draw a line through the points
First find the slope m of the line containing the two given points (1 , 3.14 ) , (2 , 6.28)
m = (y2-y1) / (x2-x1)
m= (6.28 –3.14) / (2– 1)
m = 3.14/1
m = 3.14
substitute the slope in the (8 , 4) to get point slope to form a line.
y-y1 = m (x-x1)
y – 3.14 =3.14 ( x –1)
y – 3.14 = 3.14x – 3.14
y = 3.14x – 3.14 + 3.14
y = 3.14x
So ,  y = 3.14x  is linear function.
Where x is the diameter of the circle.

c. The variables x and y represents a trapezoid
a. From the given table , Ordered pairs are (1 , 5) , (2 , 6) , (3 , 7) , (4 , 8)
First find the slope m of the line containing the two given points (1 ,5) and (2, 6)
m = (y2-y1) / (x2-x1)
m= (6 – 5) / (2 – 1)
m = 1 .
substitute the slope in the (1 ,5) to get point slope to form a line.
y-y1 = m (x-x1)
y – 5 = 1(x – 1)
y – 5 = x – 1
y = x – 1 + 5
y = x + 4
So, y = x + 4 is a linear equation

d. The variables x and y represents a cube
a. From the given table , Ordered pairs are (1 , 28) , (2 , 40) , (3 , 52) , (4 , 64)
First find the slope m of the line containing the two given points (1 ,28) and (2, 40)
m = (y2-y1) / (x2-x1)
m= (40 – 28) / (2 – 1)
m = 12 .
substitute the slope in the (1 ,28) to get point slope to form a line.
y-y1 = m (x-x1)
y – 28 = 12(x – 1)
y – 28 = 12x – 12
y = 12x – 12 + 28
y = 12x + 16
So, y = 12x + 16 is a linear equation.

Try It

Question 1.
Use the graph to write a linear function that relates y to x.

Answer: the linear function is y = $$\frac{-1}{2}$$x -1.

Explanation:
In order to write the function we have to write the ordered pairs of the graph ,
Ordered pairs are  (-4 , 1) , (-2 , 0 ) , (0 , -1) , ( 2, -2 )
First find the slope m of the line containing the two given points (0 ,-1) and (2, -2)
m = (y2-y1) / (x2-x1)
m= (-2 – (-1)) / (2 – 0)
m = -1 / 2 .
Because the line crosses the y axis at ( 0, -1 ) , The y intercept is -1.
So , the linear function is y = $$\frac{-1}{2}$$x -1.

Question 2.
Use the table to write a linear function that relates y to x.

Answer: the linear function is y = (0)x + 2.

Explanation:
Ordered pairs are (-2 , 2) , (-1 , 2) , (0 , 2) , (1 , 2)
Plot the points in the table , Draw a line through the points
First find the slope m of the line containing the two given points (0 ,2) and (1, 2)
m = (y2-y1) / (x2-x1)
m= (2 – 2) / (1 – 0)
m = 0
Because the line crosses the y axis at ( 0, 2 ) , The y intercept is 2.
So , the linear function is y = (0)x + 2.

Question 3.
WHAT IF?
The rate of descent doubles. Repeat parts (a) and (b).
Answer: a. the linear function is y = -1x + 65.
b. The slope indicates that the height decreases 1000 feet per minute.
The y intercept indicates that the descent begins at a cruising altitude of 65,000 feet.

Explanation:
a. From the Given table , The rate of descents is 5
If it doubles , then The rate of descents is 10.
The the ordered pairs will be (0 , 65) , (10 ,55) , (20 , 45) .
First find the slope m of the line containing the two given points (0 ,65) and (10, 55)
m = (y2-y1) / (x2-x1)
m= (55 – 65) / (10 – 0)
m = -10 / 10
m = -1
Because the line crosses the y axis at ( 0, 65 ) , The y intercept is 65.
So , the linear function is y = -1x + 65.

b. The slope indicates that the height decreases 1000 feet per minute.
The y intercept indicates that the descent begins at a cruising altitude of 65,000 feet.

Self-Assessment for Concepts & Skills
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 4.
WRITING A LINEAR FUNCTION
Use the graph to write a linear function that relates y to x.

Answer:  The linear function is y = -4x -2 .

Explanation:
In order to write the function we have to write the ordered pairs of the graph ,
Ordered pairs are  (-2 , 6) , (-1 , 2 ) , (0 , -2) , ( 1, -6 )
First find the slope m of the line containing the two given points (0 ,-2) and (1, -6)
m = (y2-y1) / (x2-x1)
m= (-6 – (-2)) / (1 – 0)
m = -4 .
Because the line crosses the y axis at ( 0, -2) , The y intercept is -2.
So , the linear function is y = -4x -2 .

Question 5.
INTERPRETING A LINEAR FUNCTION
The table shows the revenue R (in millions of dollars) of a company when it spends A (in millions of dollars) on advertising.

a. Write and graph a linear function that relates R to A.
b. Interpret the slope and the y-intercept.
Answer:  a. The linear function is y = 2x + 2. and the graph is shown below
b. The slope indicates that the increasing in the amount of spending on advertising by 2 million dollars
The y intercept indicates that the Revenue begins to increasing from the 2 million dollars.

Explanation:
a. From the given table ,
The the ordered pairs will be (0 , 2) , (2 ,6) , (4 , 10) , (6 , 14) , (8 ,18) .
The graph is
First find the slope m of the line containing the two given points (0 ,2) and (2, 6)
m = (y2-y1) / (x2-x1)
m= (6 – 2) / (2 – 0)
m = 4 / 2
m = 2
Because the line crosses the y axis at ( 0, 2 ) , The y intercept is 2.
So , the linear function is y = 2x + 2.

b. The slope indicates that the increasing in the amount of spending on advertising by 2 million dollars
The y intercept indicates that the Revenue begins to increasing from the 2 million dollars.

Self-Assessment for Problem Solving
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 6.
Manager A earns $15 per hour and receives a$50 bonus. The graph shows the earnings of Manager B. (a) Which manager has a greater hourly wage? (b) After how many hours does Manager B earn more money than Manager A?

Answer: a. Manager B has the greater hourly wage than Manager A .
b. As manager A receives a $50 bonus , Manager B has to work an hour extra to earn more money than Manager A . Explanation: a. Manager A earns$15 per hour and receives a $50 bonus. The ordered pairs will be (0 , 0) , (1 , 15) , (2 , 30) , (3 , 45) The graph shows the earnings of Manager B. Ordered pairs from the graph are (0 , 0) , (1 , 25) , (2 , 50) , (3 , 75) So, Manager B has the greater hourly wage than Manager A . b. As manager A receives a$50 bonus , Manager B has to work an hour extra to earn more money than Manager A .

Question 7.
Each month, you start with 2 gigabytes of data and use 0.08 gigabyte per day. The table shows the amount (in gigabytes) of data that your friend has left days after the start of each month. Who runs out of data first? Justify your answer.

Answer:  you will be run out of data first

Explanation:
a. Given , Each month, you start with 2 gigabytes of data and use 0.08 gigabyte per day.
Let x be the number of days and y be the total data in gigabytes.
So, y = -0.08x + 2 ,
You will be out of data if , -0.08x + 2 = 0 ,
-0.08x + 2 = 0
2 = 0.08x
x = $$\frac{2}{0.08}$$
x = 25.
Hence ,you will be run out of data in 25 days.
b. Daily data usage for the friend will be given by the slope of the graph.
The the ordered pairs will be (0 , 3) , (7 ,2.3) , (14 , 1.6) .
First find the slope m of the line containing the two given points (7 ,2.3) and (14, 1.6)
m = (y2-y1) / (x2-x1)
m= (1.6 – 2.3) / (14 – 7)
m = -0.7 / 7
m = -0.1
Because the line crosses the y axis at ( 0, 3 ) , The y intercept is 3.
So , the linear function is y = -0.1x + 3.
Your friend will be out of data if ,
-0.1x + 3 = 0
3 = 0.1x
x = $$\frac{3}{0.1}$$
x = 30 .
Hence ,Friend will be run out of data in 30 days

So , you will be run out of data first

Linear Functions Homework & Practice 7.3

Review & Refresh

Write a function rule for the statement. Then graph the function.
Question 1.
The output is ten less than the input.
Answer: y = x – 10.

Explanation:
Let us say x is input and y is output , then
The output is ten less than the input, will be ,
y = x – 10.

Question 2.
The output is one-third of the input.
Answer: y = $$\frac{x}{3}$$

Explanation:
Let us say x is input and y is output , then
The output is one-third of the input, will be ,
y = $$\frac{x}{3}$$ .

Solve the system.
Question 3.
y = x + 5
y = – 3x + 1
Answer: X = 0 , Y = 5

Explanation:
Y=3X+5 ——————-(1)
Y=X+5 ——————(2)
Substitute Y=X+5 in equation (1)
X+5=3X+5
Solve it for X
X+3X=55
4X=0
X=0/4=0
X = 0
Substitute X=0 in equation (1)
Y=0+5
Y=5

Question 4.
x + y = – 4
6x + 2y = 4
Answer:  X = 3 , Y= -7 .

Explanation:
2Y=−6X+4 ——————-(1)
Y= –X-4 ——————(2)
Substitute Y= –X-4 in equation (1)
2Y = −6X+4
2 ( X – 4 ) = −6X + 4
-2X – 8 = -6X + 4
6X -2X = 8 + 4
4X = 12
X = 3
Substitute X=3 in equation (2)
Y=– 3 – 4
Y= -7 .

Question 5.
– 4x + 3y = 14
y = 2x + 8
Answer:  X = -5 , Y = -2 .

Explanation:
3Y = 4X+14 ——————-(1)
Y = 2X + 8 ——————(2)
Substitute Y= 2X + 8 in equation (1)
3Y = 4X+14
3(2X + 8) = 4X+14
6X + 24 = 4X + 14
6X – 4X = 14 – 24
2X = -10
X = -5
Substitute X= -5 in equation (2)
Y= 2(-5) + 8
Y= -10 + 8
Y = -2.

Concepts, Skills, &Problem Solving

WRITING AND GRAPHING FUNCTIONS The table shows a familiar pattern from geometry. (a) Determine what the variables x and y represent. Then write a function rule that relates y to x. (b) Is the function a linear function? Explain your reasoning. (See Exploration 1, p. 289.)
Question 6.

Answer: a. The variables x and y represent a right angle triangle
b. y = 2x  is linear function.

Explanation:
In order to write the function we have to write the ordered pairs
Ordered pairs are  (1 , 2) , (2 , 4) ,  (3 , 6 ) , (4 , 8), (5 , 10 ) .
a. the variables x and y represent a right angle triangle
Plot the points in the table , Draw a line through the points
First find the slope m of the line containing the two given points (1 , 2) , (2 , 4)
m = (y2-y1) / (x2-x1)
m= (4 – 2) / (2– 1)
m = 2/1
m = 2
b. substitute the slope in the (2 , 4) to get point slope to form a line.
y-y1 = m (x-x1)
y – 4 = 2 ( x – 2)
y – 4 = 2x – 4
y = 2x – 4 + 4
y = 2x
So ,  y = 2x  is linear function.

Given side of triangle is 4 then x= 4/2 = 2
x = 2 and y = 4.

Question 7.

Answer: y = 3.14x  is linear function. and The variables x and y represent a circle

Explanation:
Ordered pairs are (1 , 3.14 ) , (2 , 6.28) , (3 , 9.42) , (4 , 12.5 )
Plot the points in the table , Draw a line through the points
First find the slope m of the line containing the two given points (1 , 3.14 ) , (2 , 6.28)
m = (y2-y1) / (x2-x1)
m= (6.28 –3.14) / (2– 1)
m = 3.14/1
m = 3.14
substitute the slope in the (8 , 4) to get point slope to form a line.
y-y1 = m (x-x1)
y – 3.14 =3.14 ( x –1)
y – 3.14 = 3.14x – 3.14
y = 3.14x – 3.14 + 3.14
y = 3.14x
So ,  y = 3.14x  is linear function.
Where x is the diameter of the circle.

WRITING LINEAR FUNCTIONS Use the graph or table to write a linear function that relates y to x.
Question 8.

Answer: The linear function is y = $$\frac{4}{3}$$x +2

Explanation:
In order to write the function we have to write the ordered pairs of the graph ,
Ordered pairs are  (-3 , -2) , (0 , 2 ) , (3 , 6) , ( 6, 10 )
First find the slope m of the line containing the two given points (3 ,6) and (6, 10)
m = (y2-y1) / (x2-x1)
m= (10 – 6) / (6 – 3)
m = 4/3 .
Because the line crosses the y axis at ( 0, 2) , The y intercept is 2.
So , the linear function is y = $$\frac{4}{3}$$x +2 .

Question 9.

Answer: The linear function is y = (0)x +3 .

Explanation:
In order to write the function we have to write the ordered pairs of the graph ,
Ordered pairs are  (-2 , 3) , (-1 , 3 ) , (0 , 3) , ( 1, 3 ) , (2 , 3)
First find the slope m of the line containing the two given points (1 ,3) and (2, 3)
m = (y2-y1) / (x2-x1)
m= (3 – 3) / (2 – 1)
m = 0 .
Because the line crosses the y axis at ( 0, 3) , The y intercept is 3.
So , the linear function is y = (0)x +3 .

Question 10.

Answer: The linear function is y = $$\frac{-1}{4}$$x + 0.

Explanation:
Ordered pairs are (-8 , 2) , (-4 , 1) , (0 , 0) , (4 , -1)
Plot the points in the table , Draw a line through the points
First find the slope m of the line containing the two given points (-8 ,2) and (-4, 1)
m = (y2-y1) / (x2-x1)
m= (1 – 2) / (-4 – (-8))
m = -1/4
Because the line crosses the y axis at ( 0, 0 ) , The y intercept is 0.
So , the linear function is y = $$\frac{-1}{4}$$x + 0.

Question 11.

Answer: The linear function is y = $$\frac{2}{3}$$x + 5.

Explanation:
Ordered pairs are (-3 , 3) , (0 , 5) , (3 , 7) , (6 , 9)
Plot the points in the table , Draw a line through the points
First find the slope m of the line containing the two given points (3 ,7) and (6, 9)
m = (y2-y1) / (x2-x1)
m= (9 – 7) / (6 – 3)
m = 2/3
Because the line crosses the y axis at ( 0, 5 ) , The y intercept is 5.
So , the linear function is y = $$\frac{2}{3}$$x + 5.

Question 12.
INTERPRETING A LINEAR FUNCTION
The table shows the length y (in inches) of a person’s hair after x months.

a. Write and graph a linear function that relates y to x.
b. Interpret the slope and the y-intercept.
Answer: a. The linear function is y = 0.5x + 11.
b. The slope indicates that the increasing in the hair length
The y intercept indicates that the increasing in hair length by time.

Explanation:
a. Given ,
The ordered pairs will be (0 , 11) , (3 ,12.5) , (6 , 14) .
The graph is
First find the slope m of the line containing the two given points (3 ,12.5) and (6 , 14)
m = (y2-y1) / (x2-x1)
m= (14 – 12.5) / (6 – 3)
m = 1.5 / 3
m = 0.5
Because the line crosses the y axis at ( 0, 11 ) , The y intercept is 11.
So , the linear function is y = 0.5x + 11.

b. The slope indicates that the increasing in the hair length
The y intercept indicates that the increasing in hair length by time.

Question 13.
INTERPRETING A LINEAR FUNCTION
The table shows the percent (in decimal form) of battery power remaining x hours after you turn on a laptop computer.

a. Write and graph a linear function that relates y to x.
b. Interpret the slope, the x-intercept, and the y-intercept.
c. After how many hours is the battery power at75%?
Answer: a. The linear function is y = -0.2x + 1.
b. given below the explanation.
c. Battery will be 75% after 1.25 hours.

Explanation:
a. Given ,
The ordered pairs will be (0 , 1) , (2 ,0.6) , (4 , 0.2) .
The graph is
First find the slope m of the line containing the two given points (2 ,0.6) and (4 , 0.2)
m = (y2-y1) / (x2-x1)
m= (0.2 – 0.6) / (4 – 2)
m = -0.4 / 2
m = -0.2
Because the line crosses the y axis at ( 0, 1 ) , The y intercept is 1.
So , the linear function is y = -0.2x + 1.

b. Slope is -0.2 which means that as time increases by 1 hour, Battery power remaining decreases by 20% .
y intercept is 1, which means initially the battery power remaining before usage was 100%.
x intercept is 5 which means the battery remaining will be 0 after 5 hours.

c. battery percent will be 75% of 0.75 if ,
-0.2x + 1 = 0.75
0.2x = 1 – 0.75
x = 0.25/0.2
x = 1.25
Battery will be 75% after 1.25 hours.

Question 14.
MODELING REAL LIFE
The number y of calories burned after x minutes of kayaking is represented by the linear function y = 4.5x. The graph shows the number of calories burned by hiking.

a. Which activity burns more calories per minute?
b. You perform each activity for 45 minutes. How many total calories do you burn? Justify your answer.
Answer: a. hiking burns more calories than kayaking .
b. In kayaking, 202.5 calories are burnt per minute. and In hiking , 225 calories are burnt per minute.

Explanation:
a. The number y of calories burned after x minutes of kayaking is represented by the linear function y = 4.5x.
So, The ordered pairs of the graph are (0 , 0) , (1 , 4.5) , (2 , 9) , (3, 13.5)
Here , In kayaking burns 4.5 calories per minute .
For hiking ,
The ordered pairs of the graph are (0 , 0) , (1 , 5) , (2 , 10) , (3, 15)
Here , In hiking burns 5 calories per minute.
Thus , hiking burns more calories than kayaking .

b. Given , perform each activity for 45 minutes.
Liner function of the kayaking is y = 4.5x
substitute x = 45 in equation
y = 4.5 (45)
y = 202.5
In kayaking, 202.5 calories are burnt per minute.
Linear function of the hiking is y = 5x
substitute x = 45 in equation
y = 5 (45)
y = 225
In hiking , 225 calories are burnt per minute.

Question 15.
DIG DEEPER!
You and a friend race each other. You give your friend a 50-foot head start. The distance y (in feet) your friend runs after x seconds is represented by the linear function y = 14x + 50. The table shows your distance at various times throughout the race. For what distances will you win the race? Explain.

Answer: you will win the race for distances greater than 190 feet

Explanation:
The distance y (in feet) your friend runs after x seconds is represented by the linear function y = 14x + 50.
The slope of the line is 14 so , your friend runs at the rate of 14 ft per second
To find your rate , the ordered pairs are (2 , 38) , (4 , 76) , (6 , 114) , (8 , 152)
First find the slope m of the line containing the two given points (2 ,38) and (4 , 76)
m = (y2-y1) / (x2-x1)
m= (76 – 38) / (4 – 2)
m = 38 / 2
m = 19
You are running at the rate of 19 ft per second.
To get the linear equation , substitute the slope in the (2 , 38) to get point slope to form a line.
Then we have , y = 19x
Now if x = 10 , to run faster then ,
y = 19(10)
y = 190 .
Your friend linear equation is y = 14x + 50 .
if x = 10 ,then
y = 14(10) + 50
y = 140 + 50
y = 190.
So , for x > 10 , means you will run farther than your friend which means you would win the race .
Therefore, you will win the race for distances greater than 190 feet.

Question 16.
REASONING
You and your friend are saving money to buy bicycles that cost $175 each. You have$45 to start and save an additional $5 each week. The graph shows the amount y(in dollars) that your friend has after x weeks. Who can buy a bicycle first? Justify your answer. Answer: your friend will but the bicycle first. Explanation: Given , your friend savings are the ordered pairs are (0,15) and (3,39) First find the slope m of the line containing the two given points (0,15) and (3,39) m = (y2-y1) / (x2-x1) m= (39 – 15) / (3 – 0) m = 24 / 3 m = 8 Because the line crosses the y axis at ( 0, 15 ) , The y intercept is 15. So , the linear function is y = 8x + 15. to buy bicycles that cost$175 each
if y = 175 , then
175 = 8x + 15
8x = 175 – 15
x = 160/8
x = 20
So, your friend need 20 weeks to buy the bicycle
Given, You have $45 to start and save an additional$5 each week
So , the linear function will be y = 5x + 45.
to buy bicycles that cost $175 each if y = 175 , then 175 = 5x + 45 5x = 175 – 45 x = 130/5 x = 26 So, you need 26 weeks to buy the bicycle. Hence, your friend will but the bicycle first. Question 17. CRITICAL THINKING Is every linear equation a linear function? Explain your reasoning. Answer: All linear equations produce straight lines when graphed, not all linear equations produce linear functions. In order to be a linear function, a graph must be both linear (a straight line) and a function (matching each x-value to only one y-value). Question 18. PROBLEM SOLVING The heat index is calculated using the relative humidity and the temperature. For every 1 degree increase in the temperature from 94°F to 97°F at 75% relative humidity, the heat index rises 4°F. On a summer day, the relative humidity is 75%, the temperature is 94°F, and the heat index is 124°F. Estimate the heat index when the relative humidity is 75% and the temperature is 100°F. Use a function to justify your answer. Answer: Heat index is 148°F Explanation: The form of linear equation is y = mx + c and the slope of the function is given by m = (y2-y1) / (x2-x1) Let y be the heat index and x be the temperature Given , (94, 124) For every 1 degree increase in the temperature from 94°F to 97°F at 75% relative humidity, the heat index rises 4°F that is m = 4 Since the line passes through (94, 124) means 124 = 4x + c 124 = 4(94) + c 124 = 376 + c c = 124 – 376 c = -252 Linear function for the heat index is y = 4x – 252 put x = 100 So, y = 4(100) – 252 y = 400 – 252 y = 148. Finally, Heat index is 148°F. Lesson 7.4 Comparing Linear and Non Linear Functions EXPLORATION 1 Comparing Functions Work with a partner. Each equation represents the height h (in feet) of a falling object after t seconds. • Graph each equation. Explain your method. • Decide whether each graph represents a or function. • Compare the falling objects. Answer: Explained below Explanation: a. Given, h = 300 – 15t can be written as y = 300 – 15x h = 300 – 15t , we know y = mx + c , where m = slope , c = constant To obtain the graph , we should have ordered pairs , So , if x = 0 , then y =300 – 15(0) = 300 . co-ordinates are (0 , 300) if x = 1 , then y = 300 – 15(1) = 285 . co-ordinates are (1 , 285) if x = 2 , then y = 300 – 15(2) = 270 , co-ordinates are (2 , 270) if x = 3 , then y = 300 – 15(3) = 255 , co-ordinates are (3 , 255) The co-ordinates (0 , 300) , (1 , 285) , (2 , 270) , (3 , 255) form a straight line . The graph is Given , h = 300- 16t2 , we know y = mx + c , where m = slope , c = constant To obtain the graph , we should have ordered pairs , So , if x = 0 , then y =300- 16(0)2 = 300 . co-ordinates are (0 , 300) if x = 1 , then y =300- 16(1)2 = 284 . co-ordinates are (1 , 284) if x = 2 , then y = 300- 16(2)2 = 236 , co-ordinates are (2 , 236) if x = 3 , then y = 300- 16(3)2 = 252 , co-ordinates are (3 , 252) The co-ordinates (0 , 300) , (1 , 284) , (2 , 236) , (3 , 252) does not form a straight line . The graph is b. For, h = 300 – 15t , The graph is linear so the so it is a function, For h = 300- 16t2 , The graph is linear so the so it is a function. c. Sky diver has the slow fall while compared to the bowling ball , because parachute can be controlled with the wind and can be divert the destination point, and bowling ball cannot be controlled while falling. Try It Does the table represent a linear or nonlinear function? Explain. Question 1. Answer: y = 2x – 12 is linear function. Explanation: Ordered pairs are (2 , -8) , (4 , -4) , (6 , 0) , (8 , 4) Plot the points in the table , Draw a line through the points First find the slope m of the line containing the two given points (6 ,0) and (8, 4) m = (y2-y1) / (x2-x1) m= (4 – 0) / (8– 6) m = 4/2 m = 2 substitute the slope in the (8 , 4) to get point slope to form a line. y-y1 = m (x-x1) y – 4 = 2 ( x – 8) y – 4 = 2x – 16 y = 2x – 16 + 4 y = 2x – 12 So , y = 2x – 12 is linear function. Question 2. Answer: y = –$$\frac{5}{3}$$x + 25 is linear function. Explanation: Ordered pairs are (0 , 25) , (3 , 20) , (7 , 15) , (12 , 10) Plot the points in the table , Draw a line through the points First find the slope m of the line containing the two given points (0 ,25) and (3, 20) m = (y2-y1) / (x2-x1) m= (20 – 25) / (3– 0) m = -5/3 Because the line crosses the y axis at ( 0, 25 ) , The y intercept is 25. So , the linear function is y = –$$\frac{5}{3}$$x + 25. So , y = –$$\frac{5}{3}$$x + 25 is linear function. Does the equation represent a linear or nonlinear function? Explain. Question 3. y = x + 5 Answer: y = x + 5 is a linear function Explanation: Given , y = x + 5 , we know y = mx + c , where m = slope , c = constant To obtain the graph , we should have ordered pairs , So , if x = 0 , then y = 0 + 5 = 5 . co-ordinates are (0 , 5) if x = 1 , then y = 1 + 5 = 6 . co-ordinates are (1 , 6) if x = 2 , then y = 2 + 5 = 7 , co-ordinates are (2 , 7) The co-ordinates (0 , 5) , (1 , 6) , (2 , 7) form a straight line . Each x input has only one y output so it is a function . And it forms a straight line when graphed . So, y = x + 5 is a linear function. Question 4. y = $$\frac{4x}{3}$$ Answer: y = $$\frac{4x}{3}$$ is a linear function. Explanation: Given , y = $$\frac{4x}{3}$$ , we know y = mx + c , where m = slope , c = constant To obtain the graph , we should have ordered pairs , So , if x = 0 , then y = $$\frac{4(0)}{3}$$ = 0 . co-ordinates are (0 , 0) if x = 1 , then y = $$\frac{4(1)}{3}$$ = $$\frac{4}{3}$$ = 1.3. co-ordinates are (1 , 1.3) if x = 2 , then y = $$\frac{4(2)}{3}$$ = $$\frac{8}{3}$$ = 2.6 , co-ordinates are (2 , 2.6) The co-ordinates (0 , 0) , (1 ,1.3 ) , (2 , 2.6) form a straight line . Each x input has only one y output so it is a function . And it forms a straight line when graphed . So, y = $$\frac{4x}{3}$$ is a linear function. Question 5. y = 1 – x2 Answer: y = 1 – x2 is not a linear function. Explanation: Given , y = 1 – x2 , we know y = mx + c , where m = slope , c = constant To obtain the graph , we should have ordered pairs , So , if x = 0 , then y = 1 – 02 = 1 . co-ordinates are (0 , 1) if x = 1 , then y = 1 – 12 = 0 . co-ordinates are (1 , 0) if x = 2 , then y = 1 – 22 = -3 , co-ordinates are (2 , -3) The co-ordinates (0 , 5) , (1 , 6) , (2 , 7) form a straight line . Each x input has only one y output so it is a function . And it does not forms a straight line when graphed . So, y = 1 – x2 is not a linear function. Does the graph represent a linear or nonlinear function? Explain. Question 6. Answer: The graph represents a nonlinear function. Explanation: In order to write the function we have to write the ordered pairs of the graph , Ordered pairs are (0 , 2) , (-1 , 0) , (-2 , -2 ) , (-3 , -4), (0 , 1 ) , (2 , -2) , ( 3, -4 ) The inputs have more than one output , And points form a straight line So , the graph is non linear function Question 7. Answer: The graph is a linear function Explanation: In order to write the function we have to write the ordered pairs of the graph , Ordered pairs are (0 , 0) , (-1 , -1) , (-2 , -2 ) , (-3 , -3), (1 , 1 ) , (2 , 2) , ( 3, 3 ) The inputs have exactly one output , And points form a straight line So , the graph is a linear function. Self-Assessment for Concepts & Skills Solve each exercise. Then rate your understanding of the success criteria in your journal. IDENTIFYING FUNCTIONS Does the table or graph represent a linear or nonlinear function? Explain. Question 8. Answer: It is not a linear function Explanation: Ordered pairs are (3 , 0) , (-1 , 2) , (-5 , 4) , (-9 , 6) Each input has exactly one output and it does not form a straight line when graphed So, it is not a linear function . Question 9. Answer: The graph is non linear function Explanation: In order to write the function we have to write the ordered pairs of the graph , Ordered pairs are (0 , -1) , (-1 , 0) , (-2 , 3 ) , (1 , 0 ) , (2 , 3) . The inputs have exactly one output , And points does not form a straight line So , the graph is non linear function Question 10. WHICH ONE DOESN’T BELONG? Which equation does not belong with the other three? Explain your reasoning. Answer: 5xy = -2 does not belong with the other three. Explanation: 15y = 6x , y = $$\frac{2}{5}$$x , 10y = 4x . These are evaluated as 5y = 2x 5xy = -2 , is different from 5y = 2x. Self-Assessment for Problem Solving Solve each exercise. Then rate your understanding of the success criteria in your journal. Question 11. The loudness of sound is measured in (dB). The graph shows the loudness y of a sound (in decibels) x meters from the source of the sound. Is the relationship between loudness and distance linear or nonlinear? Approximate the loudness of the sound 12 meters from the source. Answer: The relationship between loudness and distance is nonlinear Function. And the loudness of the sound 12 meters from the source is approximately 85dB as shown in the graph. Explanation: As shown in the graph , the plot of the points does not form a straight line , Its a parabolic decay , The amount of loudness decreases with the increase in distance, So, The relationship between loudness and distance is nonlinear Function. And the loudness of the sound 12 meters from the source is approximately 85dB as shown in the graph. Question 12. A video blogger is someone who records a video diary. A new website currently hosts 90 video bloggers and projects a gain of 10 video bloggers per month. The table below shows the actual numbers of video bloggers. How does the projection differ from the actual change? Answer: Projections are more than the actual values Explanation: So, Projections are more than the actual values Comparing Linear and Non Linear Functions Homework & Practice 7.4 Review & Refresh Write a linear function that relates y to x. Question 1. Answer: The linear function is y = x – 2 In order to write the function we have to write the ordered pairs of the graph , Ordered pairs are (0 , -2) , (1 , -1 ) , (-1 , -3) , ( 2, 0), (3 , 1) , (4 , 2) , ( 5, 3) First find the slope m of the line containing the two given points (2 ,0) and (3, 1) m = (y2-y1) / (x2-x1) m= (1 – 0) / (3 – 2) m = 1 . Because the line crosses the y axis at ( 0, -2 ) , The y intercept is -2. So , the linear function is y = x – 2 . Question 2. Answer: The linear function is y =$$\frac{-1}{1.5}$$x + 5. Explanation: Ordered pairs are (0 , 5) , (1.5 , 4) , (3 , 3) , (4.5 , 2) Plot the points in the table , Draw a line through the points First find the slope m of the line containing the two given points (1.5 ,4) and (3, 3) m = (y2-y1) / (x2-x1) m= (3 – 4) / (3 – 1.5) m = -1 /1.5 Because the line crosses the y axis at ( 0, 5 ) , The y intercept is 5. So , the linear function is y =$$\frac{-1}{1.5}$$x + 5. The vertices of a figure are given. Draw the figure and its image after a dilation with the given scale factor. Identify the type of dilation. Question 3. A (- 3, 1), B (- 1, 3), C (- 1, 1); k = 3 Answer: The New right angle triangle is larger than the original one So , its a increase . Explanation: Given , (- 3, 1), (- 1, 3), (- 1, 1) these pairs form a right angle triangle K = 3 , For the dilation figure multiply the 3 with the given ordered pairs , then (- 3, 1) × 3 = ( -9 , 3) (- 1, 3) × 3 = ( -3 , 9) (- 1, 1) × 3 = (-3 , 3) From these new ordered pairs we form a new right angle triangle The figure is The New right angle triangle is larger than the original one So , its a increase . Question 4. J (2, 4), K (6, 10), L (8, 10), M (8, 4); k = $$\frac{1}{4}$$ Answer: It is a reduction Explanation: Given , (2, 4), (6, 10), (8, 10) ,(8,4) these pairs forms a figure K = 0.25 , For the dilation figure multiply the 3 with the given ordered pairs , then (2, 4) × 0.25 = (0.5, 1) (6, 10) × 0.25 = (1.5, 2.5) (8, 10) × 0.25 = (2, 2.5) (8, 4) × 0.25 = (2 , 1) From these new ordered pairs we form a new figure The figure is The New figure is smaller than the original , So, It is a reduction . Concepts, Skills, & Problem Solving COMPARING FUNCTIONS Graph each equation. Decide whether each graph represents a linear or nonlinear function. (See Exploration 1, p. 295.) Question 5. h = 5 + 6t Equation 1 h = 5 + 6t2 Equation 2 Answer: h = 5 + 6t Equation 1 is a linear function h = 5 + 6t2 Equation 2 is a non linear function . Explanation: Given , h = 5 + 6t , we know y = mx + c , where m = slope , c = constant To obtain the graph , we should have ordered pairs , So , if x = 0 , then y = 5 + 6(0) = 5 . co-ordinates are (0 , 5) if x = 1 , then y = 5 + 6(1) = 11 . co-ordinates are (1 , 11) if x = 2 , then y = 5 + 6(2) = 17 , co-ordinates are (2 , 17) if x = 3 , then y = 5 + 6(3) = 23 , co-ordinates are (3 , 23) The co-ordinates (0 , 5) , (1 , 11) , (2 , 17) , (3 , 23) form a straight line . Given , h = 5 + 6t2 , we know y = mx + c , where m = slope , c = constant To obtain the graph , we should have ordered pairs , So , if x = 0 , then y = 5 + 6(0)² = 5 . co-ordinates are (0 , 5) if x = 1 , then y = 5 + 6(1)² = 11 . co-ordinates are (1 , 11) if x = 2 , then y = 5 + 6(2)² = 26 , co-ordinates are (2 , 26) if x = 3 , then y = 5 + 6(3)² = 59 , co-ordinates are (3 , 59) The co-ordinates (0 , 5) , (1 , 11) , (2 , 26) , (3 , 59) does not form a straight line . The graph of both equations is So, h = 5 + 6t Equation 1 is a linear function h = 5 + 6t2 Equation 2 is a non linear function . Question 6. y = – $$\frac{x}{3}$$ Equation 1 y = – $$\frac{3}{x}$$ Equation 2 Answer: y = – $$\frac{x}{3}$$ Equation 1 is a linear function y = – $$\frac{3}{x}$$ Equation 2 is a non linear function. Given , y =- $$\frac{x}{3}$$ , we know y = mx + c , where m = slope , c = constant To obtain the graph , we should have ordered pairs , So , if x = 0 , then y =- $$\frac{0}{3}$$ = 0 . co-ordinates are (0 , 0) if x = 1 , then y = – $$\frac{1}{3}$$ = – 0.3 . co-ordinates are (1 , – 0.3 ) if x = 2 , then y = – $$\frac{2}{3}$$ = – 0.6 , co-ordinates are (2 ,-0.6) if x = 3 , then y = – $$\frac{3}{3}$$ = -1 , co-ordinates are (3 , -1) The co-ordinates (0 , 0) , (1 , -0.3) , (2 , -0.6) , (3 , -1) form a straight line . Given , y =- $$\frac{3}{x}$$ , we know y = mx + c , where m = slope , c = constant To obtain the graph , we should have ordered pairs , So , if x = 0 , then y =- $$\frac{3}{0}$$ = no number if x = 1 , then y = – $$\frac{3}{1}$$ = – 3 . co-ordinates are (1 , – 1 ) if x = 2 , then y = – $$\frac{3}{2}$$ = – 1.5 , co-ordinates are (2 ,-1.5) if x = 3 , then y = – $$\frac{3}{3}$$ = -1 , co-ordinates are (3 , -1) The co-ordinates (1 , -1) , (2 , -1.5) , (3 , -1) form a straight line . The graph of both the equations is So, y = – $$\frac{x}{3}$$ Equation 1 is a linear function y = – $$\frac{3}{x}$$ Equation 2 is a non linear function. IDENTIFYING FUNCTIONS FROM TABLES Does the table represent a linear or nonlinear function? Explain. Question 7. Answer: linear function is y = 4x + 4. Explanation: Ordered pairs are (0 , 4) , (1 , 8) , (2 , 12) , (3 , 16) Plot the points in the table , Draw a line through the points First find the slope m of the line containing the two given points (2 , 12) and (3 , 16) m = (y2-y1) / (x2-x1) m= (16 – 12) / (3– 2) m = 4/1 m = 4 Because the line crosses the y axis at ( 0, 4 ) , The y intercept is 4. So , the linear equation is y = 4x + 4. And it is a linear function. The graph is Question 8. Answer: y = 4x – 6 is linear function. Explanation: Ordered pairs are (6 , 21) , (5 , 15) , (4 , 10) , (3 , 6) Plot the points in the table , Draw a line through the points First find the slope m of the line containing the two given points (4 , 10) and (3 , 6) m = (y2-y1) / (x2-x1) m= (6 – 10) / (3– 4) m = -4/-1 m = 4 substitute the slope in the(4 , 10) to get point slope to form a line. y-y1 = m (x-x1) y – 10 = 4 ( x – 4) y – 10 = 4x – 16 y = 4x – 16 + 10 y = 4x – 6 So , y = 4x – 6 is linear function. The graph is IDENTIFYING FUNCTIONS FROM EQUATIONS Does the equation represent a linear or nonlinear function? Explain. Question 9. 2x + 3y = 7 Answer: The function is linear when m = $$\frac{-2}{3}$$ and c = $$\frac{7}{3}$$ Explanation: Given ,2x + 3y = 7 3y = 7 – 2x y = $$\frac{-2}{3}$$x+ $$\frac{7}{3}$$ So, The function is linear when m = $$\frac{-2}{3}$$ and c = $$\frac{7}{3}$$ Question 10. y + x = 4x + 5 Answer: The function is linear when m = 3 and c = 5 . Explanation: Given , y + x = 4x + 5 y = 4x – x + 5 y = 3x + 5 So, The function is linear when m = 3 and c = 5 . Question 11. y = $$\frac{8}{x^{2}}$$ Answer: The function is linear when m = 8 and c = 0 . Explanation: Given , y = $$\frac{8}{x^{2}}$$ slope m = 8 c = 0 So, The function is linear when m = 8 and c = 0 . IDENTIFYING FUNCTIONS FROM GRAPHS Does the graph represent a linear or nonlinear function? Explain. Question 12. Answer: The graph is linear function Explanation: In order to write the function we have to write the ordered pairs of the graph , Ordered pairs are (0 , 1) , (2 , 0) , (4 , -1 ) , (-2 , 2), ( -4, 3 ) The inputs have exactly one output , And points form a straight line So , the graph is linear function Question 13. Answer: The graph is non linear function. Explanation: In order to write the function we have to write the ordered pairs of the graph , Ordered pairs are (0 , 0) , (-1 , -1) , (-4 , -2 ) , (1 , 1), ( 4, 2 ) The inputs have exactly one output , And points does not form a straight line So , the graph is non linear function Question 14. IDENTIFYING A FUNCTION The graph shows the volume V (in cubic feet) of a cube with an edge length of x feet. Does linear nonlinear the graph represent a linear or nonlinear function? Explain. Answer: The graph is non linear function n order to write the function we have to write the ordered pairs of the graph , Ordered pairs are (1 , 1) , (2 , 8) , (3 , 27 ) , (4 , 64) The inputs have exactly one output , And points does not form a straight line So , the graph is non linear function Question 15. MODELING REAL LIFE The frequency y (in terahertz) of a light wave is a function of its wavelength x (in nanometers). Is the function relating the wavelength of light to its frequency linear or nonlinear? Answer: The function is a non linear function Explanation: table is as follows change in x is constant but change in y is not constant , it is increasing So, the function is a non linear function . Question 16. DIG DEEPER! The table shows the cost (in dollars) of pounds of sun flower seeds. a. What is the missing -value that makes the table represent a linear function? b. Write a linear function that represents the cost of x pounds of seeds. Interpret the slope. c. Does the function have a maximum value? Explain your reasoning. Answer: a. 3 pounds =$4.2
b. y = 1.4x  is linear function.
c.  If y has maximum value then the x also has maximum value.

Explanation:
a. As per the table 1 pound = $1.4 2 pounds =$2.8
3pounds = $4.2 4 pounds =$5.6
So, the price is increasing with weight of the seeds.

b. Ordered pairs are (2 , 2.8) , (3 , 4.2) , (4 , 5.6)
Plot the points in the table , Draw a line through the points
First find the slope m of the line containing the two given points (3 , 4.2) and (4 , 5.6)
m = (y2-y1) / (x2-x1)
m= (5.6 – 4.2) / (4 – 3)
m = 1.4
substitute the slope in the (3 , 4.2) to get point slope to form a line.
y-y1 = m (x-x1)
y – 4.2 = 1.4 ( x – 3)
y – 4.2 = 1.4x – 4.2
y = 1.4x – 4.2 + 4.2
y = 1.4x
So ,  y = 1.4x  is linear function.

c. As shown in the table , and the function if y increases then x also increases with respect to the y
So, if y has maximum value then the x also has maximum value.

Question 17.
MODELING REAL LIFE
A birch tree is 9 feet tall and grows at a rate of 2 feet per year. The table shows the height h (in feet) of a willow tree after x years.

a. Does the table represent a linear or nonlinear function? Explain.
b. Which tree is taller after 10 years? Explain.
Answer: There is no linear relationship between x and y .

Explanation:
Table is as follows
Change in y is constant but change in x is increasing , not a constant
Hence, there is no linear relationship between x and y .

Question 18.
CRITICAL THINKING
In their first year, Show A has 7 million viewers and Show B has 5 million viewers. Each year, Show A has 90% of the viewers it had in the previous year. Show B loses 200,000 viewers each year.
a. Determine whether the function relating the year to the number of viewers is linear or nonlinear for each show.
b. Which show has more viewers in its sixth year?
Answer: a. The function relating the year to the number of viewers is linear
b. Both shows  has same number of viewers in the sixth year .

Explanation:
a. Given, In their first year, Show A has 7 million viewers and Show B has 5 million viewers. Each year, Show A has 90% of the viewers it had in the previous year. Show B loses 200,000 viewers each year.
For show A
So , In first year = 7
2 year = 90% of 7 = 6.3
3 year = 90% of 6.3 = 5.6
4 year = 90% of 5.6 = 5.04
5 year = 90% of 5 = 4.5
6 year = 90% of 4.5 = 4.05
So the ordered pairs are (1 , 7) , (2 , 6.3) , (3 , 5.6) , (4 , 5), (5 , 4.5) , (6 , 4)

For show B
In first year = 5 , As the viewers reduces by 2,00,000 in 5M
2 year = 5 – 0.2 = 4.8
3 year = 4.8 – 0.2 = 4.6
4 year = 4.6 – 0.2 = 4.4
5 year = 4.4 – 0.2 = 4.2
6 year = 54.2 – 0.2 = 4
So the ordered pairs are (1 , 5) , (2 , 4.8) , (3 , 4.6) , (4 , 4.4), (5 , 4.2) , (6 , 4)
As the year increases the viewers are also decreasing constantly as per the individual shows
So, The function relating the year to the number of viewers is linear .

b. As shown in part a , the ordered pairs having (6,4) represents the number of viewers to the year
So, Both shows  has same number of viewers in the sixth year .

Question 19.
NUMBER SENSE
The ordered pairs represent a function. (0,- 1), (1, 0), (2, 3), (3, 8), and (4, 15)
a. Graph the ordered pairs and describe the pattern. Is the function linear or nonlinear?
b. Write an equation that represents the function.
Answer: a. The graph is shown below and function is linear
b. The linear equation is y = 7x – 1.

Explanation:
Given, ordered pairs represent a function. (0,- 1), (1, 0), (2, 3), (3, 8), and (4, 15)
a. the graph is
Each input has exactly one output and it forms a straight line So, the graph is linear
b. First find the slope m of the line containing the two given points (3 ,8) and (4, 15)
m = (y2-y1) / (x2-x1)
m= (15 – 8) / (4– 3)
m = 7/1
m = 7
Because the line crosses the y axis at ( 0, -1 ) , The y intercept is -1.
So , the linear equation is y = 7x – 1.

Lesson 7.5 Analyzing and Sketching Graphs

EXPLORATION 1

Matching Situations to Graphs
Work with a partner. Each graph shows your speed during a bike ride. Match each situation with its graph. Explain your reasoning.

a. You increase your speed, then ride at a constant speed along a bike path. You then slow down until you reach your friend’s house. Analyze Relationships
b. You increase your speed, then go down a hill. You then quickly come to a stop at an intersection.
c. You increase your speed, then stop at a store for a couple of minutes. You then continue to ride, increasing your speed.
d. You ride at a constant speed, then go up a hill. Once on top of the hill, you increase your speed.
Answer: a – C ,
b – A ,
c – D ,
d – B ,

Explanation:
a. You increase your speed, then ride at a constant speed along a bike path. You then slow down until you reach your friend’s house. The graph C has the perfect graph representing the situation of given question.
b. You increase your speed, then go down a hill. You then quickly come to a stop at an intersection.
Because The graph A has the bike speed representing the situation for the time .
c. You increase your speed, then stop at a store for a couple of minutes. You then continue to ride, increasing your speed. Thus, The graph D is the final answer for the question
d. You ride at a constant speed, then go up a hill. Once on top of the hill, you increase your speed.
Because of the speed with respect to time the graph B is the correct answer for the question.

EXPLORATION 2

Interpreting a Graph
Work with a partner. Write a short paragraph that describe show the height changes over time in the graph shown. What situation can this graph represent?

Answer: The Graph can be representing a situation for low and high tides of the Ocean

Explanation:
As shown in the figure, The graph is plotted between the height and time,
We can take an example of an Ocean for its waves , As the time passes at the morning of a normal day, The waves of the ocean start rising higher at a period of time, and for the time being maintaining a peak height then drops to a lower height at a particular intervals of time , this process takes place for a while and vise versa.
Thus, the Graph can be representing a situation for low and high tides of the Ocean

Try It

Question 1.
The graph shows the location of a pelican relative to your location.

a. Describe the path of the pelican.
b. Write an explanation for the decrease in the vertical distance of the pelican.
Answer: Both of them are explained below.

Explanation:
a. The path of the pelican is flying in the air , As they always fly in line and the amazing thing is the deeper the prey the higher they dive.
The graph shows the relationship between the horizontal distance that is the height from the land, vertical distance is the point from where its destination point is located, so at the starting point of the flight it has more distance from the ground means flying at a higher level , as the time passes it reaches to the closer point of its destination point so the altitude of the flight decreases with the decrease in the vertical distance and at a particular distance reaches its point of destination.

b. The decrease in the vertical distance of the pelican. is due to its flight to the destination point as it requires to stop flying to reach it, so in order to have a smooth landing on the ground , the bird gradually decreases its speed by decreasing its altitude.

Question 2.
A fully-charged battery loses its charge at a constant rate until it has no charge left. You plug it in, and it fully recharges at a constant rate. Then it loses its charge at a constant rate until it has no charge left. Sketch a graph that represents this situation.
Answer:  The graph is

Explanation:
In the graph , let the x-axis be time and y-axis be the battery charge ,
A fully-charged battery loses its charge at a constant rate until it has no charge left. So, line segment starts from 100 and decreases until it touches the x-axis.
You plug it in, and it fully recharges at a constant rate. Thus, line segment increases at a constant rate until it reaches 100
Then it loses its charge at a constant rate until it has no charge left. line segment decreases again at a constant rate until it again touches the x-axis .

Self-Assessment for Concepts & Skills
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 3.
ANALYZING GRAPHS
The graph shows the growth rate of a plant over time.

a. Describe the change in growth rate.
b. Write an explanation for the decrease in growth rate and the increase in growth rate.
Answer: the answers are given below

Explanation:
a. the change in growth rate of a plant over the time is given by its size and height , So as the time passes the growth rate is constant from the the start and from a particular time the growth rate has been dropping slightly due to external or internal reasons of a plant and again at some time the growth rate is increasing at a constant rate until it reaches to its perfect growth of a plant.

b. The decrease in growth rate of the plant is due to some external causes like weather, rain, sunlight , watering, and the soil may effect its growth rate and the increase in growth rate is probably due to its soil fertility and sufficient sunlight providing sufficient chlorophyll.

Question 4.
SKETCHING GRAPHS
As you snowboard down a hill, you gain speed at a constant rate. You come to a steep section of the hill and gain speed at a greater constant rate. You then slow down at a constant rate until you come to a stop. Sketch a graph that represents this situation.
Answer: The graph is

Explanation:
In the graph , let the x-axis be time and y-axis be the speed ,
As you snowboard down a hill, you gain speed at a constant rate, line segment decreases at a constant rate
You come to a steep section of the hill and gain speed at a greater constant rate, line segment becomes steeper i.e., the line segment decreases at a high constant rate.
You then slow down at a constant rate until you come to a stop, line segment becomes flatter i.e., the constant rate of decrease becomes less until it touches its x-axis

Self-Assessment for Problem Solving
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 5.
Two rowing teams are in a race. The graph shows their distances from the finish line over time. Describe the speed of each team throughout the race. Then determine which team finishes first.

Answer: Team B will finishes race first.

Explanation:
Team A , The relationship between the time and distance from the finish line is given in the graph,
At starting point Team A has maintained a fair speed at the Beginning of the race and has been a little slow while reaching out to the destination point, and for a while they have been balancing the speed with the distance representing a curving point in the graph and directly dropping to the finish line drastically creating a slope, until it reaches in the x-axis line.
Team B , The relationship between the time and distance from the finish line is given in the graph,
As same as the Team A , Team B has a perfect start but it has been a way different them Team A because Team B has a game plan to win the race, as shown in the graph they have maintained a constant speed while reaching out to the destination and also having a smooth drift at a level of decreasing their distance from the finish line.

Team B will  finishes the race first because they are having a constant and smooth decreasing speed which comes to an end gradually at the finishing line.

Question 6.
DIG DEEPER!
The graphs show the movements of two airplanes over time. Describe the movement of each airplane.

Answer: Detailed explanation is given below.

Explanation:
As shown in the graph , x-axis is time and y-axis be the height above ground
Airplane A, the line segment at a constant rate at the time of starting of the takeoff and then drops to a point while decreasing in the height to the ground for landing and for a constant time it is at the ground level until again it takes off  having an increase in the height from the ground level , at last it maintains a constant speed.

Airplane B, the line segment at a constant rate at the time of starting of the takeoff and then drops to a point while decreasing in the height to the ground for landing and for a constant time it is at the ground level until again it takes off  having an increase in the height from the ground level , at last it maintains a constant speed.
It is as same as the airplane A.

Analyzing and Sketching Graphs Homework & Practice 7.5

Review & Refresh

Does the table or equation represent a linear or nonlinear function? Explain.
Question 1.

Answer: y = -0.5x + 11.5 is a linear function.

Explanation:
Ordered pairs are (-5 , 14) , (-1 , 12) , (3 , 10) , (7 , 8)
Plot the points in the table , Draw a line through the points
First find the slope m of the line containing the two given points (3 , 10) and (7 , 8)
m = (y2-y1) / (x2-x1)
m= (8 – 10) / (7 – 3)
m = -2/4
m = -0.5
substitute the slope in the(3 , 10) to get point slope to form a line.
y-y1 = m (x-x1)
y – 10 = -0.5 ( x – 3)
y – 10 = -0.5x + 1.5
y = -0.5x + 1.5 + 10
y = -0.5x + 11.5
So , the linear equation is y = -0.5x + 11.5
And it is a linear function.

The graph is

Question 2.
y = x2 + 8
Answer: The function is linear  when m= 1 and c = 8.

Explanation:
Given , y = x2 + 8 ,
slope m = 1
c = 8
So, the function is linear when m = 1 and c= 8.

Graph the linear equation.
Question 3.
– 4x + y = – 1
Answer: The graph is

Explanation:
we can write – 4x + y = – 1 as y = 4x – 1
Given , y = 4x – 1  , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = 4(0) – 1 = -1 . co-ordinates are (0 , -1)
if x = 1 , then y = 4(1) – 1 = 3 . co-ordinates are (1 , 3)
if x = 2 , then y = 4(2) – 1 = 7 , co-ordinates are (2 , 7)
if x = 3 , then y = 4(3) – 1= 11 , co-ordinates are (3 , 11)
The co-ordinates (0 , -1) , (1 , 3) , (2 , 7) , (3 , 11) form a straight line .

Question 4.
2x – 3y = 12
Answer: The graph is

Explanation:
we can write  2x – 3y = 12 as y = $$\frac{2x-12}{3}$$ or y = $$\frac{2}{3}$$x – 4
Given , y =$$\frac{2}{3}$$x – 4 , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = $$\frac{2}{3}$$0 – 4= – 4 . co-ordinates are (0 , -4)
if x = 1 , then y = $$\frac{2}{3}$$1 – 4 = 0.66 – 4 = -3.3 . co-ordinates are (1 , -3.3)
if x = 2 , then y = $$\frac{2}{3}$$2 – 4 =0.66(2) – 4 =1.3 – 4 = -2.6, co-ordinates are (2 , -2.6)
if x = 3 , then y = $$\frac{2}{3}$$3 – 4 = 0.66(3) – 4 = 1.98 – 4 = -2.0  , co-ordinates are (3 , -2.0)
The co-ordinates (0 , -4) , (1 , -3.3) , (2 , -2.6) , (3 , -2) form a straight line .

Question 5.
5x + 10y = 30
Answer: The graph is

Explanation:
5x + 10y = 30 can be written as y = -0.5x + 3
take 5 common on both sides we get
x + 2y = 6
y = $$\frac{-x + 6}{2}$$
y = $$\frac{-x}{2}$$ + 6
y = -0.5x + 3
Given , y =-0.5x + 3 , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = -0.5(0) + 3 = 3 . co-ordinates are (0 , 3)
if x = 1 , then y = -0.5(1) + 3= 2.5 . co-ordinates are (1 , 2.5)
if x = 2 , then y = -0.5(2) + 3 = 4 , co-ordinates are (2 , 4)
if x = 3 , then y = -0.5(3) + 3 = 4.5  , co-ordinates are (3 , 4.5)
The co-ordinates (0 , 3) , (1 , 2.5) , (2 , 4) , (3 , 4.5)does not form a straight line .

Concepts, Skills, &Problem Solving

MATCHING DESCRIPTIONS WITH GRAPHS The graph shows your speed during a run. Match the verbal description with the part of the graph it describes. (See Exploration 1, p. 301.)

Question 6.
You run at a constant speed.
Answer: C

Explanation:
Because the line segment of the graph at point C show that the running speed is constant for a particular time ,
Thus forming a straight horizontal line.

Question 7.
You slow down at a constant rate.
Answer: D

Explanation:
Because the line segment of the graph at point D show that the running speed is decreasing at a constant rate for a particular time ,
Thus forming a straight steep line down the time axis.

Question 8.
You increase your speed at a constant rate.
Answer: A

Explanation:
Because the line segment of the graph at point A show that the running speed is increasing at a constant rate at a starting point of the race on time ,
Thus forming a slope in the graph.

Question 9.
You increase your speed at a faster and faster rate.
Answer: B

Explanation:
Because the line segment of the graph at point B show that the running speed is increasing at a faster rate after starting the race and maintaining a gradual growth of the speed and after reaching the next point speed is doubled from before ,
Thus forming a slope with a curve in the graph.

ANALYZING GRAPHS Describe the relationship between the two quantities.
Question 10.

Answer: As the Time passes there will be increase in the volume.

Explanation:
The graph shows the relation between the volume and time of a Balloon , To fill up the balloon with air, with the inlet of air increases in volume and the time taken to fill the balloon with air is shown ,
The line segment starts at initial point stating the balloon at a no air state, then gradually increases with constant halts having constant volume at a particular time and vice versa.

So, As the Time passes there will be increase in the volume.

Question 11.

Answer: As the times passes Dollars are maintaining imbalance.

Explanation:
The relationship between the time and dollars is given in the graph, As we all know money is never ever constant with time , As if it only increases or decreases or having both simultaneously , in this graph the line segment is having a steep and at some point of time it is maintaining a slight growth constantly with the time.

So, As the times passes Dollars are maintaining imbalance.

Question 12.

Answer: An engine power is directly proportional to the engine speed and its horse power

Explanation:
The relationship between the engine speed and horse power is given in the graph, Generally every automobile is is defined as the best for its horse power which is the heart of the engine and it highlights the speed of the vehicle, Here engine power is defined by the horse power and the engine speed the line segment is having a curve increment in the horse power due to the increase in engine speed.

So, An engine power is directly proportional to the engine speed and its horse power

Question 13.

Answer: As time increases the process of grams decaying will be faster.

Explanation:
The relationship between grams and time is given in the graph, its obvious that every product has its own expiry date, and if it crosses that its starts to decay, the graph implies that with the increase time the quality of the gram decreases or grams start to decay . The line segment in the graph shows that the gradually decrease indicating the spoiling rate of the grams  with rate of change of time.

So, As time increases the process of grams decaying will be faster.

Question 14.

Answer: At the particular intervals of time hair growth has stopped and at regular intervals again starts growing with respect to the time.

Explanation:
The graph shows the relationship between the length of the hair and time taken to the growth of the hair, of course hair growth is not constant every time, here we have the graph with the line segment  not constant and having breaks at the times of interval.

So, At the particular intervals of time hair growth has stopped and at regular intervals again starts growing with respect to the time.

Question 15.

Answer: In a period of time the balance will be reduced gradually at regular intervals maintaining a constant balance to clear the loan.

Explanation:
The relationship between the balance of the loan with the time period of the loan to be cleared, The loan should be cleared in the time limit and should maintain a neat balance, every increase in time period the balance is debited from the loan , there will be decrease in the balance and gaps are occurred in the graph.

so, In a period of time the balance will be reduced gradually at regular intervals maintaining a constant balance to clear the loan.

Question 16.
ANALYZING GRAPHS
Write an explanation for the relationship shown in the graph in Exercise 10.
Answer: The graph shows the relation between the volume and time of a Balloon , To fill up the balloon with air, with the inlet of air increases in volume and the time taken to fill the balloon with air is shown ,
The line segment starts at initial point stating the balloon at a no air state, then gradually increases with constant halts having constant volume at a particular time and vice versa.

Question 17.
MODELING REAL LIFE
The graph shows the natural gas usage for a house.

a. Describe the change in usage from January to March.
b. Describe the change in usage from March to May.
Answer: a. The change in usage from January to March, As shown in the graph the usage is at peaks in the month of January and started to decrease after that month and continuing to decrease in the month of March.
b.  The change in usage from March to May, As shown in the graph the usage is continuing to decrease in the month of March and started to increase again in the month of May and maintaining to use highly from the month of May.

Explanation:
a. The change in usage from January to March, As shown in the graph the usage is at peaks in the month of January and started to decrease after that month and continuing to decrease in the month of March.

b.  The change in usage from March to May, As shown in the graph the usage is continuing to decrease in the month of March and started to increase again in the month of May and maintaining to use highly from the month of May.

SKETCHING GRAPHS Sketch a graph that represents the situation.
Question 18.
The value of a television decreases at a constant rate, and then remains constant.
Answer: The graph is

Explanation:
Draw the axis and label the x- axis as time and y- axis as value, then sketch the graph,
The value of the television decreases at a constant rate: line segment starts to decrease at a constant rate,
And then remains constant, after reaching a certain value : line segment becomes parallel to horizontal axis.

Question 19.
The distance from the ground changes as your friend swings on a swing.
Answer: The graph is

Explanation:
Your friend starts close to the ground and then swings up. Then she falls back down close to the ground again and swings back . When she swings back, she gets higher than when she was swinging forward, she then starts to swing forward again getting close to the ground and then going up even higher than when she was swinging backward, she continues to getting higher and higher every time she swings forwards and backwards,

Question 20.
The value of a rare coin increases at a faster and faster rate.
Answer: The graph is

Explanation:
Draw the Axis and label them as x-axis as time and y – axis as distance,
The value of a rare coin increases at a faster and faster rate , so the curve moves upwards at an increasing rate.

Question 21.
You are typing at a constant rate. You pause to think about your next paragraph and then you resume typing at the same constant rate.
Answer: The graph is

Explanation:
A constant rate means that portion of the graph is linear , pausing means the number of words stays constant, typing again at the same constant rate means the last piece of the graph is linear again with the same slope as the first portion of the graph.

Question 22.
CRITICAL THINKING
The graph shows the speed of an object over time.

a. Sketch a graph that shows the distance traveled by the object over time.
b. Describe a possible situation represented by the graphs.
Answer: a. The distance and time are directly proportional to each other.
b. As time passes the speed and time are relatively balancing each other in the graph.

Explanation:
a. The graph is
In this graph the relationship between distance and time is shown, for example , let the object be a bike, the time taken to reach the destination for the bike is directly proportional to the distance travelled , So as time passes the distance is gradually increasing from the starting point.

So, the distance and time are directly proportional to each other.

b. Th graph shown , is the relationship between the speed and the time , let the object moving be Train,
it is running between the station so it has to be halted in the stations to be listed in the stoppings , So the line segment is started with a constant speed with the time and again at the time interval dropping the speed with respect to time it has maintaining the same speed .

So, As time passes the speed and time are relatively balancing each other in the graph.

Question 23.
MODELING REAL LIFE
The graph shows the average scores of two bowlers from the start of a season to the end of the season.

a. Describe each bowler’s performance.
b. Who had a greater average score most of the season? Who had a greater average score at the end of the season?
c. Write an explanation for the change in each bowler’s average score throughout the bowling season.
Answer: All the answers are explained below

Explanation:
a.  Bowler A : As the graph represent the relationship between the score and the week, bowler A has started with the good take off and having able to grasp the same energy his performance on the goals are increasing rapidly over the week and had the good score than Bowler B.

Bowler B : As the graph represent the relationship between the score and the week, bowler B has started with the good take off but as the time passes the performance of the  bowler is has been intended to decrease his scores gradually over the week.

b.  Bowler A and Bowler B had a greater average score most of the season, but Bowler A had a greater average score at the end of the season

c. Bowler A has the same energy his performance on the goals are increasing rapidly over the week and had the good score than Bowler B. so it has a smaller change in average’s score in the bowling season .
While Bowler B has good take off but as the time passes the performance of the  bowler is has been intended to decrease his scores gradually over the week. so he has a drastic change in average’s score in the bowling season .

Question 24.
DIG DEEPER!
You can use a supply and demand model to understand how the price of a product changes in a market. The supply curve of a particular product represents the quantity suppliers will produce at various prices. The demand curve for the product represents the quantity consumers are willing to buy at various prices.

a. Describe and interpret each curve.
b. Which part of the graph represents a surplus? Explain your reasoning.
c. The curves intersect at the equilibrium point, which is where the quantity produced equals the quantity demanded. Suppose that demand for a product suddenly increases, causing the entire demand curve to shift to the right. What happens to the equilibrium point?
Answer:  All of them are explained below .

Explanation:
a. The supply curve of a particular product represents the quantity suppliers will produce at various prices, As shown in the graph, the relationship between the price and quantity is given , so if prices increases gradually Quantity increases .
The demand curve for the product represents the quantity consumers are willing to buy at various prices, As shown in the graph, the relationship between the price and quantity is given , so if prices decreases with increase in Quantity .

b. The graph does not implies any surplus because each demand and supply is given by their respective curve over the prices and quantity

c. As shown in the graph, The curves intersect at the equilibrium point, which is where the quantity produced equals the quantity demanded. Given, that demand for a product suddenly increases, causing the entire demand curve to shift to the right. Then the equilibrium point will be pointed where the two curves meet after the change in the demand graph so change in the supply graph is also possible.

Functions Connecting Concepts

Using the Problem-Solving Plan
Question 1.
The table shows the lengths x (in inches) and weights y(in pounds) of several infants born at a hospital. Determine whether weight is a function of length. Then estimate the weight of an infant that is 20 inches long.

Understand the problem.
You know the lengths and weights of several infants. You are asked to determine whether weight is a function of length and to estimate the weight of a 20-inch-long infant.

Make a plan.
Determine whether any of the lengths are paired with more than one weight. Then use a graphing calculator to find an equation that represents the data. Evaluate the equation when x = 20 to estimate the weight of a 20-inch-long infant.

Solve and check.
Use the plan to solve the problem. Then check your solution.
Answer: Weight is the function of the length

Explanation:
From the table we have , Each length has only one weight , so weight is a function of length.
First find the slope m of the line containing the two given points (19.3 , 7.3) and (18.9 , 6.5)
m = (y2-y1) / (x2-x1)
m= (6.5 – 7.3) / (18.9 – 19.3)
m = 0.2
substitute the slope in the (19.3 , 7.3) to get point slope to form a line.
y-y1 = m (x-x1)
y – 7.3 = 0.2 ( x – 19.3)
y – 7.3 = 0.2x – 3.86
y = 0.2x – 3.86 + 7.3
y = 0.2x + 3.4
So ,  y = 0.2x + 3.4 is linear function.

For x = 20 ,
y = 0.2 (20) + 3.4
y = 4 + 3.4
y = 7.4

So, The weight of an infant that is 20 inches long. is 7.4.

Question 2.
Each mapping diagram represents a linear function. At what point do the graphs of the functions intersect? Justify your answer.

Answer:  The point of intersection is (-1, -4)

Explanation:
Function 1 – Ordered pairs are ( -8 , 24 ) , ( -3 , 4 ) , ( -1 , -4 ) , ( 1 , -12) .
Function 2 – Ordered pairs are ( 6 , 17 ) , ( 10 , 29 ) , ( 13 , 38 ) , ( 15 , 44 ) .
Graph the points we get, So, The point of intersection is (-1,-4).

Performance Task

Heat Index
At the beginning of this chapter, you watched a STEAM Video called “Apparent Temperature.” You are now ready to complete the performance task related to this video, available at BigIdeasMath.com. Be sure to use the problem-solving plan as you work through the performance task.

Answer:

Functions Chapter Review

Review Vocabulary

Write the definition and give an example of each vocabulary term.

Input: Ordered pairs can be used to show inputs and outputs , inputs are represented by x

Output: Ordered pairs can be used to show inputs and outputs , Outputs are represented by y

Relation: A relation pairs inputs with outputs

Mapping diagram: A relation can be represented by ordered pairs or mapping diagrams.

Function: The relation that pairs each input with exactly one output is a function.

Function rule: it is an equation, that describes the relationship between inputs(independent variables) and outputs(dependent variables).

Linear function: A linear function is a function whose graph is a straight line i.e., non vertical line . A linear can be written in the form y = mx + c , where m is the slope and c is the y intercept

Non linear function: The graph of a linear function shows a constant rate of change, A non linear function does not have a constant rate of change, So its graph is a not a line.

Graphic Organizers
You can use an Example and Non-Example Chart to list examples and non-examples of a concept. Here is an Example and Non-Example Chart for functions.

Choose and complete a graphic organizer to help you study the concept.

1. linear functions
2. nonlinear functions
3. linear functions with positive slope
4. linear functions with negative slope

Answer: 1. linear functions

Chapter Self-Assessment

As you complete the exercises, use the scale below to rate your understanding of the success criteria in your journal.

7.1 Relations and Functions (pp. 275–280)
Learning Target: Understand the concept of a function.

List the ordered pairs shown in the mapping diagram. Then determine whether the relation is a function.
Question 1.

Answer: The ordered pairs are ( 1 , -4 ) , ( 3 , 6 ) , ( 5 , 0 ) , ( 7 , 6 ) , ( 7 , 8 ) and The relation is not a function .

Explanation:
As shown , The ordered pairs are ( 1 , -4 ) , ( 3 , 6 ) , ( 5 , 0 ) , ( 7 , 6 ) , ( 7 , 8 ) .
The input 7 has more than one output,
So, The relation is not a function .

Question 2.

Answer: ordered pairs are ( 0 , 0 ) , ( 1 , 10 ) , ( 2 , 5 ) , ( 3 , 15 ) and The relation is a function .

Explanation:
As shown , The ordered pairs are ( 0 , 0 ) , ( 1 , 10 ) , ( 2 , 5 ) , ( 3 , 15 ).
Each input has exactly one output ,
So, The relation is a function .

Question 3.

Answer: The ordered pairs are ( -1 , 0 ) , ( -1 , 1 ) , ( 0 , 1 ) , ( 1 , 2 ), ( 3 ,3 ) and The relation is not a function

Explanation:
As shown , The ordered pairs are ( -1 , 0 ) , ( -1 , 1 ) , ( 0 , 1 ) , ( 1 , 2 ), ( 3 ,3 ) .
The input -1 has more than one output ,
So, The relation is not a function .

Question 4.
For ordered pairs that represent relations, which coordinate represents the input? the output?

Answer: x coordinate is the input and y coordinate is the output

Explanation:
Ordered pairs from the given graph are ( 2 , 7 ) , ( 3 , 7 ) , ( 4 , 5 ) , ( 5 , 5 ) , ( 6 , 3 ) .
So , x coordinate is the input and y coordinate is the output

Question 5.
Draw a mapping diagram that represents the relation shown in the graph. Then determine whether the relation is a function. Explain.
Answer:

Explanation:
The mapping diagram is
each input has more than one output
So, relation is not a function.

Question 6.
The mapping diagram represents the lengths (in centimeters) of a rubber band when different amounts of force (in Newtons) are applied.

a. Is the length of a rubber band a function of the force applied to the rubber band?
b. Describe the relationship between the length of a rubber band and the force applied to the rubber band.
Answer:  a. Yes
b. For every increase in 0.7 in input there is an increment of 2 in output.

Explanation:
a. The ordered pairs are  ( 0 , 5 ) , ( 0.7 , 7 ) , ( 1.4 , 9 ) , ( 2.1 , 11 )
Each input has exactly one output
So, the length of a rubber band a function of the force applied to the rubber band.

b. For every increase in 0.7 in input there is an increment of 2 in output.

7.2 Representations of Functions (pp. 281–288)
Learning Target: Represent functions in a variety of ways.

Write a function rule for the statement.
Question 7.
The output is two less than the input.
Answer: y = x – 2

Explanation:
Let us say x is input and y is output , then
The output is two less than the input, will be
y = x – 2

Question 8.
The output is two more than one-fourth of the input.
Answer: y = $$\frac{x}{4}$$ + 2

Explanation:
Let us say x is input and y is output , then
The output is two more than one-fourth of the input, will be
y = $$\frac{x}{4}$$ + 2

Find the value of y for the given value of x.
Question 9.
y = 2x – 3; x = – 4
Answer: y = -8

Explanation:
Given, y = 2x
substitute x = -4 , we get
y = 2(-4)
y = -8.

Question 10.
y = 2 – 9x ; x = $$\frac{2}{3}$$
Answer: y = – 3.4

Explanation:
Given , y = 2 – 9x
substitute x = $$\frac{2}{3}$$ , we get
y = 2 – 9 (0.6)
y = 2 – 5.4
y = – 3.4

Question 11.
y = $$\frac{x}{3}$$ + 5; x = 6
Answer: y = 7.

Explanation:
Given, y = $$\frac{x}{3}$$ + 5
substitute x = 6 , we get
y = $$\frac{6}{3}$$ + 5
y = 2 + 5
y = 7.

Graph the function.
Question 12.
y = x + 3
Answer: The graph is

Explanation:
Given , y = x + 3  , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = 0 + 3 = 3 . co-ordinates are (0 , 4)
if x = 1 , then y = 1 + 3  = 4 . co-ordinates are (1 , 5)
if x = 2 , then y = 2 + 3 = 5 , co-ordinates are (2 , 6)
if x = 3 , then y = 3 + 3 = 6 , co-ordinates are (3 , 7)
The co-ordinates (0 , 3) , (1 , 4) , (2 , 5) , (3 , 6) form a straight line .

Question 13.
y = – 5x
Answer: The graph is

Explanation:
Given , y = – 5x  , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y =- 5(0) = 0 . co-ordinates are (0 , 0)
if x = 1 , then y = – 5(1)  = – 5 . co-ordinates are (1 , – 5)
if x = 2 , then y = – 5(2) = -10 , co-ordinates are (2 , -10)
if x = 3 , then y =- 5(3) = -15 , co-ordinates are (3 , -15)
The co-ordinates (0 , 0) , (1 , -5) , (2 , -10) , (3 , -15) form a straight line .

Question 14.
y = 3 – 3x
Answer: The graph is

Explanation:
Given , y =3 – 3x , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = 3 – 3(0) = 3 . co-ordinates are (0 , 3)
if x = 1 , then y = 3 – 3(1)  = 0 . co-ordinates are (1 , 0)
if x = 2 , then y = 3 – 3(2) = – 3 , co-ordinates are (2 , – 3)
if x = 3 , then y =3 – 3(3) = – 6 , co-ordinates are (3 , – 6)
The co-ordinates (0 , 3) , (1 , 0) , (2 , – 3) , (3 , – 6) form a straight line .

Question 15.
An online music store sells songs for $0.90 each. a. Write a function that you can use to find the cost of buying songs. b. What is the cost of buying 5 songs? Answer: a. C = 0.90s b.$4.5

Explanation:
a. The total cost is equal to the cost of each song times the number of songs, if each song is $0.90, Then the total cost C of s songs is C = 0.90s. b. Substituting s= 5 in C = 0.90s we get, C = 0.90(5) = 4.5. So, cost of 5 songs is$4.5.

7.3 Linear Functions (pp. 289–294)
Learning Target: Use functions to model linear relationships.

Use the graph or table to write a linear function that relates y to x.
Question 16.

Answer: The linear function is y = $$\frac{1}{3}$$x + 3.

Explanation:
In order to write the function we have to write the ordered pairs of the graph ,
Ordered pairs are  (3 , 4) , (0 , 3 ) , (-3 , 2) , ( -6, 1 )
First find the slope m of the line containing the two given points (0 ,3) and (-3, 2)
m = (y2-y1) / (x2-x1)
m= (2 – 3) / (-3 – 0)
m = -1 / -3 .
m = 1/3 .
Because the line crosses the y axis at ( 0, 3 ) , The y intercept is 3.
So , the linear function is y = $$\frac{1}{3}$$x + 3.

Question 17.

Answer: The linear function is y = −(0)x -7.

Explanation:
In order to write the function we have to write the ordered pairs of the graph ,
Ordered pairs are  (-2 , -7) , (0 , -7 ) , (2 , -7) , ( 4 , -7 )
First find the slope m of the line containing the two given points (0 ,-7) and (2, -7)
m = (y2-y1) / (x2-x1)
m= (-7 – (-7)) / (2 – 0)
m = 0 .
Because the line crosses the y axis at ( 0, -7 ) , The y intercept is -7.
So , the linear function is y = −(0)x -7.

Question 18.
The table shows the age x (in weeks) of a puppy and its weight y (in pounds).

a. Write and graph a linear function that relates y to x.
b. Interpret the slope and the y-intercept.
c. After how many weeks will the puppy weigh 33 pounds?
Answer: a. y = $$\frac{3}{2}$$x + 3
b. 3 pounds
c. Age is 20 weeks

Explanation:
a. In order to write the function we have to write the ordered pairs of the graph ,
Ordered pairs are  (6 , 12) , (8 , 15 ) , (10 , 18) , ( 12 , 21 )
First find the slope m of the line containing the two given points ((6 ,12) and (8 , 15)
m = (y2-y1) / (x2-x1)
m= (15 – 12) / (8 – 6)
m = 3/2 .
substitute the slope in the (6 ,12) to get point slope to form a line.
y-y1 = m (x-x1)
y – 12 = 3/2 ( x – 6)
2(y – 12) = 3(x – 6)
2y – 24 = 3x – 18
2y = 3x – 18 + 24
2y  = 3x + 6
So ,  2y  = 3x + 6 or y = $$\frac{3}{2}$$x + 3 is linear function.

b. The slope measures the rate of change of weight due to change in weeks, Here the slope of 3/2 means that as one week passes, weight of the puppy increases by 3/2 pounds.
y intercept measures the weight of the puppy, when it was born which is 3 pounds in this case measured by c.

c. put y = 33,
33 = $$\frac3}{2}$$x + 3
30 = $$\frac{3}{2}$$x
30 × 2 = 3x
x = 60/3
x = 20.
So, Age is 20 weeks.

7.4 Comparing Linear and Nonlinear Functions (pp. 295–300)
Learning Target: Understand differences between linear and nonlinear functions.

Does the table represent a linear or nonlinear function? Explain.
Question 19.

Answer: y = 3x – 8 is linear function.

Explanation:
Ordered pairs are (3 , 1 ) , (6 , 10) , (9 , 19) , (12 , 28)
Plot the points in the table , Draw a line through the points
First find the slope m of the line containing the two given points (3 , 1 ) and (6 , 10)
m = (y2-y1) / (x2-x1)
m= (10 – 1) / (6– 3)
m = 9/3
m = 3
substitute the slope in the (3 , 1) to get point slope to form a line.
y-y1 = m (x-x1)
y – 1 = 3 ( x – 3)
y – 1 = 3x – 9
y = 3x – 9 + 1
y = 3x – 8
So ,  y = 3x – 8 is linear function.

Question 20.

Answer: y = -x + 4 is linear function.

Explanation:
Ordered pairs are (1 , 3 ) , (3 , 1) , (5 , 1) , (7 , 3)
Plot the points in the table , Draw a line through the points
First find the slope m of the line containing the two given points (1 , 3 ) and (3 , 1)
m = (y2-y1) / (x2-x1)
m= (1 – 3) / (3– 1)
m = -2/2
m = -1
substitute the slope in the (1 , 3) to get point slope to form a line.
y-y1 = m (x-x1)
y – 3 = -1 ( x – 1)
y – 3 = -x + 1
y = -x + 1 + 3
y = -x + 4
So ,  y = -x + 4 is linear function.

Question 21.
Does the graph represent a linear or nonlinear function? Explain.

Answer: The graph represent a non linear function.

Explanation:
As shown in the graph linear function represents a  straight line to which not happened here,
So , the graph is non linear function

Question 22.
Does the equation y = 2.3x represent a linear or nonlinear function? Explain.
Answer: y = 2.3x is a linear function.

Explanation:
Given , y = 2.3x  , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = 2.3(0) = 0 . co-ordinates are (0 , 0)
if x = 1 , then y = 2.3(1) = 2.3 . co-ordinates are (1 , 2.3)
if x = 2 , then y = 2.3(2) = 4.6 , co-ordinates are (2 , 4.6)
The co-ordinates (0 , 0) , (1 , 2.3) , (2 , 4.6) form a straight line .
Each x input has only one y output so it is a function .
And it  forms a straight line when graphed .
So, y = 2.3x is a linear function.

7.5 Analyzing and Sketching Graphs (pp. 301–306)
Learning Target: Use graphs of functions to describe relationships between quantities.

Question 23.
Describe the relationship between the two quantities in the graph.

Answer: At certain point of time, line segment is increased with respect to time and maintained a constant change and again dropped gradually with the increase in time.

Explanation:
The relationship between the graph is population and time ,
At certain point of time, line segment is increased with respect to time and maintained a constant change and again dropped gradually with the increase in time.
So, the city population is not constant at all the time.

Sketch a graph that represents the situation.
Question 24.
You climb a climbing wall. You climb halfway up the wall at a constant rate, then stop and take a break. You then climb to the top of the wall at a greater constant rate.
Answer: The graph is

Explanation:
You start climbing a wall at a constant rate so the first portion of the graph needs to be linear with a positive slope, you then take a break which means your height is constant so the second part of the graph needs to be a horizontal line, you then start climbing again at a constant rate, so the last part of the graph needs to be linear with a positive slope.

Question 25.
The price of a stock increases at a constant rate for several months before the stock market crashes. The price then quickly decreases at a constant rate.
Answer: The graph is

Explanation:
The stock price is increasing at a constant rate so the first part of the graph needs to be linear with positive slope, Then price begins to drop quickly so the second part of the graph needs to be linear with a steep negative slope.

Question 26.
The graph shows the sales of two companies during a particular year.

a. Describe the sales of each company.
b. Which company has greater total sales for the year?
c. Give a possible explanation for the change in each company’s sales throughout the year.
Answer: All The explanation is given below

a. Company A – The sales of the company is increasing at a constant rate so the first part of the graph needs to be linear with positive slope, and decreasing with a slight negative steep and again increasing at a constant rate increasing the sales of the company

Company B – The sales of the company is increasing at a constant rate so the first part of the graph needs to be increase in curve with a slight decrease in the graph leading to decrease in sales and vise versa.

b. Company A has the greater total sales for the year compared to Company B, with maintaining the sales up to the mark without losses.

c. The change in each company’s sales throughout the year,  Company A – The sales of the company is increasing at a constant rate so the first part of the graph needs to be linear with positive slope,
Company B – The sales of the company is increasing at a constant rate so the first part of the graph needs to be increase in curve with a slight decrease in the graph leading to decrease in sales

Functions Practice Test

Question 1.
List the ordered pairs shown in the mapping diagram. Then determine whether the relation is a function.

Answer: The relation is a function

Explanation:
As shown , Ordered pairs are the combinations of input and output
So , Ordered pairs are ( 2 , 9 ) , ( 4 , 9 ) , ( 6 , 10 ) , ( 8 , 11 ) .
Each input has exactly one output ,
So , The relation is a function .

Question 2.
Draw a mapping diagram that represents the relation. Then determine whether the relation is a function. Explain.

Answer: The mapping diagram is

Explanation:
Ordered pairs from the given graph are ( -3 , 5 ) , ( -1 , 1 ) , ( -1 , 3 ) , ( 1 , 2 ) , ( 3 , 4 ) .
Each input has exactly one output ,
So , The relation is a function .

Question 3.
Write a function rule for “The output is twice the input.”
Answer: y = 2x

Explanation:
Let us say x is input and y is output , then
The output is twice the input. will be
y = 2x

Question 4.
Graph the function y = 1 – 3x.
Answer: The graph is

Explanation:
Given , y = 1 – 3x  , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y =1 – 3(0) = 1 . co-ordinates are (0 , 1)
if x = 1 , then y = 1 – 3(1)  = -2 . co-ordinates are (1 , -2)
if x = 2 , then y = 1 – 3(2) = -5 , co-ordinates are (2 , -5)
if x = 3 , then y =1 – 3(3) = -8 , co-ordinates are (3 , -8)
The co-ordinates (0 , 1) , (1 , -2) , (2 , -5) , (3 , -8) form a straight line .

Question 5.
Use the graph to write a linear function that relates y to x.

Answer: The linear function is y = 0.5x – 1

Explanation:
In order to write the function we have to write the ordered pairs of the graph ,
Ordered pairs are  (-4 , -3) , (-2 , -2 ) , (0 , -1) , ( 2 , 0 )
First find the slope m of the line containing the two given points (0 , -1) and ( 2 , 0 )
m = (y2-y1) / (x2-x1)
m= (0 – (-1)) / (2 – 0)
m = 1 / 2 .
m = 0.5 .
Because the line crosses the y axis at ( 0, -1 ) , The y intercept is -1.
So , the linear function is y = 0.5x – 1 .

Question 6.
Does the table represent a linear or nonlinear function? Explain.

Answer: The linear function is y = −4x + 8

Explanation:
In order to write the function we have to write the ordered pairs of the graph ,
Ordered pairs are  (0 , 8) , (2 , 0 ) , (4 , -8) , ( 6 , -16 )
First find the slope m of the line containing the two given points (0 , 8) and (2 , 0 )
m = (y2-y1) / (x2-x1)
m= (0 – 8) / (2 – 0)
m = -4
Because the line crosses the y axis at ( 0, 8 ) , The y intercept is 8.
So , the linear function is y = −4x + 8.

Question 7.
The table shows the number of y meters a water-skier travels in x minutes.

a. Write a function that relates y to x.
b. Graph the linear function.
c. At this rate, how many kilometers will the water-skier travel in 12 minutes?
d. Another water-skier travels at the same rate but starts a minute after the first water-skier. Will this water-skier catch up to the first water-skier? Explain.
Answer: All the answers are given below

Explanation:
Ordered pairs are  (1 , 600) , (2 , 1200 ) , (3 , 1800) , ( 4 , 2400 ) , (5 , 3000)
First find the slope m of the line containing the two given points(1 , 600) and (2 , 1200 )
m = (y2-y1) / (x2-x1)
m= (1200 – 600) / (2 – 1)
m = 600
So, the line is of the form y = 600x + c
put x= 3 and y = 1800 in the above equation we get,
1800 = 600(3) + c
c = 1800 – 1800
c = 0.
So, The line is y = 600x.

b. The graph is

c. put x = 12 in y = 600x
y = 600(12)
y = 7200
7200 meters, i.e., 7.2km

d. Another water skier travels at the same rate but starts a minute after the first water skier, Since both are travelling at the same rate , the water skier who was late will always be behind the first water skier.

Question 8.
The graph shows the prices of two stocks during one day.

a. Describe the changes in the price of each stock.
b. Which stock has a greater price at the end of the day?
c. Give a possible explanation for the change in the price of Stock B throughout the day.
Answer: Detailed Explanation is given below.

Explanation:
a. The changes in the price of each stock is Stock A has the constant increase in stock for a particular time and maintains a constant price forming a straight line in the graph, and again decreasing with a negative slope and vise versa, while Stock B is having steep negative slope that is decreasing in the prices with the time and having a constant horizontal line. and having a positive increase in the slope and same repeats again.

b. stock B has a greater price at the end of the day, having a positive increase in the slope

c. The change in the price of Stock B throughout the day, is having steep negative slope that is decreasing in the prices with the time and having a constant horizontal line. and having a positive increase in the slope and same repeats again, compared to stock A .

Question 9.
You are competing in a footrace. You begin the race by increasing your speed at a constant rate. You then run at a constant speed until you get a cramp and have to stop. You wait until your cramp goes away before you start increasing your speed again at a constant rate. Sketch a graph that represents the situation.
Answer: The graph is

Explanation:
You begin the race by increasing your speed at a constant rate so the first portion of the graph needs to be linear with a positive slope , you then run at a constant speed so the next portion of the graph needs to be horizontal line , you then stop and take a break , so your speed is zero, which means the next portion of the line needs to be
horizontal line on the x axis , you then increase your speed again at a constant rate sop that the last portion of the graph needs to be linear with a positive slope

Functions Cumulative Practice

Question 1.
What is the slope of the line?

Answer: Not in the options but the answer is m = -4/3

Explanation:
Ordered pairs are  (-4 , 5) , (1 , -3 ),
First find the slope m of the line containing the two given points
m = (y2-y1) / (x2-x1)
m= (-3 – 5) / (2 – (-4))
m = -8/6
m = -4/3.

Question 2.
Which value of a makes the equation 24 = $$\frac{a}{3}$$ – 9 true?
F. 5
G. 11
H. 45
I. 99
Answer: I. 99

Explanation:
Substitute a = 99 , in the given equation we get,
24 = $$\frac{a}{3}$$ – 9
24 = $$\frac{99}{3}$$ – 9
24 = 33 – 9
24 = 24.
So, last option is the correct answer.

Question 3.
A mapping diagram is shown.

What number belongs in the box so that the equation describes the function represented by the mapping diagram?

Answer: m = 7 , y = 7x + 5

Explanation:
Ordered pairs are  (4 , 33) , (7 , 54 ), (10 , 75) , (13 , 96 ),
First find the slope m of the line containing the two given points (4 , 33) and (7 , 54 )
m = (y2-y1) / (x2-x1)
m= (54 – 33) / (7 – 4)
m = 21/3
m = 7.
So, y = 7x + 5

Question 4.
What is the solution of the system of linear equations?
3x + 2y = 5
x = y + 5
A. (3, – 2)
B. (- 2, 3)
C. (- 1, 4)
D. (1, – 4)
Answer: A. (3, – 2)

Explanation:
Given 3x + 2y = 5
Then substitute , x = y + 5 in the above equation
3( y + 5) + 2y = 5
3y + 15 + 2y = 5
5y + 15 = 5
5( y + 3) = 5
y + 3 = 1
y = 1 – 3
y = -2,
substitute y = -2 in x = y + 5 then
x = 3
So, (3 , -2)

Question 5.
The director of a research lab wants to present data to donors. The data show how the lab uses a large amount of donated money for research and only a small amount of money for other expenses. Which type of display best represents these data?
F. box-and-whisker plot
G. circle graph
H. line graph
I. scatter plot
Answer: I. scatter plot

Explanation:
Scatter plot is the best graph for this type of data where vertical axis will show the amount of money and Horizontal axis will show research and other expenses.

Question 6.
Which graph shows a nonlinear function?

Answer: option B

Explanation:
As all the other options are representing the linear function that is forming a straight line  expect for option B , it is representing a non linear equation.

Question 7.
Which equation of a line passes through the point (—2, 3) and has a slope of $$\frac{3}{4}$$?

Answer:  F. y – 3 = $$\frac{3}{4}$$(x + 2)

Explanation:
Given, y – 3 = $$\frac{3}{4}$$(x + 2)
it is in the form of y = mx + c
so, slope m = $$\frac{3}{4}$$
Substitute the given points in this equation that is x = -2 and y = 3
3 – 3 = $$\frac{3}{4}$$(-2 + 2)
0 = 0.
So, F is the correct option.

Question 8.
The tables show the sales (in millions of dollars) for two companies over a five-year period.

Part A Does the first table show a linear function? Explain your reasoning.
Part B Does the second table show a linear function? Explain your reasoning.
Answer: Part A, the first table shows a linear function,
Part B the second table shows a linear function.

Explanation:
Part A – ordered pairs are (1 , 2) , (2 , 4) , (3 , 6) , (4 , 8) , (5 , 10)
Each input has exactly one output and forms a straight line when graphed
So, it is a linear function.

Part B – ordered pairs are (1 , 1) , (2 , 1) , (3 , 2) , (4 , 3) , (5 , 5)
Each input has exactly one output and does not form a straight line when graphed
So, it is a linear function.

Question 9.
The equations y = – x + 4 and y = $$\frac{1}{2}$$x – 8 form a system of linear equations. The table shows the values of y for given values of x.

What can you conclude from the table?
A. The system has one solution, when x = 0.
B. The system has one solution, when x = 4.
C. The system has one solution, when x = 8.
D. The system has no solution.
Answer: C. The system has one solution, when x = 8.

Explanation:
Given , y = – x + 4 and y = $$\frac{1}{2}$$x – 8
for x = 8 we have
y = -8 + 4 = -4
y = 0.5(8) – 8 = 4 – 8 = -4
Both the equations have one solution for x = 8
So, The system has one solution, when x = 8.

Question 10.
The vertices of a triangle are A (- 1, 3), B (1, 2), and C (- 1, – 1). Dilate the triangle using a scale factor of 2. What is the y-coordinate of the image of B?

Answer: The New right angle triangle is larger than the original one So , its a increase .

Explanation:
Given , (- 1, 3),  ( 1, 2 ),  (- 1, -1) these pairs form a right angle triangle
K = 2 , For the dilation figure multiply the 3 with the given ordered pairs , then
(- 1, 3) × 2 = ( -2 , 6)
( 1, 2) × 2 = ( 2 , 4)
(- 1, -1) × 2 = (-2 , -2)
From these new ordered pairs we form a new  right angle triangle

The New right angle triangle is larger than the original one So, it’s an increase.

Conclusion

Enhance your problem-solving skills and overall math proficiency using the Big Ideas Math Grade 8 Chapter 7 Functions. Give your best in exams by solving the Problems Over here aligned as per the Latest BIM Textbooks. In case of any suggestions do leave us your queries via the comment box so that we can get back to you.

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Big Ideas Math Answers Grade 1 Chapter 10 Measure and Compare Lengths

Students can find the exact solution for every question of Big Ideas Math Answers Grade 1 Chapter 10 Measure and Compare Lengths. So, the students who are searching for the Grade 1 Solution Key can get them on the Big Ideas Math Answer page. In order to excel in the exams, you have to choose the best material. Thus Download Big Ideas Math Answers Grade 1 Chapter 10 Measure and Compare Lengths from here and start practicing the problems.

Big Ideas Math 1st Grade Answer Key Chapter 10 Measure and Compare Lengths

Here we can discuss different topics on Measuring and Comparing, Order Objects by Length, Compare Lengths Indirectly, Measure Lengths, Measure More Lengths, Solve Compare Problems Involving Length, and so on. Compare the topic you learned here with the real time and understand the topic deeply. Those topics were being set up by the mathematical experts as indicated by the latest edition. Scroll down this page to get the answers to all the inquiries. Click on the links to look at the subjects shrouded in this chapter Measuring and comparing lengths.

Lesson 1 Order Objects by Length

Lesson 2 Compare Lengths Indirectly

Lesson 3 Measure Lengths

Lesson 4 Measure More Lengths

Lesson 5 Solve Compare Problems Involving Length

Performance Task

Measure and Compare Lengths Vocabulary

Organize It

Review Words:
longer
shorter

Use the review words to complete the graphic organizer.

Answer:
The first image longer and the second image is shorter.

Explanation:

Define It

Use your vocabulary cards to identify the words. Find each word in the word search.

Answer:

Lesson 10.1 Order Objects by Length

Explore and Grow

Draw an object that is shorter than the pencil and longer than the crayon.

Answer:

Chalk is smaller than crayon.

Show and Grow

Question 1.
Order from longest to shortest.

_____________, _____________, _____________
Answer:
Purple, Pink, Blue.

Explanation:
In the above image, the purple color tube is the longest after that the pink color tube is the longest, and the blue color tube is the shortest.

Question 2.
Order from shortest to longest.

_____________, _____________, _____________
Answer:
Yellow, Green, Black.

Explanation:
In the above image, the yellow brush is the longest, and then the green brush is the longest. The black brush is the shortest.

Apply and Grow: Practice

Question 3.
Order from longest to shortest.

_____________, _____________, _____________
Answer:
Purple crayon, Red crayon, Green crayon.

Explanation:
In the above image, the Purple crayon is the longest, and then the red crayon is the longest. The green crayon is the shortest.

Question 4.
Order from shortest to longest.

_____________, _____________, _____________
Answer:
The order from shortest to longest is, Pink is the shortest, green is the largest, and blue is the longest.

Explanation:
In the above image, we can see that the order from shortest to longest is, Pink is the shortest, green is the largest, and blue is the longest.

Question 5.
YOU BE THE TEACHER
Your friend ordered from shortest to longest. Is your friend correct? Explain.

Yellow   ,          green  ,           red
Answer:
No, my friend is not correct.

Explanation:
No, my friend is not correct. He represented the order from longest to shortest. In the above image, the red chill is the shortest, and the green chili is the longest. The black brush is the shortest.

Think and Grow: Modeling Real Life

Your yarn is longer than Newton’s. Descartes’s is longer than Newton’s and shorter than yours. Who has the longest yarn?

Draw a picture:

Who has the longest yarn?
You             Newton      Descartes
Answer:
My yarn is the longest yarn.

Explanation:

As my yarn is longer than Newton’s and Descartes’s is longer than Newton’s and shorter than mine. So the longest yarn is mine, and the largest yarn is Descartes. The shortest yarn is Newton’s.

Show and Grow

Question 6.
Descartes’s pencil is shorter than Newton’s. Yours is shorter than Newton’s and longer than Descartes’s. Who has the shortest pencil?
Draw a picture:

Who has the shortest pencil?
Descartes        Newton           You
Answer: Your pencil is shorted among the three.

Order Objects by Length Practice 10.1

Order from longest to shortest.

Question 1.

_____________, _____________, _____________
Answer:
The order of the bats from longest to the shortest is
Bat 2, Bat 1, Bat 3.

Explanation:
In the above image, we can see the longest bat is bat 2 and then the largest bat is bat 1. The shortest bat is bat 3.

Question 2.

_____________, _____________, _____________
Answer:
The order of the colors from longest to shortest is
Gold, Blue, Red.

Explanation:
In the above image, we can see that the longest color is gold and then the largest color is blue. The shortest color is red.

Question 3.
Order from shortest to longest.

_____________, _____________, _____________
Answer:
The order from shortest to longest is
Vine 3, Vine 1, Vine 2.

Explanation:
In the above image, we can see that the shortest vine is vine 3 after that the largest vine is vine 1 and the longest vine is Vine 2.

Question 4.
DIG DEEPER!
Use the clues to match. The red pencil is longer than the yellow pencil. The shortest pencil is blue.

Answer:
The order from shortest to longest is
Blue pencil, Yellow pencil, Red pencil.

Explanation:

As by the given clue, the Blue pencil is the shortest pencil, the Yellow pencil is the largest pencil, and the Red pencil is the longest pencil.

Question 5.
Modeling Real Life
Your jump rope is longer than Newton’s. Descartes’s is longer than Newton’s and shorter than yours. Who has the longest jump rope?

Who has the longest jump rope?
You           Newton        Descartes
Answer:
My rope is the longest jump rope.

Explanation:
As my jump rope is longer than Newton’s and Descartes’s jump rope longer than Newton’s and Newton’s jump rope is shorter than yours. So my jump rope is Longer, Newton’s jump rope is the Largest, and the shortest rope is Descartes.

Review & Refresh

Compare.

Question 6.
25 ○ 52
Answer:
25 > 52.

Explanation:
Given that to compare 25 and 52. So we can see that 52 is greater than 25.

Question 7.
41 ○ 44
Answer:
41 > 44.

Explanation:
Given that to compare 41 and 44. So we can see that 44 is greater than 41.

Lesson 10.2 Compare Lengths Indirectly

Explore and Grow

Use string to compare the keys. Which key is longer?
How do you know?

Answer:
Key 1 is longer than Key 2.

Explanation:
In the above image, we can see that key 1 is longer than key 2.

Show and Grow

Question 1.
Circle the longer object.

Answer:
The Pen is longer than Eraser.

Explanation:

In the above image, we can see that the Pen is longer than the Easer.

Question 2.
Draw a line through the shorter object.

Answer:

Explanation:

Apply and Grow: Practice

Question 3.
Draw a line through the shorter object.

Answer:
Object 2 is shorter than object 1.

Explanation:

In the above image, we can see that object two is shorter than object 1.

Question 4.
Circle the longer object.

Answer:
The spoon is longer than the brush.

Explanation:

We can see in the above image, that the spoon is longer than the brush. So we will round off the spoon.

Question 5.
DIG DEEPER
Which object is longer? Explain.

Answer:
The book’s shelf is longer than the key.

Explanation:

As we can see in the above image, that the books shelf is longer than the key. So we will circle the books shelf.

Think and Grow: Modeling Real Life

A green crayon is shorter than a blue crayon. The blue crayon is shorter than a yellow crayon. Is the green crayon longer than or shorter than the yellow crayon?

Draw a picture:

Longer              Shorter
Answer:
The green crayon is shorter than the yellow crayon.

Explanation:

Given that green crayon is shorter than blue crayon, and blue crayon is shorter than yellow crayon. So green crayon is shorter than yellow crayon.

Show and Grow

Question 6.
A yellow ribbon is longer than a pink ribbon. The pink ribbon is longer than a blue ribbon. Is the yellow ribbon longer than or shorter than the blue ribbon?

Draw a picture:

Longer            Shorter
Answer:
The Yellow ribbon is longer than the blue ribbon.

Explanation:

Given that yellow ribbon is longer than pink ribbon, and the pink ribbon is longer than a blue ribbon. So yellow ribbon is longer than the blue ribbon.

Compare Lengths Indirectly Practice 10.2

Question 1.
Circle the longer object.

Answer:
Object 1 is longer than object 2.

Explanation:
In the above image, we can see that object one is longer than image two. So we will circle object one.

Question 2.
Draw a line through the shorter object.

Answer:
Object two is shorter than object one.

Explanation:

In the above image, we can see that object two is shorter than object one. So we will circle object two.

Question 3.
DIG DEEPER
Use the clues to match.
The blue string is longer than the orange string.
The purple string is shorter than the orange string.

Answer:
The order of the strings from longest to shortest is
Blue string, Orange string, Purple string.

Explanation:

Given that the blue string is longer than the orange string, and the purple string is shorter than the orange string.
So the order of the strings is the blue string is longer, the orange string is the largest and the purple string is the shortest.

Question 4.
Modeling Real Life
A kayak is shorter than a canoe. The canoe is shorter than a paddle board. Is the kayak longer than or shorter than the paddle board?

Longer             Shorter
Answer:
The Kayak is shorter than the paddle board.

Explanation:
Given that the Kayak is shorter than a canoe, and the canoe is shorter than a paddle board. So the Kayak is shorter than the paddle board.

Review & Refresh

Question 5.
Circle the objects that have capacity as an attribute.

Answer:
The objects which have the capacity to store are circled in the below image.

Explanation:

The circled items have the capacity to store. As we can see that the lid jar is used to store any items, and chalk box is used to store chalk pieces, and glass is used to store water or any liquids, and a glue bottle is used to store glue. So these items have the capacity to store. And the remaining things do not have that much capacity to store.

Lesson 10.3 Measure Lengths

Explore and Grow

Find and measure the objects shown in your classroom.

Answer:
The length of the table is about two colored red tiles.

Explanation:
The table is measured by using the colored red tile. As each tile is equal to one unit. So the length of the table is two colored red tiles, which means two units. We must take each tile without gaps or any overlaps between them.

Answer:
The length of the pencil is about one tile.

Explanation:
The pencil is measured by using the colored red tile. As each tile is equal to one unit. So the length of the pencil is one colored red tile, which means one unit. We must take each tile without gaps or any overlaps between them.

Show and Grow

Measure

Question 1.

about ___________ color tile
Answer:
The length of the object is about one tile.

Explanation:
The object is measured by using the colored tile. As each tile is equal to one unit. So the length of the object is about one colored tile, which means one unit. We must take each tile without gaps or any overlaps between them.

Question 2.

about ___________ color tile

Answer:
The length of the object is about five colored tiles.

Explanation:
The object is measured by using the colored tile. As each tile is equal to one unit. So the length of the object is about five colored tiles, which means five units. We must take each tile without gaps or any overlaps between them.

Apply and Grow: Practice

Measure

Question 3.

about ___________ color tile
Answer:
The length of the object is about four colored tiles.

Explanation:
The object is measured by using the colored tile. As each tile is equal to one unit. So the length of the object is about four colored tiles, which means four units. We must take each tile without gaps or any overlaps between them.

Question 4.

about ___________ color tile
Answer:
The length of the object is about three colored tiles.

Explanation:
The object is measured by using the colored tile. As each tile is equal to one unit. So the length of the object is about three colored tiles, which means three units. We must take each tile without gaps or any overlaps between them.

Question 5.
MP Precision
Which picture shows the correct way to measure the straw?

Answer:
The red color tiles are the correct way to measure the straw.

Explanation:
In the above image, we can see that the correct way to measure the straw is the red color tiles.

Think and Grow: Modeling Real Life

Will the scissors fit inside a pencil case that is 7 color tiles long?

Circle:      Yes     No
Tell how you know:
Answer:
Yes, the scissors can fit inside a pencil case that is 7 color tiles long.

Explanation:
Yes, the scissors can fit inside a pencil case. As the length of the scissors is four colored tiles. As the scissors are measured by using the colored tile. And each tile is equal to one unit. So the length of the scissors is about four colored tiles, which means four units. And the pencil case is 7 colored tiles long. So the scissors can fit. We must take each tile without gaps or any overlaps between them.

Show and Grow

Question 6.
Will the cell phone fit inside a case that is 5 color tiles long?

Circle:     Yes     No
Tell how you know:
Answer:
No, the cell phone cannot fit inside a case of 5 color tiles long.

Explanation:
No, the cell phone cannot fit inside a case of 5 color tiles long. As the length of the cell phone is six colored tiles. And the cell phone is measured by using the colored tile. And each tile is equal to one unit. So the length of the cell phone is about six colored tiles, which means six units. And cannot fit inside 5 colored tiles long. We must take each tile without gaps or any overlaps between them.

Measure Lengths Practice 10.3

Measure.

Question 1.

about ___________ color tile
Answer:
The length of the object is about four colored tiles.

Explanation:
The object is measured by using the colored tile. As each tile is equal to one unit. So the length of the object is about four colored tiles, which means four units. We must take each tile without gaps or any overlaps between them.

Question 2.

about ___________ color tile
Answer:
The length of the object is about two colored tiles.

Explanation:
The object is measured by using the colored tile. As each tile is equal to one unit. So the length of the object is about two colored tiles, which means two units. We must take each tile without gaps or any overlaps between them.

Question 3.

about ___________ color tile
Answer:
The length of the glue stick is about three colored tiles.

Explanation:
The glue stick is measured by using the colored tile. As each tile is equal to one unit. So the length of the glue stick is about three colored tiles, which means three units. We must take each tile without gaps or any overlaps between them.

Question 4.
MP Reasoning
The green yarn is about 3 color tiles long. How long is the blue yarn?

about ___________ color tile
Answer:
The length of the blue yarn is about six color tiles.

Explanation:
Given that the length of the green yarn is three color tiles long, so we should find the length of the blue yarn. To measure the length of the blue yarn, we will take the colored tiles without gaps or any overlaps between them. As each tile is equal to one unit. So the length of the blue yarn is about six colored tiles, which means six units.

Question 5.
Modeling Real Life
Will the gift card fit inside an envelope that is 8 color tiles long?

Circle:      Yes      No
Tell how you know:
Answer:
Yes, the gift can fit inside an envelope that is 8 color tiles long.

Explanation:
Yes, the gif card can fit inside an envelope that is 8 color tiles long. As the length of the gift card is four colored tiles. And the gift card is measured by using the colored tile. And each tile is equal to one unit. So the length of the gift is about four colored tiles, which means four units. And the envelope is 8 colored tiles long. So the gift card can fit. We must take each tile without gaps or any overlaps between them.

Review & Refresh

Question 6.
Complete the fact family.
7 + 3 = _________                                     __________ – 3 = 7
_________ + _________ = _________             _________ – 7 = _________
Answer:
7 + 3 is 10,
3 + 7 is 10
10 – 3 is 7,
10 –  7 is 3.

Explanation:
As a fact family represents a group of math facts, equations, which are created by using the same set of digits. This fact family defines the relation between the three numbers which are involved. In addition or subtraction in a fact family, there will be four addition and subtractions will be created using these three numbers. So
7 + 3 is 10,
3 + 7 is 10
10 – 3 is 7,
10 –  7 is 3.

Lesson 10.4 Measure More Lengths

Explore and Grow

Find and measure the objects shown in your classroom two ways. What do you notice?

Answer:
The length of the table is about two colored red tiles and the length of the table which is measured using a paper clip is one paper clip.

Explanation:
The table is measured by using the colored red tile. As each tile is equal to one unit. So the length of the table is two colored red tiles, which means two units. We must take each tile without gaps or any overlaps between them. And to measure the table with the paper clip, we will place the paper clip without gaps or overlaps between them. And each paper clip is equal to about one unit. So the length of the table using the paper clip is one paper clip, which means one unit. We have noticed that if we take a colored tile, then the length of the table will be two color tiles. And if we take a paper clip then the length of the table will be one paper clip. So the length is doubled while measuring with color tile than the paper clip.

Answer:
The length of the pencil is about two colored red tiles and the length of the pencil which is measured using a paper clip is one paper clip.

Explanation:
The pencil is measured by using the colored red tile. As each tile is equal to one unit. So the length of the table is two colored red tiles, which means two units. We must take each tile without gaps or any overlaps between them. And to measure the table with the paper clip, we will place the paper clip without gaps or overlaps between them. And each paper clip is equal to about one unit. So the length of the pencil using the paper clip is one paper clip, which means one unit. We have noticed that if we take a colored tile, then the length of the table will be two color tiles. And if we take a paper clip then the length of the pencil will be one paper clip. So the length is doubled while measuring with color tile than the paper clip.

Show and Grow

Measure

Question 1.

about __________ color tiles about __________ paper clips
Answer:
The length of the brush is about four colored tiles and the length of the brush which is measured using a paper clip is two paper clips.

Explanation:
The brush is measured by using the colored tile. As each tile is equal to one unit. So the length of the table is two colored tiles, which means two units. We must take each tile without gaps or any overlaps between them. And to measure the brush with the paper clip, we will place the paper clip without gaps or overlaps between them. And each paper clip is equal to about one unit. So the length of the brush using the paper clip is one paper clip, which means one unit. We have noticed that if we take a colored tile, then the length of the table will be two color tiles. And if we take a paper clip then the length of the brush will be one paper clip. So the length is doubled while measuring with color tile than the paper clip.

Question 2.

about __________ color tiles about __________ paper clips
Answer:
The length of the skating board is about two colored tiles and the length of the pencil which is measured using a paper clip is one paper clip.

Explanation:
The skating board is measured by using the colored tile. As each tile is equal to one unit. So the length of the table is two colored tiles, which means two units. We must take each tile without gaps or any overlaps between them. And to measure the skating board with the paper clip, we will place the paper clip without gaps or overlaps between them. And each paper clip is equal to about one unit. So the length of the skating board using the paper clip is one paper clip, which means one unit. We have noticed that if we take a colored tile, then the length of the skating board will be two color tiles. And if we take a paper clip then the length of the pencil will be one paper clip. So the length is doubled while measuring with color tile than the paper clip.

Apply and Grow: Practice

Measure.

Question 3.

about __________ color tiles about __________ paper clips
Answer:
The length of the object is about six colored tiles and the length of the pencil which is measured using a paper clip is three paper clips.

Explanation:
The object is measured by using the colored tile. As each tile is equal to one unit. So the length of the object is six colored tiles, which means six units. We must take each tile without gaps or any overlaps between them. And to measure the table with the paper clip, we will place the paper clip without gaps or overlaps between them. And each paper clip is equal to about one unit. So the length of the object using the paper clip is three paper clips, which means three units. We have noticed that if we take a colored tile, then the length of the object will be six color tiles. And if we take a paper clip then the length of the pencil will be three paper clips. So the length is doubled while measuring with color tile than the paper clip.

Question 4.

about __________ color tiles about __________ paper clips
Answer:

The length of the sunscreen lotion bottle is about four colored tiles and the length of the sunscreen lotion bottle which is measured using a paper clip is two paper clips.

Explanation:
The sunscreen lotion bottle is measured by using the colored tile. As each tile is equal to one unit. So the length of the sunscreen lotion bottle is four colored tiles, which means four units. We must take each tile without gaps or any overlaps between them. And to measure the sunscreen lotion bottle with the paper clip, we will place the paper clip without gaps or overlaps between them. And each paper clip is equal to about one unit. So the length of the sunscreen lotion bottle using the paper clip is two paper clips, which means two units. We have noticed that if we take a colored tile, then the length of the sunscreen lotion bottle will be four color tiles. And if we take a paper clip then the length of the sunscreen lotion bottle will be two paper clips. So the length is doubled while measuring with color tile than the paper clip.

Question 5.
YOU BE THE TEACHER
Your friend says the pencil is more paper clips long than color tiles. Is your friend correct? Explain.

Answer:
No, my friend is not correct.

Explanation:
No, my friend is not correct. As we can see from the above problems that while measuring with the paper clip we need fewer paper clips than the color tiles. And the length is doubled while measuring with color tile than the paper clip. So my friend is not correct.

Think and Grow: Modeling Real Life

Your guitar is 33 color tiles long. Is your guitar more than or less than 33 paper clips long?

Circle:      more than 33         less than 33
Tell how you know:
Answer:
My guitar will be less than 33 paper clips.

Explanation:
My guitar will be less than 33 paper clips. As we can see from the above problems that while measuring with the paper clip we need fewer paper clips than the color tiles. And the length is doubled while measuring with color tile than the paper clip. So there will be fewer paper clips than the color tiles.

Show and Grow

Question 6.
Your mailbox is 11 paper clips long. Is your mailbox more than or less than 11 color tiles long?

Circle:       more than 11         less than 11
Tell how you know:
Answer:
My mailbox will have more than 11 color tiles long.

Explanation:
My mailbox will have more than 11 color tiles. As we can see from the above problems that while measuring with the paper clip we need fewer paper clips than the color tiles. And the length is doubled while measuring with color tile than the paper clip. So there will be more color tiles than the paper clips.

Measure More Lengths Practice 10.4

Measure

Question 1.

about __________ color tiles about __________ paper clips
Answer:
The length of the rocket is about six colored tiles and the length of the pencil which is measured using a paper clip is three paper clips.

Explanation:
The rocket is measured by using the colored tile. As each tile is equal to one unit. So the length of the rocket is six colored tiles, which means six units. We must take each tile without gaps or any overlaps between them. And to measure the rocket with the paper clip, we will place the paper clip without gaps or overlaps between them. And each paper clip is equal to about one unit. So the length of the rocket using the paper clip is three paper clips, which means three units. We have noticed that if we take a colored tile, then the length of the rocket will be six color tiles. And if we take a paper clip then the length of the pencil will be three paper clips. So the length is doubled while measuring with color tile than the paper clip.

Question 2.
YOU BE THE TEACHER
Your friend says the marker is more color tiles long than paper clips. Is your friend correct? Explain.

Answer:
Yes, the marker has more color tiles long than paper clips. So my friend is correct.

Explanation:
Yes, the marker has more color tiles long than paper clips. As we can see from the above problems that while measuring with the paper clip we need fewer paper clips than the color tiles. And the length is doubled while measuring with color tile than the paper clip. So there will be more color tiles than the paper clips for the marker. So my friend is correct.

Question 3.
Modeling Real Life
Your folder is 15 color tiles long. Is your folder more than or less than 15 paper clips long?

Circle:       more than 15         less than 15
Tell how you know:

Answer:
My folder will be less than 15 paper clips.

Explanation:
My folder will be less than 15 paper clips. As we can see from the above problems that while measuring with the paper clip we need fewer paper clips than the color tiles. And the length is doubled while measuring with color tile than the paper clip. So there will be fewer than 15 paper clips than the color tiles for my folder.

Review & Refresh

Question 4.
8 tigers swim.
5 tigers leave.
How many tigers are left?
___________ – ___________ = ___________ tigers
Answer:
the number of tigers left is
8 – 5= 3 tigers.

Explanation:
The number of tigers swim is 8 and the number of tigers which leave is 5 tigers. So the number of tigers left is
8 – 5= 3 tigers.

Question 5.
You have 6 pencils.
You lose 2 pencils.
How many pencils are left?
___________ – ___________ = ___________ pencils
Answer:
the remaining pencils are
6 – 2= 4 pencils.

Explanation:
The number of pencils I have is 6 pencils, and the number of pencils lost is 2 pencils. So the remaining pencils are
6 – 2= 4 pencils.

Lesson 10.5 Solve Compare Problems Involving Length

Explore and Grow

Draw a line that is 2 color tiles longer than the pencil.

Draw a line that is 2 color tiles shorter than the pencil.

Answer:

Show and Grow

Question 1.
Your lunch box is 6 paper clips long. Your friend’s is 3 paper clips long. How many paper clips longer is your lunch box?

__________ paper clips
Answer:
My lunch box is three paper clips longer than my friend’s and my friend’s lunch box is three times shorter than my friend’s lunch box.

Explanation:

Given that my lunch box is six paper clips long and my friend’s lunch box is three paper clips long. So there will be three paper clips longer than my friend’s. And my friend’s lunch box is three paper clips shorter than my lunch box.

Apply and Grow: Practice

Question 2.
Your scarf is 10 paper clips long. Your friend’s is 7 paper clips long. How many paper clips longer is your scarf?

Answer:
Myscarf is three paper clips longer than my friend’s and my friend’s scarf is three times shorter than my friend’s scarf.

Eplanation:

Given that my scarf is ten paper clips long and my friend’s scarf is seven paper clips long. So there will be three paper clips longer than my friend’s. And my friend’s scarf is three paper clips shorter than my scarf.

Question 3.
Your marker is 6 color tiles long. Your friend’s is 7 color tiles long. How many tiles shorter is your marker?

Answer:
My marker is one tile shorter than my friend’s marker.

Explanation:

Given that my marker is 6 color tiles long and my friend’s marker is 7 color tiles. So my marker is 7 – 6 which is one color tile shorter than my friend’s. And my friend’s marker is one color tile longer than mine.

Question 4.
MP Reasoning
Your pencil is 4 color tiles long. Your
friend’s is 2 color tiles long. Complete the sentences.

Your pencil is ________ color tiles ________ than your friend’s.
Your friend’s pencil is ________ color tiles ________ than yours.
Answer:
Your pencil is two color tiles longer than your friend’s.
Your friend’s pencil is two color tiles shorter than yours.

Explanation:
Given that my pencil is 4 color tiles long and my friend’s pencil is 2 color tiles. So my pencil is 4 – 2 which is two color tiles longer than my friend’s. And my friend’s pencil is two color tiles shorter than mine.

Think and Grow: Modeling Real Life

Your friend’s paper chain is 6 paper clips shorter than yours. Your chain is 12 paper clips long. How long is your friend’s?

Model:

Equation:

____________ paper clips long
Answer:
My paper chain is six color tiles longer than my friend’s.

Explanation:

Given that my paper chain is 12 color tiles long and my friend’s paper chain is 6 color tiles. So my paper chain is 12 – 6 which is six color tile shorter than my friend’s. And my friend’s paper chain is six color tile longer than mine.

Show and Grow

Question 5.
Your paper airplane is 9 color tiles shorter than your friend’s. Your friend’s paper airplane is 16 color tiles long. How long is yours?

Model:

Answer:
Equation:

____________ color titles long
Answer:
My airplane is seven color tiles longer than my friend.

Explanation:

Given that my airplane is 9 color tiles long and my friend’s airplane is 16 color tiles. So my marker is 16 – 9 which is seven color tile shorter than my friend’s. And my friend’s airplane is seven color tile longer than mine.

Solve Compare Problems Involving Length Practice 10.5

Question 1.
Your backpack is 15 paper clips long. Your friend’s is 12 paper clips long. How many paper clips longer is your backpack?

Answer:
Three paper clips longer than my friend’s

Explanation:

Given that my backpack is 15 paper clips long and my friend’s backpack is 12 paper clips long. So there will be three paper clips longer than my friend’s.

Question 2.
MP Reasoning
Your baseball miff is 8 paper clips long. Your friend’s is 7 paper clips long. Complete the sentences.

Your friend’s baseball mitt is ________ paper clip _________ than yours.
Your baseball mitt is _________ paper clip _________ than your friends.
Answer:
Your friend’s baseball mitt is 1 paper clip shorter than yours.
Your baseball mitt is 1 paper clip longer than your friends.

Explanation:
Given that my baseball miff is 8 paper clips long and my friend’s baseball miff is 7 paper clips. So my baseball miff is 8 – 7 which is 1 paper clip longer than my friend’s. And my friend’s baseball miff is 1 paper clip shorter than mine.

Question 3.
Modeling Real Life
Your desk is 7 paper clips longer than your friend’s. Your friend’s desk is 14 paper clips long. How long is yours?

___________ paper clips long
Answer:
7 paper clips long.

Explanation:

As my desk is 7 paper clips longer than my friend’s and my friend’s desk is 14 paper clips long. So my desk will be 14 – 7= 7 paper clips long.

Review & Refresh

Use the picture to complete the number bond.

Question 4.

Answer:
By number bond, the total number of flowers is 5 + 3= 8 flowers.

Explanation:

This number bond explains to us how the numbers are joined together and how they break down into a certain number of parts. In the above image, we can see that there are five yellow flowers and three red flowers. By number bond, we will add all the flowers. So the total number of flowers is 5 + 3= 8 flowers. Now they break down by five yellow flowers in one circle and three red flowers in another circle.

Question 5.

Answer:
By number bond, the total number of cans is 3 + 3= 8 cans.

Explanation:

This number bond explains to us how the numbers are joined together and how they break down into a certain number of parts. In the above image, we can see that there are three green cans and three red cans. By number bond, we will add all the cans. So the total number of cans is 3 + 3= 6 flowers. Now they break down by three green cans in one circle and three red cans in another circle.

Measure and Compare Lengths Performance Task

Question 1.
Use a piece of string to compare the routes from your house to the library, the post office, and the school.Order the routes from shortest to longest.
____________, ____________, ____________
Answer:
The order from shortest to longest is the post office, the library, and the school.

Explanation:
The route from my house to the library is three meters and the route from my house to the post office is two meters and the route from my house to the school is four meters. So the order from shortest to longest is the post office, the library, and the school.

Question 2.
Use a piece of string to measure the different routes from your house to your friend’s house. Color the route you would use to ride your bike to your friend’s house.
Answer:

Question 3.
a. The bakery is farther from your house than the pool. The park is closer to your house than the pool. Which place is closest to your house?
Park             Bakery              Pool
Answer:

b. Label the park, bakery, and pool on the map.
Answer:

Order Objects by Length Homework & Practice 10.1

Question 1.
Order from longest to shortest.

____________, _____________, ______________
Answer:
The order from longest to shortest is
Shark, Lobster, Fish.

Explanation:
In the above image, the longest is the shark, the largest is the lobster and the shortest is the fish.

Question 2.
MP Problem Solving
A green snake is shorter than a black snake. A brown snake is shorter than a black snake. Which snake is the longest?
green        block         brown
Answer:
The black snake is the longest.

Explanation:
Given that a green snake is shorter than a black snake and a brown snake is shorter than a black snake. So the longest snake is a black snake.

Compare Lengths Indirectly Homework & Practice 10.2

Question 3.
Circle the longer object.

Answer:
We will circle the second image.

Explanation:

In the above image, the longest object is the second image. So we will circle the second image.

Question 4.
Circle the longer Object.

Answer:
We will circle the second image.

Explanation:

In the above image, the longest object is the second image. So we will circle the second image.

Question 5.
Draw a line through the shorter object.

Answer:
The first image is the shorter image.

Explanation:

In the above image, the shorter object is the first image. So we will circle the first image.

Measure Lengths Homework & Practice 10.3

Measure

Question 6.

about ___________ color tiles
Answer:
The length of the object is about four colored tiles.

Explanation:
The object is measured by using the colored tile. As each tile is equal to one unit. So the length of the object is about four colored tiles, which means four units. We must take each tile without gaps or any overlaps between them.

Question 7.

about ___________ color tiles
Answer:
The length of the object is about five colored tiles.

Explanation:
The object is measured by using the colored tile. As each tile is equal to one unit. So the length of the object is about five colored tiles, which means five units. We must take each tile without gaps or any overlaps between them.

Measure More Lengths Homework & Practice 10.4

Measure

Question 8.

about ___________ color tiles
about ___________ paper clips
Answer:
The length of the object is about four colored tiles and the length of the object which is measured using a paper clip is two paper clips.

Explanation:
The object is measured by using the colored tile. As each tile is equal to one unit. So the length of the object is four colored tiles, which means four units. We must take each tile without gaps or any overlaps between them. And to measure the object with the paper clip, we will place the paper clip without gaps or overlaps between them. And each paper clip is equal to about one unit. So the length of the object using the paper clip is two paper clips, which means two units. We have noticed that if we take a colored tile, then the length of the object will be four color tiles. And if we take a paper clip then the length of the pencil will be two paper clips. So the length is doubled while measuring with color tile than the paper clip.

Question 9.
Modeling Real Life
Your hockey stick is 18 paper clips long. Is your hockey stick more than or less than 18 color tiles long?

Circle: more than 18 less than 18
Tell how you know:
Answer:
My hockey stick has more than 18 color tiles.

Explanation:
My hockey stick has more than 18 color tiles. As we can see from the above problems that while measuring with the paper clip we need fewer paper clips than the color tiles. And the length is doubled while measuring with color tile than the paper clip. So there will be more than 18 color tiles than the paper clips for the hockey stick.

Solve Compare Problems Involving Length Homework & Practice 10.5

Question 10.
Your water bottle is 5 paper clips long. Your Friend’s is 4 paper clips long. How many paper clips longer is your water bottle?

Answer:
One paper clip long.

Explanation:

Given that my water bottle is 5 paper clips long and my friend’s water bottle is 4 paper clips long. So there will be one paper clip longer than my friend’s.

Question 11.
Your bookshelf is 19 color tiles long. Your friend’s is 15 color tiles long. How many tiles longer is your bookshelf?

Answer:
My bookshelf is four tiles longer.

Explanation:

Given that my bookshelf is 19 paper clips long and my friend’s bookshelf is 15 paper clips long. So there will be four paper clips longer than my friend’s.

Question 12.
MP Reasoning
Your pencil is 6 color tiles long. Your friend’s is 3 color tiles long. Complete the sentences.

Your pencil is ____________ color tiles ____________ than your friend’s.
Your friend’s pencil is ____________ color tiles ____________ than yours.
Answer:
Your pencil is three color tiles longer than your friend’s.
Your friend’s pencil is three color tiles shorter than yours.

Explanation:
As my pencil is six color tiles long, and my friend’s pencil is three color tiles long. So my pencil will be three color tiles longer than my friend’s pencil. And my friend’s pencil is three color tiles shorter than my pencil.

Conclusion:

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Big Ideas Math Answers Grade 3 Chapter 15 Find Perimeter and Area

Find Big Ideas Math Grade 3 Chapter 15 Find Perimeter and Area Questions with Solutions here. The students can get the answer along with the detailed explanation in the following sections. This BIG Math Answers Grade 3 Chapter 15 Find Perimeter and Area is helpful to complete the homework in time. Hence, download Big Ideas Math Book Grade 3 Chapter 15 Find Perimeter and Area Solutions pdf for free of cost.

Big Ideas Math Book 3rd Grade 15th Chapter Find Perimeter and Area Answer Key

Students have to prepare all the topics of Big Ideas Math Answers 3rd Grade 15th Chapter Find Perimeter and Area to score good marks in the exam. The different topics included in BIM Book Grade 3 Chapter 15 Find Perimeter and Area Answers are Understand Perimeter, Find Perimeters of Polygons, Find Unknown Side Lengths, Same Perimeter, Different Areas, and Same Area, Different Perimeters. After preparing these topics, check your skills at the performance task provided at the end.

The success criteria of Find Perimeter and Area is to understand the area and perimeter of polygons. The topic wise quick links to access Big Ideas Math Book Grade 3 Chapter 15 Find Perimeter and Area Answer Key is provided below. So, just tap on them and get the solutions easily.

Lesson 1 Understand Perimeter

Lesson 2 Find Perimeters of Polygons

Lesson 3 Find Unknown Side Lengths

Lesson 4 Same Perimeter, Different Areas

Lesson 5 Same Area, Different Perimeters

Performance Task

Lesson 15.1 Understand Perimeter

Explore and Grow

Question 1.
Model a rectangle on your geoboard. Draw the rectangle and label its side lengths. What is the distance around the rectangle?

Answer:
The units around the rectangle are 15 units.

Explanation:

In the above figure, the length of the rectangle is 4 units, and
the breadth of the rectangle is 3 units.
The units around the rectangle are 15 units.

Structure
Change the side lengths of the rectangle on your geoboard. What do you notice about the distance around your rectangle compared to the distance around the rectangle above? Explain

Answer:
The distance around the rectangle is 15 units

Explanation:

In the above figure, we can see the length of the rectangle is 3 units, and
the breadth of the rectangle is 4 units.
There is no change in the distance around the rectangle which is 15 units.

Think and Grow: Understand Perimeter
Perimeter is the distance around a ﬁgure. You can measure perimeter using standard units, such as inches, feet, centimeters, and meters.
Example
Find the perimeter of the rectangle.
Choose a unit to begin counting. Count each unit around the rectangle.

Answer:
The perimeter of the rectangle is 20 in.

Explanation:
To find the perimeter of the triangle, we need to know the length and the breadth of the rectangle.
Here, the length of the rectangle is 7 in,
and the breadth is 3 in,
So the perimeter of the rectangle is 2(Length + Breadth)
= 2(7+3)
= 2(10)
= 20 in.

Show and Grow

Question 1.
Find the perimeter of the figure.

There are ___ units around the figure.
Answer:
There are 22 units around the figure.

Explanation:
To find the perimeter of the rectangle, we need to know the length and the breadth of the rectangle.
Here, the length of the rectangle is 6 m,
and the breadth is 5 m,
So the perimeter of the rectangle is 2(Length + Breadth)
= 2(6+5)
= 2(11)
= 22 m.

Question 2.

There are ___ units around the figure.
So, the perimeter is ____ feet.
Answer:
The perimeter of the figure is 24 ft.

Explanation:
To find the perimeter of the figure, we will add all the sides of the figure
So the sides of the figure are 5,3,2,4,3,7.
and the perimeter of the figure is 5+3+2+4+3+7
= 24 ft.

Question 3.
Draw the figure that has a perimeter of 16 centimeters.

Answer:
The sides of the figure are 4 cm.

Explanation:
As the perimeter of the figure is 16 cm,
So let the sides of the figure be 4 cm,
by that, we can get the perimeter as 16 cm, i.e
4+4+4+4= 16 cm.

Apply and Grow: Practice

Question 4.
Find the perimeter of the ﬁgure.

There are ___ units around the figure.
So, the perimeter is ___
Answer:
There are 20 units around the figure.
So, the perimeter is 20 cm.

Explanation:
To find the perimeter of the figure, we will add all the sides of the figure
So the sides of the figure are 3,3,2,2,5,5.
and the perimeter of the figure is 3+3+2+2+5+5
= 20 units,
So the perimeter is 20 cm.

Question 5.

There are ___ units around the figure.
So, the perimeter is ___
Answer:
There are 26 units around the figure.
So, the perimeter is 26 ft.

Explanation:
To find the perimeter of the figure, we will add all the sides of the figure
So the sides of the figure is 5,5,2,3,1,3,2,5
and the perimeter of the figure is 5+5+2+3+1+3+2+5
= 26 units,
So the perimeter is 26 ft.

Question 6.

Perimeter = ___
Answer:
The perimeter of the figure is 22 in.

Explanation:
To find the perimeter of the figure, we will add all the sides of the figure
So the sides of the figure is 4,3,2,3,3,3,1,3
and the perimeter of the figure is 4+3+2+3+3+3+1+3
= 22 units,
So the perimeter is 22 in.

Question 7.

Perimeter = ___
Answer:
The perimeter is 24 m.

Explanation:
To find the perimeter of the figure, we will add all the sides of the figure
So the sides of the figure is 5,7,2,2,1,2,1,1,1,2
and the perimeter of the figure is 5+7+2+2+1+2+1+1+1+2
= 24 units,
So the perimeter is 24 m.

Question 8.
Draw a ﬁgure that has a perimeter likely measurement for the of 14 centimeters.

Answer:
The length of the rectangle is 2 cm, and
the breadth is 5 cm.

Explanation:

In the above figure, we can see the length of the rectangle is 2 cm
and the breadth of the rectangle is 5 cm
perimeter = 2(length + breadth)
= 2(5+2)
= 2(7)
= 14 cm.
so the perimeter of the rectangle is 14 cm.

Question 9.
Precision Which is the most likely measurement for the perimeter of a photo?
20 inches
100 meters
5 centimeters
2 inches
Answer:
2 inches.

Explanation:
The most likely measurement for the perimeter of a photo is 2 inches.

Question 10.
You be the teacher Your friend counts the units around the figure and says the perimeter is 12 units. Is your friend correct? Explain.

Answer:
No, he is not correct.

Explanation:
No, he is not correct. As there are 2+2+1+1+4+1+1+2= 14 units, but not 12 units. So he is not correct.

Think and Grow: Modeling Real Life
Use a centimeter ruler to ﬁnd the perimeter of the bookmark.

The perimeter is ___
Answer:
The perimeter of the bookmark is 26 cm.

Explanation:
On measuring, the length of the sides of the bookmark are 4 cm, 9 cm, 2 cm, 2 cm, 9 cm
to find the perimeter of the bookmark, we will add all the length of the sides
so the perimeter of the bookmark is
p = 4 cm+ 9 cm+ 2 cm+ 2 cm+ 9 cm
= 26 cm.
The perimeter of the bookmark is 26 cm.

Show and Grow

Question 11.
Use an inch ruler to find the perimeter of the decal.

Answer:
The perimeter of the decal is 17 in.

Explanation:
On measuring, the length of the sides of the decal is 5in, 2in, 5in, 5in
to find the perimeter of the decal, we will add all the length of the sides
so the perimeter of the decal is
p= 5in+2in+5in+5in
= 17 in
The perimeter of the decal is 17 in.

Question 12.
How much greater is the perimeter of your friend’s desk than the perimeter of your desk?

Answer:
Friend’s perimeter is 22-16= 6 times greater.

Explanation:
As we can see figure 1 is a square and the sides of the square are 4 ft.
So the perimeter of the square is a+a+a+a,
= 4+4+4+4
= 16 ft.
And now let’s find the perimeter of the friend’s figure,
So the sides of the figure is 2,6,5,2,3,4
and the perimeter is 2+6+5+2+3+4= 22 ft.
By this, we can see that the friend’s figure has a greater perimeter,
and friend’s perimeter is 22-16= 6 times greater.

Question 13.
DIG DEEPER!
Explain how you might use a centimeter ruler and string to estimate the perimeter of the photo of the window.

Answer:
Here, we will use the ruler to find the length of the bottom part of the window and the sides which are straight we can find the length using the ruler. And the curve sides we will measure using the string.

Understand Perimeter Homework & Practice 15.1

Question 1.

There are ___ units around the figure.
So, the perimeter is ___ inches.
Answer:
There are 20 units around the figure,
So the perimeter is 20 in.

Explanation:
To find the perimeter of the figure, we will add all the sides of the figure,
As we can see the above image is a square,
and the perimeter of the square is s+s+s+s
= 5+5+5+5
= 20 in.
As there are 20 units around the figure,
So the perimeter is 20 in.

Question 2.

There are ___ units around the figure.
So, the perimeter is ___ feet.

Answer:
There are 32 units around the figure,
So the perimeter is 32 ft.

Explanation:
To find the perimeter of the figure, we will add all the sides of the figure,
The sides of the above figure is 2,3,4,3,2,5,8,5
and the perimeter of the figure is 2+3+4+3+2+5+8+5
= 32 units.
As there are 32 units around the figure,
So the perimeter is 32 ft.

Question 3.

Perimeter = ___
Answer:
There are 26 units around the figure,
So the perimeter is 26 cm.

Explanation:
To find the perimeter of the figure, we will add all the sides of the figure,
The sides of the above figure is 2,2,2,3,1,3,5,8
and the perimeter of the figure is 2+2+2+3+1+3+5+8
= 26 units.
As there are 26 units around the figure,
So the perimeter is 26 cm.

Question 4.

Perimeter = ___
Answer:
There are 38 units around the figure,
So the perimeter is 38 m.

Explanation:
To find the perimeter of the figure, we will add all the sides of the figure,
The sides of the above figure is 2,8,2,3,1,3,2,8,2,3,1,3
and the perimeter of the figure is 2+8+2+3+1+3+2+8+2+3+1+3
= 38 units.
As there are 38 units around the figure,
So the perimeter is 38 m.

Question 5.
Draw a figure that has a perimeter of 18 inches.

Answer:
The length of the rectangle is 5 in
and the breadth of the rectangle is 4 in.

Explanation:

Let the length of the rectangle be 5 in
and the breadth of the rectangle be 4 in
so the perimeter of the rectangle is
p = 2(length+breadth)
= 2(5+4)
= 2(9)
= 18 in.

Question 6.
Reasoning
Which color represents the perimeter of the rectangle? What does the other color represent?

Answer:
Blue color represents the perimeter of the rectangle and
the other color yellow represents the area of the figure.

Explanation:
In the above figure, the blue color represents the perimeter of the rectangle.
Because the perimeter represents the distance around the edge of the shape.
And the other color yellow represents the area of the figure,
as area represents the amount of space inside a shape.

Question 7.
Modeling Real Life
Use a centimeter ruler toﬁnd the perimeter of the library card.

Answer:
The perimeter of the library card is 24 cm.

Explanation:
On measuring, the length of the sides of the library card is 5cm, 7cm, 5cm, 7cm
to find the perimeter of the library card, we will add all the length of the sides
so the perimeter of the library card is
p= 5cm+7cm+5cm+7cm
= 24 cm
The perimeter of the library card is 24 cm.

Question 8.
Modeling Real Life
How much greater is the perimeter of your piece of fabric than the perimeter of your friend’ spiece of fabric?

Answer:
My fabric is 4 inches greater than my friend’s fabric.

Explanation:
To find the perimeter of the figure 1, we will add all the sides of the figure,
The sides of the above figure is 2,1,1,2,2,2,8,2,2,2,1,1
and the perimeter of the figure is 2+1+1+2+2+2+8+2+2+2+1+1
= 26 in.
So the perimeter of the figure is 26 in.
And now let’s find the perimeter of the friend’s figure,
So the sides of the figure is 1,2,2,4,1,5,4,3
and the perimeter is 1+2+2+4+1+5+4+3
= 22 in.
By this, we can see that my fabric has the highest perimeter than the friend’s fabric
which is 26-22= 4 in greater.

Review & Refresh

Write two equivalent fractions for the whole number.

Question 9.

Answer:
1= 4/4 = 6/6

Explanation:
Here, the equivalent fraction for the whole number means if the numerator was divided by the denominator without any reminder then the fraction is equivalent to a whole number. So 1= 4/4 = 6/6.

Question 10.

Answer:
4= 4/1 = 8/2.

Explanation:
Here, the equivalent fraction for the whole number means if the numerator was divided by the denominator without any reminder then the fraction is equivalent to a whole number. So 4= 4/1 = 8/2.

Question 11.

Answer:
6= 24/4 = 36/6.

Explanation:
Here, the equivalent fraction for the whole number means if the numerator was divided by the denominator without any reminder then the fraction is equivalent to a whole number. So 6= 24/4 = 36/6.

Find Perimeters of Polygons 15.2

Explore and Grow

Model a rectangle on your geoboard. Draw the rectangle and label its side lengths. Then ﬁnd the perimeter in more than one way.

Answer:
The perimeter of the rectangle is 14 units.

Explanation:

In  the above figure, we can see the length of the rectangle is 4 units
and the breadth of the rectangle is 3 units,
so the perimeter of the rectangle is
p =2(length + breadth)
= 2(4+3)
= 2(7)
= 14 units.
The perimeter of the rectangle is 14 units.
Critique the Reasoning of Others
Compare your methods of ﬁnding the perimeter to your partner’s methods. Explain how they are alike or different.
Answer:
There are two methods to find the perimeter explained below.

Explanation:
There are two ways to find the perimeter.
The first method is
Perimeter = 2(length + breadth)
here, we will add length and breadth, and then we will multiply the result by 2.
and the second method is
Perimeter = a+b+c+d
here, we will add all the sides of the figure.

Think and Grow: Find Perimeter

Example
Find the perimeter of the trapezoid.

You can find the perimeter of a figure by adding all of the side lengths.

Write an equation. The letter P represents the unknown perimeter. Add the side lengths.
So, the perimeter is ___.
Answer:
So, the perimeter is 36 in.

Explanation:
To find the perimeter of the trapezoid, we will add all the sides of the trapezoid,
so the sides of the trapezoid is 6 in,12 in,8 in,10 in
The perimeter is 6+12+8+10
= 36 in.

Example
Find the perimeter of the rectangle

Because a rectangle has two pairs of equal sides, you can also use multiplication to solve.

One Way:

Another Way:

So, the perimeter is ___.

Answer:
Perimeter = 7+9+7+9
= 32 cm.
Perimeter= 2×9 + 2× 7
= 32 cm.
So, the perimeter is 32 cm.

Explanation:
To find the perimeter of a rectangle, we have two ways,
One way is to add all the sides of the rectangle, which is
7+9+7+9= 32 cm.
And the other way is, as the two sides of the rectangle are equal, we wil use formula
Perimeter = 2( Length + Breadth)
= 2× Length + 2 × Breadth
= 2×9 + 2× 7
= 18+ 14
= 32 cm.

Show and Grow

Find the perimeter of the polygon.
Answer:

Explanation:
To find the perimeter of the polygon, we will add the length of the all sides.

Question 1.

The perimeter is ___.
Answer:
The perimeter is 12 m.

Explanation:
To find the perimeter of the polygon, we will add the length of all sides of the polygon.
So the length of the sides is 5 m, 3 m, 3 m, 2 m
The perimeter is 5+3+3+2
= 12 m.

Question 2.

The perimeter is ___.

Answer:
The perimeter is 24 ft.

Explanation:
As we can see in the above figure which has all sides are equal,
so the perimeter of the square is 4s
= 4×6
= 24 ft.

Apply and Grow: Practice

Find the perimeter of the polygon.

Question 3.

Perimeter = ___
Answer:
The perimeter of the polygon is 28 m.

Explanation:
To find the perimeter of the polygon, we will add the length of the all sides of the polygon.
So the length of the sides is 4 m, 9 m, 10 m, 5 m
The perimeter is 4+9+10+5
= 28 m.

Question 4.

Perimeter = ___
Answer:
The perimeter of the figure is 40 in.

Explanation:
To find the perimeter of the figure, we will add the length of all sides of the figure.
So the length of the sides is 12 in, 4 in, 8 in, 7 in,9 in.
The perimeter is 12+4+8+7+9
= 40 in.

Question 5.
Rectangle

Perimeter = ___
Answer:
The perimeter of the rectangle is 36 cm.

Explanation:
To find the perimeter of the rectangle, we need to know the length and the breadth of the rectangle.
Here, the length of the rectangle is 9 cm,
and the breadth is 10 cm,
So the perimeter of the rectangle is 2(Length + Breadth)
= 2(10+9)
= 2(19)
= 38 cm.

Question 6.
Rhombus

Perimeter = ___
Answer:
The perimeter of the Rhombus is 4 in.

Explanation:
As all sides of the Rhombus are equal, so the formula of the Rhombus is 4a
and the side of the Rhombus is 1 in,
so the perimeter is 4a
= 4×1
= 4 in.

Question 7.
Parallelogram

Perimeter = ___
Answer:
The perimeter of the parallelogram is 22 cm.

Explanation:
As the opposite sides of the parallelogram are equal,
so the perimeter of the parallelogram is 2(a+b)
the length of the parallelogram is 3 cm,
and the breadth of the parallelogram is 8 cm
So the perimeter of the parallelogram is
= 2(3+8)
= 2(11)
= 22 cm.

Question 8.
Square

Perimeter = ___
Answer:
The perimeter of the square is 16 ft.

Explanation:
As we can see in the above figure which has all sides are equal,
so the perimeter of the square is 4s
= 4×4
= 16 ft.

Question 9.
You build a pentagon out of wire for a social studies project. Each side is 8 centimeters long. What is the perimeter of the pentagon?
Answer:
The perimeter of the pentagon is 40 cm.

Explanation:
The length of the pentagon is 8 cm,
so the perimeter of the pentagon is 5a,
which is 5×8
= 40 cm

Question 10.
Number Sense
The top length of the trapezoid is 4 feet. The bottom length is double the top. The left and right lengths are each 2 feet less than the bottom. Label the side lengths and ﬁnd the perimeter of the trapezoid.

Answer:
The sides of the trapezoid are 4ft, 6ft, 8ft, 6ft, and the perimeter is 24 ft.

Explanation:
As the top length of the trapezoid is 4 feet and the bottom length is double the top,
which is 4×2= 8 feet. And the left and right lengths are each 2 feet less than the bottom,
which means 8 – 2 = 6 ft each.

So to find the perimeter of the trapezoid is we will add the length of all sides.
The perimeter of the trapezoid is 4+6+8+6= 24 ft.

Question 11.
Writing
Explain how finding the perimeter of a rectangle is different from finding its area.
Answer:
Perimeter= 2(length + breadth)
Area = length × breadth.

Explanation:
To find the perimeter of the rectangle,
we will add the length and breadth and will multiply the result with 2
Perimeter= 2(length + breadth)
and to find the area of the rectangle,
we will multiply the length and breadth of the rectangle.
Area = length × breadth.

Question 12.
Dig Deeper!
A rectangle has a perimeter of 12 feet. What could its side lengths be ?
Answer:
The length of the rectangle is 4 feet
and the breadth of the rectangle is 2 feet

Explanation:
Given the perimeter is 12 feet,
so the let the length be 4 feet
and the breadth be 2 feet
Let’s check on the length and breadth is correct are not
perimeter = 2( length + breadth)
= 2( 4+2)
= 2(6)
= 12 feet.

Think and Grow: Modeling Real Life

The rectangular sign is 34 feet longer than it is wide. What is the perimeter of the sign?

Understand the problem:
Make a plan:
Solve:
The perimeter is ___.
Answer:
The perimeter of the rectangular sign is 124 ft.

Explanation:
Given the rectangular sign is 34 feet longer than it’s wide
and the wide is 14 ft,
so the length is 34ft + 14ft= 48 ft.
the perimeter of the rectangular sign is
P= 2(48ft + 14ft)
= 2(62ft)
= 124 ft.
The perimeter of the rectangular sign is 124 ft.

Show and Grow

Question 13.
A city has a rectangular sidewalk in a park. The sidewalk is 4 feet wide and is 96 feet longer than it is wide. What is the perimeter of the sidewalk?
Answer:
The perimeter of the sidewalk is 208 feet.

Explanation:
As given the rectangular sidewalk’s wide is 4 feet and the length is 96 feet longer than it’s wide,
which means 96+4= 100 feet is the length of the rectangular sidewalk,
so the perimeter of the rectangular sidewalk is 2( length + breadth)
= 2( 4+100)
= 2(104)
= 208 feet.

Question 14.
A team jogs around a rectangular field three times. The field is 80 yards long and 60 yards wide. How many yards does the team jog?
Answer:
The number of yards does the team jog is 3×280= 840 yards.

Explanation:
The length of the rectangular field is 80 yards,
The breadth of the rectangular field is 60 yards,
So, the perimeter of the rectangular field is 2(length + breadth)
= 2(80+60)
= 2(140)
= 280 yards.
As the team jogs around a rectangular field three times,
so the number of yards does the team jog is 3×280= 840 yards.

Question 15.
Each side of the tiles is 8 centimeters long. What is the sum of the perimeters?

You put the tiles together as shown. Is the perimeter of this new shape the same as the sum of the perimeters above? Explain.

Answer:
The sum of the perimeters is 96 cm.
Yes, there will be a change in the perimeter of the new figure. The perimeter is 80 cm.

Explanation:
In the above figure, we can see a hexagon that has six sides.
So the formula for the perimeter of a hexagon is 6a,
the perimeter of the tiles is 6×8
= 48 cm.
So the sum of the tiles 48+48= 96 cm.
Yes, there will be a change in the perimeter of the new figure. As there are ten sides in the new figure, so the perimeter of the new figure is 10×8= 80 cm.

Find Perimeters of Polygons Homework & Practice 15.2

Find the perimeter of the polygon

Question 1.

Perimeter = ___
Answer:
The perimeter of the polygon is 29 in.

Explanation:
The perimeter of the polygon is the sum of the length of its sides,
So the sides of the polygon are 9 in, 4 in, 6 in, 10 in,
and the perimeter of the polygon is 9+4+6+10= 29 in.

Question 2.

Perimeter = ___
Answer:
The perimeter of the figure is 38 cm.

Explanation:
The perimeter of the figure is the sum of the length of its sides,
So the sides of the figure are 6 cm, 5 cm, 7 cm, 8 cm, 7 cm, 5 cm,
and the perimeter of the figure is 6+5+7+8+7+5= 38 cm.

Question 3.
Square

Perimeter = ___
Answer:
The perimeter of the square is 8 ft.

Explanation:
The length of the square is 2 ft,
and the perimeter of the square is 4a
= 4×2
= 8 ft.
so the perimeter of the square is 8 ft.

Question 4.
Parallelogram

Perimeter = ___
Answer:
The perimeter of the parallelogram is 8m.

Explanation:
The length of the parallelogram is 1 m,
and the breadth of the parallelogram is 3 m,
so the perimeter of the parallelogram is 2(length + breadth)
= 2(1+3)
= 2(4)
= 8 m.

Question 5.
Rhombus

Perimeter = ___
Answer:
The perimeter of the rhombus is 40 ft.

Explanation:
The length of the side of the rhombus is 10 ft,
and the perimeter of the rhombus is 4a
= 4×10
= 40 ft.

Question 6.

Perimeter = ___
Answer:
The perimeter of the rectangle is 22 in.

Explanation:
To find the perimeter of the rectangle, we need to know the length and the breadth of the rectangle.
Here, the length of the rectangle is 7 in,
and the breadth is 4 in,
So the perimeter of the rectangle is 2(Length + Breadth)
= 2(7+4)
= 2(11)
= 22 in.

Question 7.
Each side of a triangle is 5 centimeters long. What is the perimeter of the triangle?
Answer:
The perimeter of the triangle is 15 cm.

Explanation:
Given the length of the side of the triangle is 5 cm,
and the perimeter of the triangle is 3a
= 3×5
= 15 cm.

Question 8.
You Be The Teacher
Descartes says that a square will always have a greater perimeter than a triangle because it has more sides. Is he correct? Explain.
Answer:
Yes, Descartes is correct.

Explanation:
Yes, Descartes is correct. As the square has four sides and the perimeter of the square is 4a.
Whereas the triangle has 3 sides and the perimeter of the triangle is 3a.
So the square will always have a greater perimeter than a triangle.

Question 9.
Structure
Draw a pentagon and label its sides so that it has the same perimeter as the rectangle.

Answer:
The length of the side of the pentagon is 4m.

Explanation:
Given the length of the rectangle is 7 m,
and the breadth of the rectangle is 3 m,
So the perimeter of the rectangle is 2( length + breadth)
= 2(7+3)
= 2(10)
= 20 m.
Here we have the perimeter of the pentagon which is 20 m,
so we should find the sides of the pentagon,
As we know the perimeter of the pentagon is 5a
5a= 20
a= 20/5
= 4 m.
So the length of the side of the pentagon is 4m.

Question 10.
Modeling Real Life
An Olympic swimming pool is 82 feet longer than it is wide. What is the perimeter of the swimming pool?

Answer:
The perimeter of the swimming pool is 492 ft.

Explanation:
Given the Olympic swimming pool is 82 feet longer than it’s wide
and the wide is 82 ft,
so the length of the swimming pool is 82+82= 164 ft
the perimeter of the swimming pool is
p= 2(length+breadth)
= 2(164+82)
= 2(246)
= 492 ft.
The perimeter of the swimming pool is 492 ft.

Question 11.
Modeling Real Life
You put painter’s tape around two rectangular windows. The windows are each 52 inches long and 28 inches wide. How much painter’s tape do you need?
Answer:
The painter’s tape 160 inches.

Explanation:
Given the length of the window is 52 inches
and the width of the window is 28 inches
so the perimeter of the window is
p= 2(length + breadth)
= 2(52+28)
= 2(80)
= 160 inches.
So the painter’s tape 160 inches.

Review & Refresh

Question 12.

Answer:
737.

Explanation:
On adding 590+147 we will get 737.

Question 13.

Answer:
894.

Explanation:
On adding 636+258 we will get 894

Question 14.

Answer:
805.

Explanation:
On adding 476+329 we will get 805

Lesson 15.3 Find Unknown Side Lengths

Explore and Grow

You have a map with the three side lengths shown. The perimeter of the map is 20 feet. Describe how you can ﬁnd the fourth side length of your map without measuring.

Answer:
The fourth side of the map is 4ft.

Explanation:
We can find the value of the fourth side in two methods
First method:
Given the perimeter of the map is 20 feet,
and the sides of the map is 6 ft, 4 ft, 6ft, X ft.
As we know the perimeter of the rectangle is
6+4+6+X= 20 feet
16+X= 20
X= 20- 16
= 4 ft.
Second method:
As we know that the opposite sides of the rectangle are equal, as we know that the length of the side is 4 ft so the other side will also be 4 ft.

Repeated Reasoning
How is finding the unknown side length of a square different from finding the unknown side length of a rectangle?
Answer:
Refer below for a detailed explanation.

Explanation:
To find the unknown side length of the square
if we know the perimeter of the square then
the perimeter of the square is
p= 4a
we will substitute the value of p, on solving we will get the length of the square.
and to find the unknown side length of the rectangle,
we need to know the area or perimeter of the rectangle
and the other side of the rectangle.
so the formula of the perimeter of the rectangle is
p = 2(length + breadth)
we will substitute the perimeter value and the other side value
then we can find the length of the rectangle.

Think and Grow: Find Unknown Side Lengths
Example
The perimeter of the trapezoid is 26 feet. Find the unknown side length.

Write an equation for the perimeter.
Add the known side lengths.
What number plus 16 equals 26?
The unknown side length is ___.
Answer:
K= 10,
The number 16+10 equals 26.
The unknown side length is 10.

Explanation:
Given the perimeter of the trapezoid is 26 ft,
So the perimeter of the trapezoid is
K+5+6+5= 26
K+16= 26
K= 10.
The number 16+10 equals 26.
The unknown side length is 10.

Example

The perimeter of the square is 32 centimeters. Find the length of each side of the square.

Write an equation for the perimeter 4 times what number equals 32?
So, the length of each side is ___.
Answer:
n= 8.
The length of each side is 8 cm.

Explanation:
The perimeter of the square is 32 cm
So to find the sides of the square
4a= 32
a= 32/4
= 8 cm.
So, the length of each side is 8 cm.

Show and Grow

Find the unknown side length.

Question 1.
Perimeter = 34 inches

y = ___
Answer:
y = 13 in.

Explanation:
Given the perimeter is 34 inches,
and the sides of the figure are 10 in, 7 in, 4 in, y in.
so the perimeter of the figure is
34 in = 10+7+4+y
34 = 21+ y
y = 34 – 21
y = 13.

Question 2.
Perimeter = 20 meters

j = ___
Answer:
The length of the sides of the square is 5 m.

Explanation:
As we can see the above figure is a square and the perimeter of the square 20 meters,
so the sides of the square are
perimeter = 4a
20 = 4 j
j= 5 m.

Apply and Grow: Practice

Find the unknown side length.

Question 3.
Perimeter = 19 feet

y = ___
Answer:
The perimeter of the figure is 8 ft.

Explanation:
The perimeter of the figure is 19 feet,
and the length of the sides of the figure is 8 ft, 3 ft, y ft.
so perimeter = 8 ft + 3 ft + y ft
19= 11 ft + y ft
y= 19 ft – 11 ft
= 8 ft.

Question 4.
Perimeter = 26 centimeters

d = ___
Answer:
d= 4 cm.

Explanation:
The perimeter of the figure is 26 cm,
and the length of the sides of the figure is 10 cm, 5 cm, 7 cm, d cm.
so the perimeter of the figure is
p = 10+5+7+d
26 = 22 + d
d= 26-22
= 4

Question 5.
Perimeter = 30 feet

k = ___
Answer:
k = 11 ft.

Explanation:
Given the perimeter of the figure is 30 feet,
and the length of the sides is 5ft, 12 ft, 2 ft, k ft
So the perimeter of the figure is
p = 5 + 12 + 2 + k
30 ft = 19 ft + k
k = 11 ft.

Question 6.
Perimeter = 32 inches

k = ___
Answer:
k= 4 in.

Explanation:
Given the perimeter of the figure is 32 inches,
and the lengths of all sides is 10 in, 4 in, 5 in, 4 in, 5 in, k in.
So the perimeter of the figure is
p= 10+4+5+4+5+k
32 in =  28 in + k
k= 4 in.

Question 7.
Perimeter = 8 meters

y = ___
Answer:
y = 2 m.

Explanation:
Given the perimeter of the rhombus is 8 feet,
and the length of the side is y m,
So the perimeter of the rhombus is
p = 4a
8 m = 4×y
y = 2 m.

Question 8.
Perimeter = 48 inches

d = ___
Answer:
d = 8 in.

Explanation:
Given the perimeter of the Hexagon is 48 inches,
and the length of the sides is d in
So the perimeter of the hexagon is
p = 6 a
48 in = 6 × d in
d= 8 in.

Question 9.
Number Sense
A rectangle has a perimeter of 30 centimeters. The left side is 7 centimeters long. What is the length of the top side?
Answer:
The length of the top side is 8 cm.

Explanation:
Given the perimeter of the rectangle is 30 cm,
and the length of the left side of the rectangle is 7 cm,
So let the length of the top side be X,
Perimeter of the rectangle is
P = 2 (Length + breadth)
30 = 2 ( 7 cm + X cm)
30 / 2 = 7 cm + X cm
15 = 7 cm + X cm
X = 15 cm – 7 cm
X = 8 cm.
so, the length of the top side is 8 cm.

Question 10.
Writing
A triangle has three equal sides and a perimeter of 21 meters. Explain how to use division to ﬁnd the side lengths.
Answer:
The length of the side is 7 m.

Explanation:
Given the perimeter of the triangle is 21 m,
and we need to find the side of the lengths,
so the perimeter of the triangle is
p = 3a
21 m = 3×a
a = 21/3
= 7 m
So the length of the side is 7 m.

Question 11.
DIG DEEPER!
Newton draws and labels the square and rectangle below. The perimeter of the combined shape is 36 feet. Find the unknown side length.

Answer:
The unknown side of the length is 14 ft.

Explanation:
As the perimeter of the combined shape is 36 feet,
and the length of the side of the rectangle is 4 ft, and the other side be X ft
and the perimeter of the rectangle is 36 ft,
so perimeter = 2 (length + breadth)
36 ft  = 2( 4 ft + X ft)
36/2 = 4 ft + X ft
18 = 4 ft + X ft
X= 14 ft.
The unknown side of the length is 14 ft.

Think and Grow: Modeling Real Life

The perimeter of the rectangular vegetable garden is 30 meters. What are the lengths of the other three sides?

Understand the problem:
Make a plan:
Solve:
The lengths of the other three sides are ___, ___, and ___.
Answer:
The lengths of other three sides is 6 m, 9 m, 9 m.

Explanation:
The perimeter of the rectangular vegetable garden is 30 m
as it is in a rectangular shape, so the opposite sides are equal,
and the length of the side of the rectangular vegetable garden is 6m,
let the other side be X m
so the perimeter is
p = 2( length + breadth)
30 m = 2( 6 m+ X m)
30/2 = 6 + X
15 = 6 + X
X= 15 – 6
= 9 m.
So the lengths of other three sides is 6 m, 9 m, 9 m.

Show and Grow

Question 12.
The perimeter of the rectangular zoo enclosure is 34 meters. What are the lengths of the other three sides?

Answer:
The lengths of the other three sides are 12 m, 11 m, 11 m.

Explanation:
The perimeter of the rectangular zoo is 34 m
as it is in rectangular shape, so the opposite sides are equal,
and the length of the side of the rectangular zoo is 12 m,
let the other side be X m
so the perimeter is
p = 2( length + breadth)
34 m = 2( 12 m+ X m)
34/2 = 12 + X
17 = 6 + X
X= 17 – 6
= 11 m.
So the lengths of the other three sides is 12 m, 11 m, 11 m.

Question 13.
The floor of an apartment is made of two rectangles. The Perimeter is 154 feet. What are the lengths of the other three sides?

Answer:
The other length of the side of the small rectangle is 7 ft.
The other length of the side of the big rectangle is 30 ft.

Explanation:
Given the perimeter of the apartment is 154 feet,
first, we will take the big rectangle,
as the opposite sides of the rectangle are equal and the length of the big rectangle is 30 ft
so the other length is also 30 ft.
as the perimeter of the small rectangle is 38 ft
and the length of the one side of the rectangle is 12 ft
so the other length of the small rectangle is
p = 2(length+breadth)
38 = 2(12 + breadth)
38/2 = 12+ breadth
19= 12 + breadth
breadth= 19 – 12
= 7 ft.
The other length of the side is 7 ft.

Question 14.
DIG DEEPER!
You want to make a flower bed in the shape of a pentagon. Two sides of the flower bed are each 7 inches long, and two sides are each 16 inches long. The perimeter is 57 inches. Sketch the flower bed and label all of the side lengths.
Answer:
The length of the other side is 11 ft.

Explanation:

Given the perimeter of the flower bed shaped pentagon is 57 inches
and the two sides of the flower bed each is 7 inches long
and the other two sides of the flower bed each is 16 inches long
the other side of the flower bed be X
the perimeter of the flower bed is
p = 7+7+16+16+X
57= 46+X
X= 11 in.
The length of the other side is 11 ft.

Find Unknown Side Lengths Homework & Practice 15.3

Find the unknown side length.

Question 1.
Perimeter = 24 feet

d = ___
Answer:
The length of the unknown side is 8 ft.

Explanation:
The perimeter of the triangle is 24 ft,
and the lengths of the sides is 10 ft, 6 ft, d ft
so the perimeter of the triangle is
p = 10+6+d
24 = 10+6+d
24 = 16+d
d = 24 – 16
=  8 ft.
The length of the unknown side is 8 ft.

Question 2.
Perimeter = 46 inches

k = ___
Answer:
The length of the unknown side is 15 in.

Explanation:
The perimeter of the figure is 46 inches,
and the lengths of the sides is 13 in, 5 in, 13 in, k in
so the perimeter of the figure is
p = 13+5+13+k
46 = 31+k
46-31 = k
k = 15
=  15 in.
The length of the unknown side is 15 in.

Question 3.
Perimeter = 21 centimeters

y = ___
Answer:
The length of the unknown side is 7 cm.

Explanation:
The perimeter of the figure is 21 cm,
and the lengths of the sides is 4 cm, 1 cm, 9 cm, y cm
so the perimeter of the figure is
p = 4 cm+ 1 cm+ 9 cm+ y cm
21 = 14+ y
y = 21 – 14
y = 7
=  7 cm.
The length of the unknown side is 7 cm.

Question 4.
Perimeter = 41 meters

y = ___
Answer:
The length of the unknown side is 8 m.

Explanation:
The perimeter of the figure is 41 m,
and the lengths of the sides is 3 m, 12 m, 10 m, 8 m, y m
so the perimeter of the figure is
p = 3+12+10+8+y
41 = 33 + y
y = 41-33
y = 8
=  8 m.
The length of the unknown side is 8 m.

Question 5.
Perimeter = 12 feet

d = ___
Answer:
The length of the sides of the triangle is 4 ft.

Explanation:
The perimeter of the triangle is 12 feet,
and the length of the side of the triangle is d ft,
so the perimeter of the triangle is
p = 3 a
12 = 3 × d
d = 12/3
= 4 ft.
The length of the sides of the triangle is 4 ft.

Question 6.
Perimeter = 50 inches

k = ___
Answer:
The length of the sides of the Hexagon is 4 in.

Explanation:
The perimeter of the Hexagon is 50 inches,
and the length of the side of the Hexagon is k in,
so the perimeter of the hexagon is
p = 5 a
50 = 5 × k
k = 50/5
= 10 in.
The length of the sides of the triangle is 10 in.

Question 7.
DIG DEEPER!
Each polygon has equal side lengths that are whole numbers. Which polygon could have a perimeter of 16 centimeters? Explain.

Answer:
The length of the sides of the octagon is 2 cm.

Explanation:
In the above three polygons, the second figure is an octagon, which has eight sides.
and the perimeter of the octagon is
p = 8 a
16 cm = 8 a
a = 16 /8
= 2 cm.

Question 8.
Number Sense
The area of a square is 25 square inches. What is its perimeter?
Answer:
The perimeter of the square is 20 inches.

Explanation:
The area of the square is 25 square inches, so
area = s^2
25 = s^2
s= 5 inches
so the perimeter of the square is
p = 4s
= 4×5
= 20 inches.

Question 9.
Modeling Real Life
The perimeter of the rectangular side walk is 260 meters. What are the lengths of the other three sides?

Answer:
The length of the other three sides is 10 m,10m,120 m.

Explanation:
The perimeter of the rectangular side walk is 260 meters,
and the length of the one side of the side walk is 120 m,
so the perimeter of the rectangular side walk is
p = 2( length + breadth)
260 = 2 ( 120 + breadth)
260/2 = 120 + breadth
130 = 120 + breadth
breadth = 130 – 120
= 10 m.
The length of the other three sides is 10 m,10m,120 m.

Question 10
Modeling Real Life
Two rectangular tables are pushed together. The perimeter is 40 feet. What are the lengths of the other three sides?

Answer:
The other length of the side of the small rectangle is 3 ft.
The other length of the side of the big rectangle is 5 ft.

Explanation:
Given the perimeter of the apartment is 40 feet,
first, we will take the big rectangle,
as the opposite sides of the rectangle are equal
and the length of the big rectangle is 5 ft
so the other length is also 5 ft.
as the perimeter of the small rectangle is 10 ft
and the length of the one side of the rectangle is 2 ft
so the other length of the small rectangle is
p = 2(length+breadth)
10 = 2(2 + breadth)
10/2 = 2+ breadth
5= 2 + breadth
breadth= 5 – 2
= 3 ft.
The other length of the side is 3 ft.

Review & Refresh

Write the time. Write another way to say the time.

Question 11.

Answer:
06: 48

Explanation:
Another way to say time is 06: 48

Question 12.

Answer:
03: 24

Explanation:
Another way to say time is 03: 24

Question 13.

Answer:

Explanation:
Another way to say the time is 11: 48

Lesson 15.4 Same Perimeter, Different Areas

Use color tiles to create two different rectangles that each have a perimeter of 16 units. Then draw your rectangles and label their dimensions. Do the rectangles have the same area? Explain how you know.

Answer:
No, the area of rectangle 1 and rectangle 2 is not the same.

Explanation:

Given the perimeter of the rectangle is 16 units
The length of rectangle 1 is 5 units
and the breadth of rectangle 1 is 3 units
so the area of rectangle 1 is
area = length × breadth
= 5×3
= 15 square units.
The length of rectangle 2 is 6 units
and the breadth of rectangle 2 is 2 units
so the area of rectangle 2 is
area = length × breadth
= 6×2
= 12 square units.
No, the area of rectangle 1 and rectangle 2 is not the same.

Repeated Reasoning
Draw another rectangle that has the same perimeter but different dimensions. Compare the area of the new rectangle to the rectangles above. What do you notice?
Answer:
No, the area of rectangle 1 and rectangle 2 is not the same.

Explanation:

The length of rectangle 1 is 4 units
and the breadth of rectangle 1 is 3 units
the perimeter of rectangle 1 is
p= 2(length+breadth)
= 2(4+3)
= 2(7)
= 14 units.
so the area of rectangle 1 is
area = length × breadth
= 4×3
= 12 square units.
The length of rectangle 2 is 5 units
and the breadth of rectangle 2 is 2 units
the perimeter of rectangle 1 is
p= 2(length+breadth)
= 2(5+2)
= 2(7)
= 14 units.
so the area of rectangle 2 is
area = length × breadth
= 5×2
= 10 square units.
No, the area of rectangle 1 and rectangle 2 is not the same.

Think and Grow : Same Perimeter, Different Areas

Example :
Find the perimeter and the area of Rectangle A. Draw a different rectangle that has the same perimeter. Which rectangle has the greater area?

Rectangle ___ has a greater area.
Answer:
The perimeter of the rectangle A is 20 m
and the area of the rectangle A is 24 m2
The perimeter of rectangle B is 20 m
and the area of the rectangle B is 16 m2
The rectangle A has a greater area.

Explanation:
Given the length of the rectangle is 6m
and the breadth of the rectangle is 4m,
so the perimeter of the rectangle is
p = 2( length + breadth)
= 2(6 + 4)
= 2(10)
= 20 m.
And the area of the rectangle is
a = length × breadth
= 6 × 4
= 24 m2

we can see in the above figure another rectangle with a length of 8m,
and the breadth is 2m,
so the perimeter of the rectangle is
p = 2(length + breadth)
= 2(8 + 2)
= 2(10)
= 20 m.
And the area of the rectangle is
area = length × breadth
= 8×2
= 16 m2
So the rectangle A has a greater area.

Show and Grow

Question 1.
Find the perimeter and area of Rectangle A. Draw a different rectangle that has the same perimeter. Which rectangle has the greater area?

Perimeter = ___
Area = ___

Perimeter = ___
Area = ___
Rectangle ___ has the greater area.
Answer:The perimeter of the rectangle A is 14 in
and the area of the rectangle A is 10 in2
The perimeter of rectangle B is 14 in
and the area of the rectangle B is 12 in2
The rectangle B has a greater area.

Explanation:
Given the length of the rectangle is 5 in
and the breadth of the rectangle is 2 in,
so the perimeter of the rectangle is
p = 2( length + breadth)
= 2(5 + 2)
= 2(7)
= 14 in.
And the area of the rectangle is
a = length × breadth
= 5×2
= 10 in2

we can see in the above figure another rectangle with a length of 4 in,
and the breadth is 3 in,
so the perimeter of the rectangle is
p = 2(length + breadth)
= 2(4 + 3)
= 2(7)
= 14 in.
And the area of the rectangle is
area = length × breadth
= 4×3
= 12 in2
So the rectangle B has a greater area.

Apply and Grow: Practice

Find the perimeter and area of Rectangle A. Draw a different rectangle that has the same perimeter. Which rectangle has the greater area?

Question 2.
Rectangle A

Perimeter = ___
Area = ___

Perimeter = ___
Area = ___
Rectangle ___ has the greater area.
Answer:
The perimeter of the rectangle A is 22 cm
and the area of the rectangle A is 11 cm2
The perimeter of rectangle B is 22 cm
and the area of the rectangle B is 30 cm2
The rectangle B has a greater area.

Explanation:
Given the length of the rectangle is 10 cm,
and the breadth of the rectangle is 1 cm,
so the perimeter of the rectangle is
p = 2( length + breadth)
= 2(10 + 1)
= 2(11)
= 22 cm.
And the area of the rectangle is
a = length × breadth
= 11 × 1
= 11 cm2

we can see in the above figure another rectangle with a length of 6 cm,
and the breadth is 5 cm,
so the perimeter of the rectangle is
p = 2(length + breadth)
= 2(6 + 5)
= 2(11)
= 22 cm.
And the area of the rectangle is
area = length × breadth
= 6×5
= 30 cm2
So the rectangle B has a greater area.

Question 3.
Rectangle A

Perimeter = ___
Area = ___

Rectangle B

Perimeter = ___
Area = ___
Rectangle ___ has the greater area
Answer:
The perimeter of the rectangle A is 20 m
and the area of the rectangle A is 21 m2
The perimeter of rectangle B is 20 m
and the area of the rectangle B is 24 m2
The rectangle B has a greater area.

Explanation:
Given the length of the rectangle is 7 m,
and the breadth of the rectangle is 3 m,
so the perimeter of the rectangle is
p = 2( length + breadth)
= 2(7 + 3)
= 2(10)
= 20 m.
And the area of the rectangle is
a = length × breadth
= 7 × 3
= 21 m2

we can see in the above figure another rectangle with a length of 6 m,
and the breadth is 4 m,
so the perimeter of the rectangle is
p = 2(length + breadth)
= 2(6 + 4)
= 2(10)
= 20 m.
And the area of the rectangle is
area = length × breadth
= 6×4
= 24 m2
So the rectangle B has a greater area.

Question 4.
MP Structure
Draw a rectangle that has the same perimeter as the one shown, but with a lesser area. What is the area ?

Answer:
The perimeter of the rectangle A is 26 ft
and the area of the rectangle A is 40 ft2
The perimeter of rectangle B is 26 ft
and the area of the rectangle B is 30 ft2
The rectangle B has a greater area.

Explanation:
Given the length of the rectangle is 5 ft,
and the breadth of the rectangle is 8 ft,
so the perimeter of the rectangle is
p = 2( length + breadth)
= 2(5 + 8)
= 2(13)
= 26 ft.
And the area of the rectangle is
a = length × breadth
= 5 × 8
= 40 ft

we can see in the above figure another rectangle with a length of 10 ft,
and the breadth is 3 ft,
so the perimeter of the rectangle is
p = 2(length + breadth)
= 2(10 + 3)
= 2(13)
= 26 ft.
And the area of the rectangle is
area = length × breadth
= 10×3
= 30 ft2
So the rectangle A has a greater area.

Think and Grow: Modeling Real Life

A paleontologist has 12 meters of twine to rope off a rectangular section of the ground. How long and wide should she make the roped-off section so it has the greatest possible area?
Draw to show:
She should make the roped-off section ___ meters long and ___ meters wide.
Answer:
She should make the roped-off section 4 meters long and 2 meters wide.

Explanation:
Given that a paleontologist has 12 meters of twine to rope off a rectangular section,
so if we take the length as 4 m and width as 2 m then we can get the greatest possible area,
so the area of the rectangular section is
area = length × breadth
= 4 m ×2 m
= 8 m2

Show and Grow

Question 5.
Newton has 16 feet of wood to make a rectangular sandbox. How long and wide should he make the sandbox so it has the greatest possible area?

Answer:
The greatest possible area of the rectangular sandbox is 15 ft2

Explanation:
As Newton has 16 feet of wood to make a rectangular sandbox,
so let the length be 5 ft and the wide be 3 ft to get the greatest possible area,
so the area of the rectangular sandbox is
area = length × breadth
= 5 ft × 3 ft
= 15 ft2
The greatest possible area of the rectangular sandbox is 15 ft2

Question 6.
DIG DEEPER!
You and Newton are building forts. You each have the same length of rope to make a rectangular perimeter for the forton the ground. Your roped-off section is shown. Newton’s section has a greater area than yours. Draw one way Newton could rope off his fort.

Descartes also builds a fort. He has the same length of rope as you to make a perimeter around his fort. Descartes’s roped-off section has a lesser area than yours. Draw one way Descartes could rope off his fort.

Answer:
Refer the below for detailed explanation.

Explanation:
The length of the rope is 7 ft
and the breadth of the rope is 3 ft
the perimeter is
p = 2(length+breadth)
= 2(7+3)
= 2(10)
= 20 ft.
The area of the rectangle is
area= length×breadth
= 7×3
= 21 square feet.

Let the length of the Newton’s rope is 6 ft
and the breadth of the Newton’s rope is 4 ft
the perimeter is
p = 2(length+breadth)
= 2(6+4)
= 2(10)
= 20 ft.
The area of the rectangle is
area= length×breadth
= 6×4
= 24 square feet.
And here Newton’s area is greater.

Let the length of the Descarte’s rope is 8 ft
and the breadth of the Descarte’s rope is 2 ft
the perimeter is
p = 2(length+breadth)
= 2(8+2)
= 2(10)
= 20 ft.
The area of the rectangle is
area= length×breadth
= 8×2
= 16 square feet.
And here Descartes area is lesser.

Same Perimeter, Different Areas Homework & Practice 15.4

Question 1.
Find the perimeter and the area of Rectangle A. Draw a different rectangle that has the same perimeter? Which rectangle has the greater area?

Perimeter = ___
Area = ___

Perimeter = ___
Area = ___
Rectangle __ has the greater area.
Answer:
The perimeter of the rectangle A is 7 cm
and the area of the rectangle A is 24 cm2
The perimeter of rectangle B is 26 ft
and the area of the rectangle B is 30 ft2
The rectangle B has a greater area.

Explanation:
Given the length of the rectangle is 7 cm,
and the breadth of the rectangle is 5 cm,
so the perimeter of the rectangle is
p = 2( length + breadth)
= 2(7 + 5)
= 2(12)
= 24 cm.
And the area of the rectangle is
a = length × breadth
= 7 × 5
= 35 cm2.

we can see in the above figure rectangle with a length of 6.5 cm,
and the breadth is 5.5 cm,
so the perimeter of the rectangle is
p = 2(length + breadth)
= 2(6.5 + 5.5)
= 2(12)
= 24 ft.
And the area of the rectangle is
area = length × breadth
= 6.5×5.5
= 35.75 square feet
So the rectangle B has a greater area.

Question 2.
Patterns
Complete the pattern. Find the area of each rectangle.

Each rectangle has the same perimeter. As the area increases, what do you notice about the shape of the rectangle?
Answer:
As the area increases the shape of the figure was changed, we can see the figure was changed from rectangle to square.

Explanation:
As we know that the perimeter of the above figure is the same,
so the perimeter of the above figures is
p = 2 (length + breadth)
= 2 (1 m+9 m)
= 2(10 m)
= 20 m.
So, the perimeter of the above figures is 20 m.
The area of figure 1 is
area = length × breadth
= 1 m × 9 m
= 9 m2.
The area of figure 2 is
= 8m × 2m
= 16 m2.
The area of figure 3 is
= 7m × 3m
= 21 m2.
Let the length of figure 4 be 6m and the breadth be 4m,
The area of figure 4 is
= 6m × 4m
= 24 m2.
Let the length of figure 5 be 5m and the breadth be 5m,
The area of figure 5 is
= 5m × 5m
= 25 m2.

Question 3.
Modeling Real Life
You are making a card with a 36-centimeter ribbon border. How long and wide should you make the card so you have the greatest possible area to write?

Answer:
The length and the breadth of the card is 9 cm.

Explanation:
Given the perimeter of the card with a ribbon border is 36 cm
so the length of the card is
p =4a
36 = 4a
a= 36/4
= 9 cm.
The length and the breadth of the card is 9 cm.
The area of the card is
a = length×breadth
= 9×9
= 81 square cm.

Question 4.
DIG DEEPER!
A school has two rectangular playgrounds that each have the same perimeter. The first playground is shown. The second has a lesser area than the first. Draw one way the second playground could look.

The school builds another playground. It has the same perimeter as the ﬁrst. The third playground has a greater area than the ﬁrst. Draw one way the third playground could look
Answer:

Explanation:

Review & Refresh

Question 5.
2 × 30 = ___
Answer:
60

Explanation:
On multiplying 2 × 30 we will get 60.

Question 6.
6 × 20 = ___
Answer:
120

Explanation:
On multiplying 6 × 20 we will get 120.

Question 7.
3 × 90 = ___
Answer:
270

Explanation:
On multiplying 3 × 90 we will get 270.

Lesson 15.5 Same Area, Different Perimeters

Explore and Grow

Use color tiles to create two different rectangles that each have an area of 18 square units. Then draw your rectangles and label their dimensions. Do the rectangles have the same perimeter? Explain how you know

Answer:
By comparing the perimeters of both rectangles, we can see that the perimeters are not the same.

Explanation:

In the above figure, we can see two colored rectangles
the length of rectangle 1 is 6 units
and the breadth of rectangle 1 is 3 units
so the perimeter of rectangle 1 is
p = 2(length+breadth)
= 2(6+3)
= 2(9)
= 18 units
and the area of rectangle 1 is
area= length×breadth
= 6×3
= 18 square units
The length of rectangle 2 is 9 units
and the breadth of rectangle 2 is 2 units
so the perimeter of rectangle 2 is
p = 2(length+breadth)
= 2(9+2)
= 2(11)
= 22 units.
and the area of rectangle 2 is
area= length×breadth
= 9×2
= 18 square units.
By comparing the perimeters of both rectangles, we can see that the perimeters are not the same.

Repeated Reasoning
As the perimeter increases and the area stays the same, what do you notice about the shape of the rectangle?

Think and Grow : Same Area, Different Perimeters

Example
Find the area and the perimeter of Rectangle A. Draw a different rectangle that has the same area. Which rectangle has the lesser perimeter?

Area = 2 × 6
= _____
Perimeter = 6 + 2 + 6 + 2
= ______

Area = ___ × ___
= ____
Perimeter = ___ + ___ + ___ + ___
= ____
Rectangle ___ has the lesser perimeter.
Answer:
The perimeter of the rectangle A is 16 ft
and the area of the rectangle A is 12 ft2
The perimeter of rectangle B is 14 ft
and the area of the rectangle B is 12 ft2
The rectangle A has a lesser area.

Explanation:
Given the length of the rectangle is 6 ft,
and the breadth of the rectangle is 2 ft,
so the perimeter of the rectangle is
p = 2( length + breadth)
= 2(6 + 2)
= 2(8)
= 16 ft.
And the area of the rectangle is
a = length × breadth
= 6 × 2
= 12 ft2

we can see in the above figure rectangle B and the length be 4 ft,
and the breadth be 3ft,
so the perimeter of the rectangle is
p = 2(length + breadth)
= 2(4 + 3)
= 2(7)
= 14 ft.
And the area of the rectangle is
area = length × breadth
= 4×3
= 12 ft2
So the rectangle A has a lesser area.

Show and Grow

Question 1.
Find the area and the perimeter of Rectangle A. Draw a different rectangle that has the same area. Which rectangle has the lesser perimeter?

Area = ___
Perimeter = ___

Area = ___
Perimeter = ___
Rectangle __ has the lesser perimeter.
Answer:
The perimeter of the rectangle A is 16 ft
and the area of the rectangle A is 12 ft2
The perimeter of rectangle B is 14 ft
and the area of the rectangle B is 12 ft2
The rectangle A has a lesser area.

Explanation:
Given the length of the rectangle is 6 cm,
and the breadth of the rectangle is 6 cm,
so the perimeter of the rectangle is
p = 2( length + breadth)
= 2(6 + 6)
= 2(12)
= 24 cm.
And the area of the rectangle is
a = length × breadth
= 6 × 6
= 36 cm2

we can see in the above figure rectangle B and the length be 9 cm,
and the breadth be 4 cm,
so the perimeter of the rectangle is
p = 2(length + breadth)
= 2(9 + 4)
= 2(13)
= 26 cm.
And the area of the rectangle is
area = length × breadth
= 9×4
= 36 cm2
So the rectangle A has a lesser perimeter.

Apply and Grow: Practice

Find the area and the perimeter of Rectangle A. Drawa different rectangle that has the same area. Which rectangle has the lesser perimeter?
Answer:

Question 2.
Rectangle A

Area = ___
Perimeter = ___
Rectangle B

Area = ___
Perimeter = ___
Rectangle ___ has the lesser perimeter.
Answer:
The perimeter of the rectangle A is 24 in
and the area of the rectangle A is 20 in2
The perimeter of rectangle B is 18 in
and the area of the rectangle B is 20 in2
The rectangle B has a lesser perimeter.

Explanation:
Given the length of the rectangle is 10 in,
and the breadth of the rectangle is 2 in,
so the perimeter of the rectangle is
p = 2( length + breadth)
= 2(10 + 2)
= 2(12)
= 24 in.
And the area of the rectangle is
a = length × breadth
= 10 × 2
= 20 in2

we can see in the above figure rectangle B and the length be 5 in,
and the breadth be 4 in,
so the perimeter of the rectangle is
p = 2(length + breadth)
= 2(5 + 4)
= 2(9)
= 18 in.
And the area of the rectangle is
area = length × breadth
= 5×4
= 20 in2
So the rectangle B has a lesser perimeter.

Question 3.
Rectangle A

Area = ___
Perimeter = ___
Rectangle B

Area = ___
Perimeter = ___
Rectangle ___ has the lesser perimeter.
Answer:
The perimeter of the rectangle A is 12 m
and the area of the rectangle A is 8 m2
The perimeter of rectangle B is 18 m
and the area of the rectangle B is 8 m2
The rectangle A has a lesser perimeter.

Explanation:
Given the length of the rectangle is 4 m,
and the breadth of the rectangle is 2 m,
so the perimeter of the rectangle is
p = 2( length + breadth)
= 2(4 + 2)
= 2(6)
= 12 m.
And the area of the rectangle is
a = length × breadth
= 4 × 2
= 8 m2

we can see in the above figure rectangle B and the length be 8 m,
and the breadth be 1 m,
so the perimeter of the rectangle is
p = 2(length + breadth)
= 2(8+ 1)
= 2(9)
= 18 m.
And the area of the rectangle is
area = length × breadth
= 8×1
= 8 m2
So the rectangle A has a lesser perimeter.

Question 4.
DIG DEEPER!
The perimeter of a blue rectangle is 10 feet. The perimeter of a green rectangle is 14 feet. Both rectangles have the same area. Find the area and the dimensions of each rectangle.
Answer:

Explanation:
Given the perimeter of the blue rectangle is 10 ft and
the perimeter of the green rectangle is 14 ft

Think and Grow: Modeling Real Life

You have 40 square patio bricks that are each 1 foot long and 1 foot wide. You want to make a rectangular patio with all of the bricks. How long and wide should you make the patio so it has the least possible perimeter?

Draw to show:
You should make the patio ___ feet long and ___ feet wide.
Answer:
You should make the patio 8 feet long and 5 feet wide.
The least possible perimeter is 26 feet.

Explanation:

As there are 40 square patio bricks and each brick is 1 foot long and 1 foot wide
so to make a rectangular patio we need
the length of the rectangular patio be 8 feet
and the breadth of the rectangular patio be 5 feet
so the perimeter of the rectangular patio is
p= 2(length+breadth)
= 2(8+5)
= 2(13)
= 26 feet.
The least possible perimeter is 26 feet.

Show and Grow

Question 5.
Your friend has 16 square foam tiles that are each 1 foot long and 1 foot wide. He wants to make a rectangular exercise space with all of the tiles. How long and wide should he make the exercise space so it has the least possible perimeter?

Answer:
The least possible perimeter is 16 feet.

Explanation:
As there are 16 square foam tiles and each foam tile is 1 foot long and 1 foot wide
so to make an exercise space we need
the length of the exercise space be 4 feet
and the breadth of the exercise space be 4 feet
so the perimeter of the exercise space is
p= 2(length+breadth)
= 2(4+4)
= 2(8)
= 16 feet.
The least possible perimeter is 16 feet.

Question 6.
DIG DEEPER!
You and your friend each use fencing to make a rectangular playpen for a puppy. Each pen has the same area. Your pen is shown. Your friend’s pen uses less fencing than yours. Draw one way your friend could make her pen.

Your cousin makes a playpen for a puppy. His pen has the same area as your pen. Your cousin’s pen uses more fencing than yours. Draw one way your cousin could make his pen.
Answer:

Same Area, Different Perimeters Homework & Practice 15.5

Question 1.
Find the area and the perimeter of Rectangle A. Drawa different rectangle that has the same area. Which rectangle has the lesser perimeter?

Area = ___
Perimeter = ___

Area = ___
Perimeter = ___
Rectangle __ has the lesser perimeter.
Answer:
The perimeter of rectangle A is 4 in
and the area of rectangle A is 20 square inches.
It is not possible to draw a rectangle that has the same area and different perimeter.

Explanation:
Given the length of the rectangle is 4 in,
and the breadth of the rectangle is 4 in,
so the perimeter of the rectangle is
p = 2( length + breadth)
= 2(4+4)
= 2(8)
= 16 in.
And the area of the rectangle is
a = length × breadth
= 4 × 4
= 16 square inches.

Question 2.
Structure
The dimensions of a rectangle are 4 feet by 10 feet. Which shape has the same area, but a different perimeter?

Answer:
The yellow shape rectangle has the same area and different perimeter.

Explanation:
Given the dimensions of the rectangle are 4 feet by 10 feet
so the perimeter of the rectangle is
p= 2(length+breadth)
= 2(4+10)
= 2(14)
= 28 feet.
The area of the rectangle is
area = length×breadth
= 10×4
= 40 square feet.
Here, we can see the yellow rectangle has a length of 8 feet
and the breadth of the rectangle is 5 feet
so the perimeter of the rectangle is
p = 2(length+beadth)
= 2(8+5)
= 2(13)
= 26 feet.
and the area of the rectangle is
area = length×breadth
= 8×5
= 40 square feet.

Question 3.
MP Reasoning
The two ﬁelds have the same area. Players run one lap around each ﬁeld. At which ﬁeld do the players run farther?

Answer:
As the perimeter of field A is greater than field B so in field A the players run father.

Explanation:
Let the length of field A is 10m
and the breadth of field A is 2m
so the perimeter of field A is
p = 2(length+breadth)
= 2(10+2)
= 2(12)
= 24 m.
and the area of field A is
area= length×breadth
= 10×2
= 20 square meters.
Let the length of field B is 5m
and the breadth of field B is 4m
so the perimeter of field B is
p = 2(length+breadth)
= 2(5+4)
= 2(9)
= 18 m.
and the area of field A is
area= length×breadth
= 5×4
= 20 square meters.
As the perimeter of field A is greater than field B so in field A the players run father.

Question 4.
Modeling Real Life
You have 24 square pieces of T-shirt that are each 1 foot long and 1 foot wide. You want to make a rectangular T-shirt quilt with all of the pieces. How long and wide should you make the quilt so it has the least possible perimeter?

Answer:
The least possible perimeter is 20 feet.

Explanation:
As there are 24 square pieces of T-shirt and each was 1 foot long and 1 foot wide
so to make a rectangular T-shirt quilt with all of the pieces we need
the length of the rectangular T-shirt quilt be 6 feet
and the breadth of the rectangular T-shirt quilt be 4 feet
so the perimeter of the rectangular T-shirt quilt is
p= 2(length+breadth)
= 2(6+4)
= 2(10)
= 20 feet.
The least possible perimeter is 20 feet.

Question 5.
DIG DEEPER!
You and Descartes each have40 cobblestone tiles to arrange in to a rectangular pathway. Your pathway is shown. Descartes’s pathway has a lesser perimeter than yours. Draw one way Descartes could make his pathway.

Newton also makes a rectangular pathway with 40 cobblestone tiles. His pathway has a greater perimeter than yours. Draw one way Newton could make his pathway.
Answer:

Review & Refresh

Identify the number of right angles and pairs of parallel sides.

Question 6.

Right angles: ___
Pairs of Parallel sides: ___
Answer:
Right angles: 1
Pairs of Parallel sides: 2.

Explanation:
In the above figure, we can see there are the right angle is 1, and the pairs of parallel sides are 2.

Question 7.

Right angles: ___
Pairs of parallel sides: ____
Answer:
Right angles: 4.
Pairs of Parallel sides: 2.

Explanation:
In the above figure, we can see there are the right angle is 4, and the pairs of parallel sides are 2.

Find Perimeter and Area Performance Task

You and your cousin build a tree house.

Question 1.
The ﬂoor of the tree house is in the shape of a quadrilateral with parallel sides that are 4 feet long and 10 feet long. The other 2 sides are equal in length. The perimeter is 24 feet. Sketch the ﬂoor and label all of the side lengths.

Answer:
The length of the sides is 6 feet.

Explanation:
The perimeter of the floor is 24 feet
so the length of the floor be 6 feet
as the two other sides are also equal
so the other side length also be 6 feet
and let’s check the perimeter
p = 2(length+breadth)
= 2(6+6)
= 2(12)
= 24 feet.

Question 2.
Each rectangular wall of the tree house is 5 feet tall. How many square feet of wood is needed for all of the walls?
Answer:

Explanation:

Question 3.
You cut out a door in the shape of a rectangle with sides that are whole numbers. Its area is 8 square feet. What is the height of the door?
Answer:
The height of the rectangular door is 4 feet.

Explanation:
The area of the rectangular shape door is 8 square feet
as the sides of the rectangular door are whole numbers
so the length rectangular door be 4 feet
and the breadth be 2 feet
then we can get the area 8 square feet
let’s check the area
area = length×breadth
= 4×2
= 8 square feet.
So the height of the rectangular door is 4 feet.

Question 4.
You want to paint the ﬂoor and walls on the inside of your tree house. The area of the ﬂoor is 28 square feet. Each quart of paint covers 100 square feet.
a. How many quarts of paint do you need to buy?
b. Do you have enough paint to paint the outside walls of the tree house? Explain.

Answer:
a. 2,800 square feet quarts of paint we need to buy.

Explanation:
a. The area of the ﬂoor is 28 square feet and each quart of paint covers 100 square feet, so we need to buy
28×100= 2,800 square feet quarts of paint.

b.

Find Perimeter and Area Activity

Perimeter Roll and Conquer

Directions:
1. Players take turns rolling two dice.
2. On your turn, draw a rectangle on the board using the numbers on the dice as the side lengths. Your rectangle cannot cover another rectangle.
3. Write an equation toﬁnd the perimeter of the rectangle.
4. If you cannot ﬁt a rectangle on the board, then you lose your turn. Play 10 rounds, if possible.
5. Add all of your rectangles’ perimeters together. The player with the greatest sum wins!

Answer:

Explanation:

Find Perimeter and Area Chapter Practice

15.1 Understand Perimeter

Find the perimeter of the figure

Question 1.

Perimeter = ___
Answer:
The perimeter of the rectangle is 18 cm.

Explanation:
In the above figure, we can see the rectangle
with a length of 5 cm,
and the breadth of 4 cm
the perimeter of the rectangle is
p = 2 (length + breadth)
= 2 (5+4)
= 2(9)
= 18 cm.

Question 2.

Perimeter = ___
Answer:
The perimeter of the figure is 26 ft.

Explanation:
To find the perimeter of the above figure,
we will add the lengths of all sides of the figure
the sides of the above figure is 2 ft, 8 ft, 3 ft, 2 ft, 2 ft, 2 ft, 1 ft, 1 ft, 2 ft, 3 ft
the perimeter of the above figure is
p = 2+8+3+2+2+2+1+1+2+3
= 26 ft.

Question 3.
Draw a figure that has a perimeter of 10 inches.

Answer:

15.2 Find Perimeter of Polygons

Find the perimeter of the polygon

Question 4.

Perimeter = ___
Answer:
The perimeter of the polygon is 33 cm.

Explanation:
To find the perimeter of the polygon, we will add all the sides of the polygon
so the sides of the polygon are 9 cm, 6 cm, 8 cm, 10 cm
the perimeter of the polygon is
p = 9 cm +6 cm +8 cm +10 cm
= 33 cm.

Question 5.

Perimeter = ___
Answer:
The perimeter of the figure is 27 ft.

Explanation:
To find the perimeter of the figure, we will add all the sides of the figure
so the sides of the perimeter is 5 ft, 11 ft, 7 ft, 3 ft, 1 ft
the perimeter of the figure is
p = 5 ft+11 ft+ 7 ft+3 ft+1 ft
= 27 ft.

Question 6.
Parallelogram

Perimeter = ___
Answer:
The perimeter of the parallelogram is 12 m.

Explanation:
Given the length of the parallelogram is 4 m
and the breadth of the parallelogram is 2 m
the perimeter of the parallelogram is
p = 2 (length + breadth)
= 2( 4 m+ 2 m)
= 2(6 m)
= 12 m

Find the perimeter of the polygon

Question 7.
Rhombus

Perimeter = ___
Answer:
The perimeter of the rhombus is 36 cm.

Explanation:
Given the length of the side of the rhombus is 9 cm
and the perimeter of the rhombus is
p = 4a
= 4× 9 cm
= 36 cm.

Question 8.
Rectangle

Perimeter = ___
Answer:
The perimeter of the rectangle is 26 in.

Explanation:
The length of the rectangle is 5 inch
and the breadth of the rectangle is 8  inch
the perimeter of the rectangle is
p = 2( length + breadth)
= 2( 5 in+ 8 in)
= 2(13 in)
= 26 in.
So the perimeter of the rectangle is 26 in.

Question 9.
Square

Perimeter = ___
Answer:
The perimeter of the square is 28 ft.

Explanation:
The length of the square is 7 ft
so the perimeter of the square is
perimeter= 4a
= 4×7
= 28 ft.

Question 10.
Modeling Real Life
You want to put lace around the tops of the two rectangular lampshades. How many centimeters of lace do you need?

Answer:
We need 1,120 square centimeters.

Explanation:
The length of the rectangular lampshades is 35 cm
The breadth of the rectangular lampshades is 32 cm
and the area of the rectangular lampshades is
area= length×breadth
= 32×35
= 1,120 square cm.
So 1,120 square centimeters of lace you need.

15.3 Find Unknown Side Lengths

Find the unknown side length.

Question 11.
Perimeter = 22 feet

d = ____
Answer:
The length of the other side is 9 ft.

Explanation:
Given the perimeter of the above figure is 22 feet
and the length of the sides of the figure is 6 ft, 7 ft, d ft
so the perimeter of the figure is
p = 6 ft+ 7 ft+ d ft
22  ft = 13 ft + d ft
d = 22 ft – 13 ft
= 9 ft.
So, the length of the other side is 9 ft.

Question 12.
Perimeter = 31 inches

k = ___
Answer:
The length of the other side is 5 in.

Explanation:
Given the perimeter of the above figure is 31 inches
and the length of the sides of the figure is 10 in, 4 in, 12 in and k in.
so the perimeter of the figure is
p = 10 in+ 4 in+ 12 in+k in
31 in = 26 in + k in
k = 31 in – 26 in
= 5 in.
So, the length of the other side is 5 in.

Question 13.
Perimeter = 34 meters

y = ___
Answer:
The length of the other side is 7 m.

Explanation:
Given the perimeter of the above figure is 34 meters
and the length of the sides of the figure is 11 m, 8 m, 2 m, 1 m, 5 m, y m.
so the perimeter of the figure is
p = 11 m+ 8 m+ 2 m+ 1 m+ 5 m+ y m
34 m = 27 m + y m
y = 34 m – 27 m
= 7 m.
So, the length of the other side is 7 m.

Find the unknown side length.

Question 14.
Perimeter = 24 feet

k = ___
Answer:
The length of the sides of the triangle is 8 feet.

Explanation:
The perimeter of the triangle is 24 feet
and the perimeter of the triangle is
p = 3a
24 feet = 3a
a= 24/3
= 8 feet.
So, the length of the sides of the triangle is 8 feet.

Question 15.
Perimeter = 16 meters

y = ___
Answer:
The length of the sides of the square is 4 meters.

Explanation:
The perimeter of the triangle is 16 meters
and the perimeter of the triangle is
p = 4a
16 meters = 4a
a= 16/4
= 4 meters.
So, the length of the sides of the triangle is 4 meters.

Question 16.
Perimeter = 30 inches

d = ___
Answer:
The length of the sides of the pentagon is 6 inches.

Explanation:
The perimeter of the pentagon is 30 inches
and the perimeter of the pentagon is
p = 5a
30 inches = 5a
d= 30/5
= 6 inches.
So, the length of the sides of the pentagon is 6 inches.

Question 17.

Number Sense
A rectangle has a perimeter of 38 centimeters. The left side length is 10 centimeters. What is the length of the top side?
Answer:
The length of the top side is 9 cm.

Explanation:
Given the perimeter of the rectangle is 38 cm and
the left side length is 10 cm
Let the length of the top side be X, so
perimeter of the rectangle is
p = 2( length +breadth)
38 = 2(10+X)
38/2 = 10 + X
19 = 10 + X
X = 19 – 10
= 9 cm.
So the length of the top side is 9 cm.

15.4 Same Perimeter, Different Area

Question 18.
Find the perimeter and area of Rectangle A. Drawa different rectangle that has the same perimeter. Which rectangle has the greater area?

Perimeter = ____
Area = ___

Perimeter = ____
Area = ___
Rectangle ___ has the greater area.
Answer:
The perimeter of the rectangle A is 14 m
The area of the rectangle A is 10 square meters
The perimeter of the rectangle B is 14 m
The area of the rectangle B is 12 square meters
The rectangle B has greater area.

Explanation:
The length of the rectangle is 5m
and the breadth of the rectangle is 2m
the perimeter of the rectangle is
p= 2(length+breadth)
= 2(5+2)
= 2(7)
= 14 m
and the area of the rectangle is
area = length×breadth
= 2×5
= 10 square meters.

In the above image, we can see the length of the rectangle is 4 m
and the breadth of the rectangle is 3 m
so the perimeter of the rectangle is
p= 2(length+breadth)
= 2(4+3)
= 2(7)
= 14 m.
and the area of the rectangle is
area = length×breadth
= 4×3
= 12 square meters.
The rectangle B has greater area.

Question 19.
Patterns
Each Rectangle has the same perimeter. Are the areas increasing or decreasing ? Explain.

Answer:
As we can see in the above images the length of the images was increasing one by one and the breadth is decreasing, so the areas increasing or decreasing will depend upon the breadth of the rectangle. So, if the breadth is also increasing then the area will also be increasing. And if the breadth was decreasing then the area will also be decreasing.

15.5 Same Area, Different Perimeters

Question 20.
Find the area and the perimeter of Rectangle A. Drawa different rectangle that has the same area. Which rectangle has the lesser perimeter?

Area = ___
Perimeter = ___

Area = ___
Perimeter = ___
Rectangle ___ has the lesser perimeter.
Answer:
The perimeter of the rectangle A is 14 m
The area of the rectangle A is 10 square meters
The perimeter of the rectangle B is 14 m
The area of the rectangle B is 12 square meters
The rectangle B has greater area.

Explanation:
The length of the rectangle is 10 in
and the breadth of the rectangle is 5 in
the perimeter of the rectangle is
p= 2(length+breadth)
= 2(10+5)
= 2(15)
= 30 in
and the area of the rectangle is
area = length×breadth
= 10×5
= 50 square inches.

Question 21.
Reasoning
The two dirt-bike parks have the same area. Kids ride dirt bikes around the outside of each park. At which park do the kids ride farther ? Explain.

Answer:
As the length of the park B is longer, so at the park B kids rid farther than the park A.

Find Perimeter and Area Cumulative practice 1 – 15

Question 1.
A mango has a mass that is 369 grams greater than the apple. What is the mass of the mango?
A. 471 grams
B. 369 grams
C. 267 grams
D. 461 grams

Answer:
B.

Explanation:
The mass of the mango is 369 grams greater than apple

Question 2.
Which term describes two of the shapes shown, but all three of the shapes?

A. polygon
B. rectangle
C. square
D. parallelogram
Answer:
B, C, D.

Explanation:
In the above figures, we can see the parallelogram and the square. As the square is also known as a rectangle so we will choose option B also.

Question 3.
A rectangular note card has an area of 35 square inches. The length of one of its sides is 7 inches. What is the perimeter of the note card?
A. 5 inches
B. 24 inches
C. 84 inches
D. 12 inches
Answer:
The breadth of the rectangular note card is 24 inches.

Explanation:
The area of the rectangular note card is 35 square inches and the length of one of its sides is 7 inches
so the breadth of the rectangular note card is
area = length × breadth
35 = 7 × breadth
breadth = 35/7
= 5 inches.
The perimeter of the rectangular note card is
p = 2(length+breadth)
= 2(7+5)
= 2(12)
= 24 inches.
The breadth of the rectangular note card is 24 inches.

Question 4.
How many minutes are equivalent to4 hours?
A. 400 minutes
B. 240 minutes
C. 24 minutes
D. 40 minutes
Answer:
B

Explanation:
The number of minutes is equivalent to 4 hours is
4× 60= 240 minutes.

Question 5.
A balloon artist has 108 balloons. He has 72 white balloons, and an equal number of red, blue, green, and purple balloons. How many purple balloons does he have?
A. 36
B. 180
C. 9
D. 32

Answer:
9 balloons.

Explanation:
As a balloon artist has 108 balloons and he has 72 white balloons
and the remaining balloons are 108 – 72= 36 balloons
and an equal number of red, blue, green, and purple balloons
which means 36 balloons are equally divided by 4 colors of balloons, so
36÷4 = 9 balloons.
So the purple balloons are 9.

Question 6.
Which statements about the figures are true?

Answer:
The shapes have different perimeters.
The shapes have the same area.

Explanation:
The length of the side of the square is 6 in,
and the perimeter of the square is
p = 4a
= 4×6 in
= 24 in
The area of the square is a^2
= 6 in×6 in
36 in^2.

Question 7.
The graph show many students ordered each lunch option.

Part A How many students ordered lunch?
Part B Choose a lesser value for the key. How will the graph change?
Answer:
Part A: 60 students ordered lunch.
Part B: Turkey hot dog has a lesser value.

Explanation:
Part A:
The number of students who ordered lunch is
the grilled chicken was ordered by 21 students
Turkey hot dog was ordered by 9 students
A peanut butter and jelly sandwich was ordered by 12 students
the salad bar was ordered by 18 students
so the number of students who ordered lunch is
21+9+12+18= 60 students.

Part B:
The turkey hot dog was ordered by 9 students which is a lesser value.

Question 8.
Find the sum

Answer:
935

Explanation:
The sum of the above given numbers is 935

Question 9.
What is the perimeter of the figure?

A. 26 units
B. 22 units
C. 20 units
D. 16 unit
Answer:
B

Explanation:
The sides of the figure is 2,4,1,2,1,1,2,2,1,1,2,1,1,1
and the perimeter of the figure is
p = 2+4+1+2+1+1+2+2+1+1+2+1+1+1
= 22 units.

Question 10.
Which bar graph correctly shows the data?

Answer:
Graph B.

Explanation:
Graph B shows the correct graph data.

Question 11.
Which polygons have at least one pair of parallel sides?

Answer:
Red color polygon.

Explanation:
The red color polygon has one pair of parallel sides, as it is a trapezoid.

Question 12.
The perimeter of the polygon is 50 yards. What is the missing side length?
A. 41 yards
B. 10 yards
C. 91 yards
D. 9 yards

Answer:
The missing side length is 9 yards.

Explanation:
Given the perimeter of the polygon is 50 yards
Let the missing side length be X yd
and the lengths of the sides of the polygon is 15 yds, 6 yds, 13 yds, 7 yds, and X yd,
So the perimeter of the polygon is
p = 15 yd+6 yd+13 yd+ 7 yd+ X yd
50 yards = 41 yards + X Yards
X = 9 yards.
So the missing side length is 9 yards.

Question 13.
Which line plot correctly shows the data?

Answer:
A

Explanation:
Option A line plot shows the correct data.

Question 14.
Your friend is asked to draw a quadrilateral with four right angles. She says it can only be a square. Is she correct?
A. Yes, there is no other shape it can be.
B. No, it could also be a rectangle.
C. No, it could also be a hexagon.
D. No, it could also be a trapezoid.
Answer:
Yes, there is no other shape it can be.

Explanation:
Yes, she is correct. There is no other shape than the square with four right angles.

Question 15.
Which numbers round to480 when rounded to the nearest ten?

Answer:
484, 480, 478

Explanation:
The numbers that are rounded to 480 to the nearest ten is 484, 480, 478.

Find Perimeter and Area Cumulative Steam Performance Task 1 – 15

Question 1.
Use the Internet or some other resource to learn more about crested geckos.
a. Write three interesting facts about geckos.
b. Geckos need to drink water every day. Is this amount of water milliliters or liters? Explain.
c. Geckos can live in a terrarium. Is the capacity of this terrarium milliliters liters measured in or?

Answer:
a) The three interesting facts about geckos are:
i) Geckos are a type of lizards and their toes help them to stick to any surface except Teflon.
ii) Gecko’s eyes are 350 times more sensitive than human eyes to light.
iii) Some of the pieces of Geckos have no legs and look more like snakes.

b) The number of water Geckos will have is in milliliters only as Geckos will not often drink water.

c)Yes, geckos can live in a terrarium and the capacity of this terrarium is between 120 liters to 200 liters.

Question 2.
Your class designs a terrarium for a gecko.
a. The base of the terrarium is a hexagon. Each side of the hexagon is 6 inches long. What is the perimeter of the base?
b. The terrarium is 20 inches tall. All of the side walls are made of glass. How many square inches of glass is needed for the terrarium?
c. Another class designs a terrarium with a rectangular base. All of its sides are equal in length. The base has the same perimeter as the base your class designs. What is the perimeter of the base? What is the area?

Answer:
a. 36 inches.
b.

Explanation:

a.
Given the length of the sides of the hexagon is 6 inches, and
the perimeter of the hexagon is
p = 6a
= 6 × 6
= 36 inches.

Question 3.
An online store sells crested geckos. The store owner measures the length of each gecko in the store. The results are shown in the table.

a. Use the table to complete the line plot.

b. How many geckos did the store owner measure?
c. What is the difference in the lengths of the longest gecko and the shortest gecko?
d. How many geckos are shorter than 6$$\frac{1}{4}$$ inches?
e.The length of a gecko’s tail is about 3 inches. How would the line plot change if the store owner measured the length of each gecko without its tail?
Answer:
b. The number of geckos the store owner measures is 24.

d. 12

Explanation:

a.

b. The number of geckos the store owner measures is 24.

d. The number of geckos shorter than 6$$\frac{1}{4}$$ inches are 12.

Conclusion:

We hope that the details provided here regarding Big Ideas Math Answers Grade 3 Chapter 15 Find Perimeter and Area is helpful to the students for better preparation. One who wants to become a pro in maths can download BIM Book Grade 3 Chapter 15 Find Perimeter and Area Answers in pdf format. Bookmark our site to get the answers for other chapters of grade 3.

Big Ideas Math Answers Grade 7 Chapter 6 Percents

Download Big Ideas Math Book 7th Grade Answer Key Chapter 6 Percents Pdf for free of cost. If you are willing to be perfect in all concepts and math skills, then you must follow the syllabus given here. To score the best marks in the exam, students must learn all the conceptualized lessons on Simple Interest, Discounts, and also the Subject knowledge which will be helpful to improve practical skills.

Big Ideas Math Answers Grade 7 Chapter 6 Percents helps you to cross all the hurdle times while studying. You can easily grasp various concepts and problems on all the concepts given below. Get all the skills and different methods of solving problems easily and quickly. It helps to solve real-life calculations in a very dignified and smooth manner. Follow the below sections to get BIM Grade 7 Chapter 6 Percents problems and solutions.

Big Ideas Math Book 7th Grade Answer Key Chapter 6 Percents

Big ideas Math Book 7th Grade Answer Key Chapter 6 percents pdf is given here. It gives the most accurate answers to all the problems which are related to the given chapter. This chapter of percents contains various topics like Fractions, Decimals and Percents, The Percent Proportion, The Percent Equation, Percents of Increase and Decrease, Discounts and Markups, Simple Interest, and so on.

With the help of Big Ideas Math Book 7th Grade Solution Key Chapter 6 Percents, you can prepare the timetable. With the help of the timetable prepared, you can easily manage your syllabus and include all the topics into it as per the time remaining. In the further sections, you will know all the concepts that are present in Grade 7 Chapter 6 Percents. Check out all the information and kickstart your preparation.

STEAM Video/Performance Task

Getting Ready for Chapter 6

Lesson 1 : Fractions, Decimals, and Percents

Lesson 2 : The Percent Proportion

Lesson 3 : The Percent Equation

Lesson 4 : Percents of Increase and Decrease

Lesson 5 : Discounts and Markups

Lesson 6 : Simple Interest

Percents Connecting Concepts

STEAM Video/Performance Task

STEAM Video

Tornado!
More tornadoes occur each year in the United States than in any other country. How can you use a percent to describe the portion of tornadoes in the United States that occur in your state?

Watch the STEAM Video “Tornado!” Then answer the following questions.
1. The map below shows the average annual number of tornadoes in each state. Which regions have the most tornadoes? the fewest tornadoes?

2. Robert says that only Alaska, Hawaii, and Rhode Island average less than 1 tornado per year. What percent of states average more than 1 tornado per year?

Performance Task

Tornado Alley
After completing this chapter, you will be able to use the concepts you learned to answer the questions in the STEAM Video Performance Task. You will be given information about the average annual numbers of tornadoes in several states over a 25-year period. For example:

You will be asked to solve various percent problems about tornadoes. Why is it helpful to know the percent of tornadoes that occur in each state?

Getting Ready for Chapter 6

Chapter Exploration
Work with a partner. Write the percent of the model that is shaded. Then write the percent as a decimal.

Answer : The Percent and decimal for the given models are
1. percent = 30%, decimal = 0.3
2. percent = 100%, decimal = 1
3. percent = 33%, decimal = 0.33
4. percent = 50%, decimal = 0.5
5. percent = 40%, decimal = 0.4
6. percent = 64%, decimal = 0.64
7. percent = 60%, decimal = 0.6

Explanation:
All the models given here are 10 by 10 grid forming a Square of 100 equal sections.
This entire square represents a whole and the shaded part is fraction.
Each of these shaded squares represents 1/100. So by using this data we have ,
1.    =
From the shaded part and the whole, we have the fraction of $$\frac{30}{100}$$,
Then the percent will be $$\frac{30}{100}$$ = 30%,
By rewriting it in decimal form we have 0.3

2.  =
From the shaded part and the whole, we have the fraction of $$\frac{100}{100}$$,
Then the percent will be $$\frac{100}{100}$$ = 100%,
By rewriting it in decimal form we have 1

3.  =
From the shaded part and the whole, we have the fraction of $$\frac{33}{100}$$,
Then the percent will be $$\frac{33}{100}$$ = 33%,
By rewriting it in decimal form we have 0.33

4.  =
From the shaded part and the whole, we have the fraction of $$\frac{50}{100}$$,
Then the percent will be $$\frac{50}{100}$$ = 50%,
By rewriting it in decimal form we have 0.5

5.  =
From the shaded part and the whole, we have the fraction of $$\frac{40}{100}$$,
Then the percent will be $$\frac{40}{100}$$ = 40%,
By rewriting it in decimal form we have 0.4

6.  =
From the shaded part and the whole, we have the fraction of $$\frac{64}{100}$$,
Then the percent will be $$\frac{64}{100}$$ = 64%,
By rewriting it in decimal form we have 0.64

7.  =
From the shaded part and the whole, we have the fraction of $$\frac{60}{100}$$,
Then the percent will be $$\frac{60}{100}$$ = 60%,
By rewriting it in decimal form we have 0.6

8. WRITE A PROCEDURE Work with a partner. Write a procedure for rewriting a percent as a decimal. Use examples to justify your procedure.

Answer:
Let us say that the fraction be $$\frac{44}{100}$$,
Then its percentage will be 44%,
To rewrite it as decimal, we divide 44 by 100 we get 0.44 (a decimal number). So, to convert from percent to decimal divide by 100 and remove the “%” sign.
We get 0.44 as decimal of 44%.

Vocabulary
The following vocabulary terms are defined in this chapter. Think about what each term might mean and record your thoughts.
percent of change
percent of decrease
discount
percent of increase
percent error
markup

Answer:
percent of change:
Percentage Change is all about comparing old values to new values.

percent of decrease:
percent of decrease is a measure of percent change, which is the extent to which something loses value. or
A negative percent of change indicates a decrease from the original value to the second value.

discount: A reduction of price  is known as discount .Sometimes discounts are in percent, such as a 10% discount, and then you need to do a calculation to find the price reduction.

percent of increase :
Percent increase is a measure of percent change, which is the extent to which something gains value. or
A positive percent of change indicates an increase from the original value to the second value.

percent error :
Percent error is the difference between estimated value and the actual value in comparison to the actual value and is expressed as a percentage.

markup:
Markup is all about how much a retailer increases the price over what they paid for it to buy the product or item in order to which is how they make money to pay for all their costs and hopefully make a profit.

Lesson 6.1 Fractions, Decimals, and Percents

EXPLORATION 1

Comparing Numbers in Different Forms
Work with a partner. Determine which number is greater. Explain your method.

Answer:
a. 7% sales tax is greater than 5% sales tax
b. 0.37 cup of flour is greater than 0.33 cup of flour
c. 0.625 inch wrench is greater than 0.375 inch wrench
d. 12.6 dollars are greater than 12.56 dollars
e. 5.83 fluid ounces is greater than 5.6 fluid ounces

Explanation:
a. 7% sales tax or $$\frac{1}{20}$$ sales tax
By using the method of converting fraction into Percent and comparing ,
We can rewrite $$\frac{1}{20}$$ as  0.05 in decimal form,
To get get the percent , multiply 100 to 0.05, then we get 5%.
So, $$\frac{1}{20}$$ can be write as 5%,
Finally, by comparing two values 7% sales tax is greater than 5% sales tax

b. 0.37 cup of flour or $$\frac{1}{3}$$ cup for flour
By using the method of converting fraction into decimal and comparing ,
We can rewrite $$\frac{1}{3}$$ as  0.33 in decimal form
Finally, by comparing two values 0.37 cup of flour is greater than 0.33 cup of flour.

c. $$\frac{5}{8}$$ inch wrench or 0.375 inch wrench
By using the method of converting fraction into decimal and comparing ,
We can rewrite $$\frac{5}{8}$$ by dividing 5 by 8, we have  0.625 in decimal form
Finally, by comparing two values 0.625 inch wrench is greater than 0.375 inch wrench.

d. $$12$$ $${\Large\frac{3}{5}}$$ dollars or 12.56 dollars
By using the method of converting fraction into decimal and comparing ,
We can rewrite fraction$$\frac{63}{5}$$ by dividing 63 by 5, we have 12.6 in decimal form
Finally, by comparing two values 12.6 dollars are greater than 12.56 dollars

e.  $$5$$ $${\Large\frac{5}{6}}$$ fluid ounces or 5.6 fluid ounces
By using the method of converting fraction into decimal and comparing ,
We can rewrite fraction $$\frac{35}{6}$$  by dividing 35 by 6, we have 5.83 in decimal form
Finally, by comparing two values 5.83 fluid ounces is greater than 5.6 fluid ounces

EXPLORATION 2

Work with a partner and follow the steps below.

• Write five different numbers on individual slips of paper. Include at least one decimal, one fraction, and one percent.
• On a separate sheet of paper, create an answer key that shows your numbers written from least to greatest.
• Exchange slips of paper with another group and race to order the numbers from least to greatest. Then exchange answer keys to check your orders.

Answer: 78%, 0.95, $$\frac{83}{45}$$, 6, 21

Explanation:
Let the numbers be 6,  21, 0.95, $$\frac{83}{45}$$, 78%
In order to put them in an ascending order we have to calculate the fraction, percent, decimal in their respective form, So Fraction $$\frac{83}{45}$$ can be rewrite as 1.84 in decimal form
Then 78% can be rewrite as 0.78 in decimal form,
As we can see 0.78 is less than 0.95 , 0.95 is less than $$\frac{83}{45}$$,$$\frac{83}{45}$$ is less than 6, 6 is less than 21,
Finally, we have the ascending order as 78%, 0.95, $$\frac{83}{45}$$, 6, 21.

Try It

Write the percent as a decimal or the decimal as a percent. Use a model to represent the number.
Question 1.
39%
Answer: 0.39

Explanation:
To write percent into decimal we have to remove percent symbol and divide by 100, which moves the decimal point two places to the left.
So, 39% in decimal form is 0.39

Question 2.
12. 6 %
Answer: 0.126

Explanation:
To write percent into decimal we have to remove percent symbol and divide by 100, which moves the decimal point two places to the left.
So, 12.6% in decimal form is 0.126

Question 3.
0.05
Answer: 5%

Explanation:
To write decimal into percent we should multiply the number by 100, which moves the decimal point two places to the right and add a percent symbol.
So, 0.05 can be rewrite as 5%

Question 4.
1.25
Answer:  125%

Explanation:
To write decimal into percent we should multiply the number by 100, which moves the decimal point two places to the right and add a percent symbol.
So, 1.25 can be rewrite as 125%

Write the fraction as a decimal and a percent.
Question 5.
$$\frac{5}{8}$$
Answer: 0.625 or 62.5%

Explanation:
To get the percent or decimal from  fraction $$\frac{5}{8}$$ we have to divide 5 by 8 ,
Then, we get 0.625,
To get the percent of 0.625 multiply by 100 , it will be 62.5%
So, $$\frac{5}{8}$$ can be written as  0.625 or 62.5%

Question 6.
$$\frac{1}{6}$$
Answer: 0.166 or 16.6%

Explanation:
To get the percent or decimal from  fraction $$\frac{1}{6}$$ we have to divide 1 by 6 ,
Then, we get 0.166,
To get the percent of 0.166 multiply by 100 , it will be 16.6%
So, $$\frac{1}{6}$$ can be written as  0.166 or 16.6%

Question 7.
$$\frac{11}{3}$$
Answer:  3.66 or 366%

Explanation:
To get the percent or decimal from  fraction $$\frac{11}{3}$$ we have to divide 11 by 3 ,
Then, we get 3.66,
To get the percent of 3.66 multiply by 100 , it will be 366%
So, $$\frac{11}{3}$$ can be written as  3.66 or 366%

Question 8.
$$\frac{3}{1000}$$
Answer: 0.003 or 0.3%

Explanation:
To get the percent or decimal from  fraction $$\frac{3}{1000}$$ we have to divide 3 by 1000 ,
Then, we get 0.003,
To get the percent of 0.003 multiply by 100 , it will be 0.3%
So, $$\frac{3}{1000}$$ can be written as  0.003 or 0.3%

Self-Assessment for Concepts & Skills
Solve each exercise. Then rate your understanding of the success criteria in your journal.

CONVERTING BETWEEN PERCENTS AND DECIMALS Write the percent as a decimal or the decimal as a percent. Use a model to represent the number.
Question 9.
46%
Answer: 0.46

Explanation:
To write percent into decimal we have to remove percent symbol and divide by 100, which moves the decimal point two places to the left.
So, 46% in decimal form is 0.46

Question 10.
$$66 . \overline{6} \%$$
Answer: $$0 .66 \overline{6}$$

Explanation:
To write percent into decimal we have to remove percent symbol and divide by 100, which moves the decimal point two places to the left.
So, $$66 . \overline{6} \%$$ in decimal form is $$0 .66 \overline{6}$$

Question 11.
0.18
Answer: 18%

Explanation:
To write decimal into percent we should multiply the number by 100, which moves the decimal point two places to the right and add a percent symbol.
So, 0.18 can be rewrite as 18%

Question 12.
$$2 . \overline{3}$$
Answer: $$233 . \overline{3} \%$$

Explanation:
To write decimal into percent we should multiply the number by 100, which moves the decimal point two places to the right and add a percent symbol.
So, $$2 . \overline{3}$$ can be rewrite as $$233 . \overline{3} \%$$

WRITING FRACTIONS AS DECIMALS AND PERCENTS Write the fraction as a decimal and a percent.
Question 13.
$$\frac{7}{10}$$
Answer: decimal = 0.7, percent = 70%

Explanation:
By using the method of converting fraction into decimal and comparing ,
We can rewrite $$\frac{7}{10}$$ as  0.7 in decimal form,
To write decimal into percent we should multiply the number by 100, which moves the decimal point two places to the right and add a percent symbol , then 0.7 can be rewrite as 70%
So, $$\frac{7}{10}$$ in decimal = 0.7, percent = 70%

Question 14.
$$\frac{5}{9}$$
Answer: decimal = $$0 .\overline{5}$$, percent = $$55 . \overline{5} \%$$

Explanation:
By using the method of converting fraction into decimal and comparing ,
We can rewrite $$\frac{5}{9}$$ as  $$0 .\overline{5}$$ in decimal form,
To write decimal into percent we should multiply the number by 100, which moves the decimal point two places to the right and add a percent symbol , then $$0 .\overline{5}$$ can be rewrite as $$55 . \overline{5} \%$$
So, $$\frac{5}{9}$$ in decimal = $$0 .\overline{5}$$, percent = $$55 . \overline{5} \%$$

Question 15.
$$\frac{7}{2000}$$
Answer: decimal = 0.0035, percent = 0.35%

Explanation:
By using the method of converting fraction into decimal and comparing ,
We can rewrite $$\frac{7}{2000}$$ as  0.0035 in decimal form,
To write decimal into percent we should multiply the number by 100, which moves the decimal point two places to the right and add a percent symbol , then 0.0035 can be rewrite as 0.35%
So, $$\frac{7}{2000}$$ in decimal = 0.0035, percent = 0.35%

Question 16.
$$\frac{17}{15}$$
Answer: decimal = $$1.1 \overline{3}$$ , percent = $$113 . \overline{3} \%$$

Explanation:
By using the method of converting fraction into decimal and comparing ,
We can rewrite $$\frac{17}{15}$$ as $$1.1 \overline{3}$$ in decimal form,
To write decimal into percent we should multiply the number by 100, which moves the decimal point two places to the right and add a percent symbol , then $$1.1 \overline{3}$$ can be rewrite as $$113 . \overline{3} \%$$
So, $$\frac{17}{15}$$ in decimal = $$1.1 \overline{3}$$, percent = $$113 . \overline{3} \%$$

Self-Assessment for Problem Solving
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 17.
An astronaut spends 53% of the day working, 0.1 of the day eating, $$\frac{3}{10}$$ of the day sleeping, and the rest of the day exercising. Order the events by duration from least to greatest. Justify your answer.

Answer:  An astronaut spends the day 7% by exercising, 10% by eating, 30% by sleeping, 53% by working.

Explanation:
An astronaut spends 53% of the day working,
0.1 of the day eating, in terms of percent we can write it as 10%,
$$\frac{3}{10}$$ of the day sleeping, in decimals we ca rewrite as 0.3 and in percent it will be 30%,
Let us say that the whole day be 100% , The sum of the works he is doing in percent we get,
53% + 10% + 30% = 93%, and
Given that the rest of the day exercising, so 100% – 93% = 7%, A whole day is completed with these works.
To put them in Order the events by duration from least to greatest, we have , 7%, 10%, 30%, 53%.
An astronaut spends the day 7% by exercising, 10% by eating, 30% by sleeping, 53% by working.

Question 18.
DIG DEEPER!
A band plays one concert in Arizona, one concert in California, and one concert in Georgia. In California, the band earned $$\frac{3}{2}$$ the profit that they earned in Arizona. Of the total profit earned by the band, 32% is earned in Arizona. How many times more money did the band earn at the most profitable concert than at the least profitable concert? Justify your answer.
Answer:

Explanation:

Fractions, Decimals, and Percents Homework & Practice 6.1

Review & Refresh

Find the missing dimension. Use the scale 1 : 15.

Answer: The model height of the Figure skater is 4.5inches and The Actual length of pipe is 75 feet .

Explanation:
Given , to use the scale oof 1 : 15 , Let the model height  be x
The model height of Figure skater is $$\frac{1}{15}$$ = $$\frac{x}{67.5}$$
15x = 67.5
x = $$\frac{67.5}{15}$$
x = 4.5
So , The model height of the Figure skater is 4.5inches

Let the Actual length is y
The Actual length of pipe is $$\frac{1}{15}$$ = $$\frac{5}{y}$$
y = 15 × 5 = 75
So , The Actual length of pipe is 75 feet .

Simplify the expression.
Question 3.
2(3p – 6) + 4p
Answer: p = 1.2

Explanation:
Let us say that whole expression is equal to 0
2(3p – 6) + 4p = 0
[2(3p) – 2(6)] + 4p = 0
6p – 12 + 4p = 0
10p – 12 = 0
p = 12/10 = 1.2
So, p = 1.2

Question 4.
5n – 3(4n + 1)
Answer: n = -0.42

Explanation:
Let us say that whole expression is equal to 0
5n – 3(4n + 1) = 0
5n – [ 3(4n) + 3(1) ] = 0
5n – 12n – 3 = 0
– 3 – 7n = 0
7n = – 3
n = -3/7 = -0.42
So, n = -0.42

Question 5.
What is the solution of 2n – 4 > – 12?
A. n < – 10
B. n < – 4
C. n > – 2
D. n > – 4
Answer:  D . n > -4

Explanation:
Given, 2n – 4 > – 12
add 4 in both sides,
2n – 4 + 4 > – 12 + 4
2n > – 8
divide both sides by 2
2n/2 > -8/2
n > – 4 .

Concepts, Skills, & Problem Solving
COMPARING NUMBERS IN DIFFERENT FORMS Determine which number is greater. Explain your method. (See Exploration 1, p. 235.)
Question 6.
4$$\frac{2}{5}$$ tons or 4.3 tons
Answer: 4$$\frac{2}{5}$$ tons is greater than 4.3 tons

Explanation:
By using the method of converting fraction into decimal and comparing ,
We can rewrite fraction $$\frac{22}{5}$$  by dividing 22 by 5, we have 4.4 in decimal form
Finally, by comparing two values 4$$\frac{2}{5}$$ tons is greater than 4.3 tons

Question 7.
82% success rate or $$\frac{5}{6}$$ success rate
Answer: $$\frac{5}{6}$$ success rate is greater than 82% success rate

Explanation:
By using the method of converting fraction into Percent and comparing ,
We can rewrite $$\frac{5}{6}$$  as  0.833 in decimal form,
To get get the percent , multiply 100 to 0.833, then we get 83.3%.
So, $$\frac{5}{6}$$  can be write as 83.3%,
Finally, by comparing two values $$\frac{5}{6}$$ success rate is greater than 82% success rate

CONVERTING BETWEEN PERCENTS AND DECIMALS Write the percent as a decimal or the decimal as a percent. Use a model to represent the number.
Question 8.
26%
Answer: 0.26

Explanation:
To write percent into decimal we have to remove percent symbol and divide by 100, which moves the decimal point two places to the left.
So, 26% in decimal form is 0.26

Question 9.
0.63
Answer: 63%

Explanation:
To write decimal into percent we should multiply the number by 100, which moves the decimal point two places to the right and add a percent symbol.
So, 0.63 can be rewrite as 63%

Question 10.
9%
Answer: 0.09

Explanation:
To write percent into decimal we have to remove percent symbol and divide by 100, which moves the decimal point two places to the left.
So, 9% in decimal form is 0.09

Question 11.
0.6
Answer: 60%

Explanation:
To write decimal into percent we should multiply the number by 100, which moves the decimal point two places to the right and add a percent symbol.
So, 0.6 can be rewrite as 60%

Question 12.
44.7%
Answer: 0.447

Explanation:
To write percent into decimal we have to remove percent symbol and divide by 100, which moves the decimal point two places to the left.
So, 44.7% in decimal form is 0.447

Question 13.
55%
Answer: 0.55

Explanation:
To write percent into decimal we have to remove percent symbol and divide by 100, which moves the decimal point two places to the left.
So, 55% in decimal form is 0.55

Question 14.
$$39 . \overline{2} \%$$
Answer: $$0.39 \overline{2}$$

Explanation:
To write percent into decimal we have to remove percent symbol and divide by 100, which moves the decimal point two places to the left.
So, $$39 . \overline{2} \%$$ in decimal form is $$0.39 \overline{2}$$

Question 15.
3.554
Answer: 355.4%

Explanation:
To write decimal into percent we should multiply the number by 100, which moves the decimal point two places to the right and add a percent symbol.
So, 3.554  can be rewrite as 355.4%

Question 16.
123%
Answer: 1.23

Explanation:
To write percent into decimal we have to remove percent symbol and divide by 100, which moves the decimal point two places to the left.
So, 123% in decimal form is 1.23

Question 17.
0.041
Answer: 4.1%

Explanation:
To write decimal into percent we should multiply the number by 100, which moves the decimal point two places to the right and add a percent symbol.
So, 0.041 can be rewrite as 4.1%

Question 18.
0.122
Answer: 12.2%

Explanation:
To write decimal into percent we should multiply the number by 100, which moves the decimal point two places to the right and add a percent symbol.
So, 0.122 can be rewrite as 12.2%

Question 19.
$$49 . \overline{92} \%$$
Answer: $$0.49 \overline{92}$$

Explanation:
To write percent into decimal we have to remove percent symbol and divide by 100, which moves the decimal point two places to the left.
So, $$49 . \overline{92} \%$$ in decimal form is $$0.49 \overline{92}$$

Question 20.
YOU BE THE TEACHER
Your friend writes $$49 . \overline{8} \%$$ as a decimal. Is your friend correct? Explain your reasoning.

Answer: He wrote the decimal for $$4. \overline{8} \%$$ instead of $$49 . \overline{8} \%$$

Explanation:
To write percent into decimal we have to remove percent symbol and divide by 100, which moves the decimal point two places to the left.
So, $$49 . \overline{8} \%$$  in decimal form is $$0.49 \overline{8}$$

WRITING FRACTIONS AS DECIMALS AND PERCENTS Write the fraction as a decimal and a percent.
Question 21.
$$\frac{29}{100}$$
Answer: decimal = 0.29, percent = 29%

Explanation:
By using the method of converting fraction into decimal and comparing ,
We can rewrite $$\frac{29}{100}$$ as  0.29 in decimal form,
To write decimal into percent we should multiply the number by 100, which moves the decimal point two places to the right and add a percent symbol , then 0.29 can be rewrite as 29%
So, $$\frac{29}{100}$$ in decimal = 0.29, percent = 29%

Question 22.
$$\frac{3}{4}$$
Answer: decimal = 0.75, percent = 75%

Explanation:
By using the method of converting fraction into decimal and comparing ,
We can rewrite$$\frac{3}{4}$$ as  0.75 in decimal form,
To write decimal into percent we should multiply the number by 100, which moves the decimal point two places to the right and add a percent symbol , then 0.75 can be rewrite as 75%
So, $$\frac{3}{4}$$ in decimal = 0.75, percent = 75%

Question 23.
$$\frac{7}{8}$$
Answer: decimal = 0.875, percent = 87.5%

Explanation:
By using the method of converting fraction into decimal and comparing ,
We can rewrite $$\frac{7}{8}$$ as  0.875 in decimal form,
To write decimal into percent we should multiply the number by 100, which moves the decimal point two places to the right and add a percent symbol , then 0.875 can be rewrite as 87.5%
So, $$\frac{7}{8}$$ in decimal = 0.875, percent = 87.5%

Question 24.
$$\frac{2}{3}$$
Answer: decimal = $$0. \overline{6}$$, percent = $$66. \overline{6} \%$$

Explanation:
By using the method of converting fraction into decimal and comparing ,
We can rewrite $$\frac{2}{3}$$ as $$0. \overline{6}$$ in decimal form,
To write decimal into percent we should multiply the number by 100, which moves the decimal point two places to the right and add a percent symbol , then$$0. \overline{6}$$ can be rewrite as $$66. \overline{6} \%$$
So, $$\frac{2}{3}$$ in decimal =$$0. \overline{6}$$, percent = $$66. \overline{6} \%$$

Question 25.
$$\frac{7}{9}$$
Answer: decimal = 0.77, percent = 77.7%

Explanation:
By using the method of converting fraction into decimal and comparing ,
We can rewrite$$\frac{7}{9}$$ as  0.77 in decimal form,
To write decimal into percent we should multiply the number by 100, which moves the decimal point two places to the right and add a percent symbol , then 0.77 can be rewrite as 77.7%
So, $$\frac{7}{9}$$ in decimal = 0.77, percent = 77.7%

Question 26.
$$\frac{12}{5}$$
Answer: decimal = 2.4, percent = 240%

Explanation:
By using the method of converting fraction into decimal and comparing ,
We can rewrite $$\frac{12}{5}$$ as  2.4 in decimal form,
To write decimal into percent we should multiply the number by 100, which moves the decimal point two places to the right and add a percent symbol , then 2.4 can be rewrite as 240%
So, $$\frac{12}{5}$$ in decimal =2.4, percent = 240%

Question 27.
$$\frac{9}{2}$$
Answer: decimal = 4.5, percent = 450%

Explanation:
By using the method of converting fraction into decimal and comparing ,
We can rewrite$$\frac{9}{2}$$ as 4.5 in decimal form,
To write decimal into percent we should multiply the number by 100, which moves the decimal point two places to the right and add a percent symbol , then 4.5 can be rewrite as 450%
So, $$\frac{9}{2}$$ in decimal = 4.5, percent = 450%

Question 28.
$$\frac{1}{1000}$$
Answer: decimal = 0.0010, percent = 0.10%

Explanation:
By using the method of converting fraction into decimal and comparing ,
We can rewrite $$\frac{1}{1000}$$ as  0.0010 in decimal form,
To write decimal into percent we should multiply the number by 100, which moves the decimal point two places to the right and add a percent symbol , then 0.0010 can be rewrite as 0.10%
So, $$\frac{1}{1000}$$ in decimal = 0.0010, percent = 0.10%

Question 29.
$$\frac{17}{6}$$
Answer: decimal = $$2.8 \overline{3}$$, percent = $$283 . \overline{3} \%$$

Explanation:
By using the method of converting fraction into decimal and comparing ,
We can rewrite $$\frac{17}{6}$$ as $$2.8 \overline{3}$$ in decimal form,
To write decimal into percent we should multiply the number by 100, which moves the decimal point two places to the right and add a percent symbol , then$$2.8 \overline{3}$$ can be rewrite as $$283 . \overline{3} \%$$
So, $$\frac{17}{6}$$ in decimal = $$2.8 \overline{3}$$, percent = $$283 . \overline{3} \%$$

Question 30.
$$\frac{3}{11}$$
Answer: decimal = 0.27, percent = 27%

Explanation:
By using the method of converting fraction into decimal and comparing ,
We can rewrite $$\frac{3}{11}$$ as  0.27 in decimal form,
To write decimal into percent we should multiply the number by 100, which moves the decimal point two places to the right and add a percent symbol , then 0.27 can be rewrite as 27%
So, $$\frac{3}{11}$$ in decimal = 0.27, percent = 27%

Question 31.
$$\frac{1}{750}$$
Answer: decimal = $$0.001 \overline{3}$$, percent =$$0.1 \overline{3} \%$$

Explanation:
By using the method of converting fraction into decimal and comparing ,
We can rewrite $$\frac{1}{750}$$ as  $$0.001 \overline{3}$$ in decimal form,
To write decimal into percent we should multiply the number by 100, which moves the decimal point two places to the right and add a percent symbol , then $$0.001 \overline{3}$$ can be rewrite as $$0.1 \overline{3} \%$$
So, $$\frac{1}{750}$$ in decimal = $$0.001 \overline{3}$$, percent = $$0.1 \overline{3} \%$$

Question 32.
$$\frac{22}{9}$$
Answer: decimal = $$2. \overline{4}$$, percent = $$244 . \overline{4} \%$$

Explanation:
By using the method of converting fraction into decimal and comparing ,
We can rewrite $$\frac{22}{9}$$ as  $$2. \overline{4}$$ in decimal form,
To write decimal into percent we should multiply the number by 100, which moves the decimal point two places to the right and add a percent symbol , then $$2. \overline{4}$$ can be rewrite as $$244 . \overline{4} \%$$
So, $$\frac{22}{9}$$ in decimal = $$2. \overline{4}$$, percent = $$244 . \overline{4} \%$$

PRECISION Order the numbers from least to greatest.
Question 33.
66.1%, 0.66, $$\frac{2}{3}$$, 0.667
Answer: 0.66, 66.1%, $$\frac{2}{3}$$, 0.667

Explanation:
In order to put them in an ascending order we have to calculate the fraction, percent, decimal in their respective form, So Fraction $$\frac{2}{3}$$ can be rewrite as $$0. \overline{6}$$ in decimal form
Then 66.1% can be rewrite as 0.661 in decimal form,
As we can see 0.66 is less than 66.1% , 66.1% is less than $$\frac{2}{3}$$, $$\frac{2}{3}$$  is less than 0.667,
Finally, we have the ascending order as 0.66, 66.1%, $$\frac{2}{3}$$, 0.667.

Question 34.
$$\frac{2}{9}$$, 21%, $$0.2 \overline{1}$$, $$\frac{11}{50}$$
Answer: 21%, $$0.2 \overline{1}$$, $$\frac{11}{50}$$, $$\frac{2}{9}$$

Explanation:
In order to put them in an ascending order we have to calculate the fraction, percent, decimal in their respective form, So Fraction $$\frac{11}{50}$$ can be rewrite as 0.22 in decimal form,
$$\frac{2}{9}$$ can be rewrite as $$0.22 \overline{2}$$in decimal form,
Then 21% can be rewrite as 0.21 in decimal form,
As we can see 21%  is less than $$0.2 \overline{1}$$ ,$$0.2 \overline{1}$$ is less than $$\frac{11}{50}$$,  $$\frac{11}{50}$$  is less than $$\frac{2}{9}$$,
Finally, we have the ascending order as 21%, $$0.2 \overline{1}$$, $$\frac{11}{50}$$, $$\frac{2}{9}$$

MATCHING Tell which letter shows the graph of the number.
Question 35.
$$\frac{7}{9}$$
Answer: decimal = 0.777 , it is in the graph at the point A

Explanation:
By using the method of converting fraction into decimal ,
We can rewrite $$\frac{7}{9}$$ as 0.777 in decimal form,
So, looking at the graph given, it is at point A.

Question 36.
0.812
Answer:  it is at the point C in the given graph.

Question 37.
$$\frac{5}{6}$$
Answer: decimal = 0.833 , it is in the graph at the point D

Explanation:
By using the method of converting fraction into decimal ,
We can rewrite $$\frac{5}{6}$$ as 0.833 in decimal form,
So, looking at the graph given, it is at point D.

Question 38.
79.5%
Answer: 0.795, it is in the graph at the point B

Explanation:
To write percent into decimal we have to remove percent symbol and divide by 100, which moves the decimal point two places to the left.
So, 79.5% in decimal form is 0.795
0.795 is in the graph given , at the point B vb

Question 39.
PROBLEM SOLVING
The table shows the portion of students in each grade that participate in School Spirit Week. Order the grades by portion of participation from least to greatest.

Answer: The grades by portion of participation from least to greatest are 7 , 6 , 8.

Explanation:
To write percent into decimal we have to remove percent symbol and divide by 100, which moves the decimal point two places to the left. So, 65% in decimal form is 0.65.
By using the method of converting fraction into Percent  ,
We can rewrite $$\frac{5}{6}$$ as 0.6 in decimal form,
Then grade 6 = 0.64 , grade 7 = 0.6 , grade 8 = 0.65 ,
So, The grades by portion of participation from least to greatest are 7 , 6 , 8.

Question 40.
MODELING REAL LIFE
The table shows the portion of gold medals that were won by the United States in five summer Olympic games. In what year did the United States win the least portion of gold medals? the greatest portion? Justify your answers.

Answer: The United States win the least portion of gold medals in the year 2004 and the greatest portion of gold medals won in the year 2012

Explanation:
By using the method of converting fraction into Percent and comparing ,
We can rewrite $$\frac{36}{301}$$  as  0.119 in decimal form,
$$\frac{23}{150}$$ as 0.153 in decimal form,
$$\frac{46}{307}$$ as 0.149 in decimal form,
To write percent into decimal we have to remove percent symbol and divide by 100, which moves the decimal point two places to the left.
$$12. \overline{3} \%$$ as $$0.12 \overline{3}$$ in decimal form.
So, according to their years and portions of gold medals we have ,
year 2000 – $$0.12 \overline{3}$$,
year 2004 – 0.119,
year 2008 – $$0. \overline{12}$$,
year 2012 – 0.153,
year 2016 – 0.149,
Finally, The United States win the least portion of gold medals in the year 2004 and the greatest portion of gold medals won in the year 2012.

Question 41.
PROBLEM SOLVING
You, your friend, and your cousin have a basketball competition where each person attempts the same number of shots. You make 70% of your shots, your friend makes of her shots, $$\frac{7}{9}$$ and your cousin makes $$0.7 \overline{2}$$ of his shots. How many times more shots are made by the first place finisher than the third place finisher?

Answer: The first place finisher made 0.077 more shorts than the third place finisher.

Explanation:
The shorts made by 1st person are 70%, that is 0.7 in decimal,
The shorts made by 2nd person are $$\frac{7}{9}$$ rewriting as 0.777 in decimal,
The shorts made by 3rd person are $$0.7 \overline{2}$$
The shorts made by  the first place finisher is  0.777 and by the Third place finisher is 0.7,
To know how many more shorts are made by first place finisher than third place finisher is the difference between 0.777 and 0.7 , that is 0.777 – 0.7 = 0.077.
So, the first place finisher made 0.077 more shorts than the third place finisher.

Question 42.
DIG DEEPER!
Three different mixtures contain small amounts of acetic acid. Mixture A is 0.036 acetic acid, Mixture B is 4.2% acetic acid, and Mixture C is $$\frac{1}{22}$$ acetic acid. Explain how to use this information to determine which mixture contains the greatest amount of acetic acid.

Answer: 0.045 , Mixture C has more amount of acetic acid as compared to Mixture A and Mixture B.

Explanation:
Mixture A is 0.036 acetic acid,
Mixture B is 4.2% acetic acid, in decimal we can write it as 0.042,
Mixture C is $$\frac{1}{22}$$ acetic acid, rewriting as 0.045 in decimal form,
So, 0.036 is less than 0.042 , 0.042 is less than 0.045,
Finally,  0.045 Mixture c has the more amount of acetic acid as compared to Mixture A and Mixture B.

Question 43.
MODELING REAL LIFE
Over 44% of the 30 students in a class read a book last month. What are the possible numbers of students in the class who read a book last month? Justify your answer.
Answer: 13 number of students in the class read the book last month.

Explanation:
Given, 44% of 30 students
=(44%) × 30
= $$\frac{44}{100}$$ × 30
= $$\frac{44 × 30}{100}$$
= $$\frac{1320}{100}$$
=13.2
So, 13 number of students in the class read the book last month.

Question 44.
NUMBER SENSE
Fill in the blanks using each of the numbers 0 – 7 exactly once, so that the percent, decimal, and fraction below are ordered from least to greatest. Justify your answer.

Answer:  The percent, decimal, and fraction are ordered from least to greatest are 12.3%, 0.57, $$\frac{4}{6}$$ by using 0 – 7 numbers only once .

Explanation:
Given , using each of the numbers 0 – 7 exactly once,
Estimating on the numbes
Using numbers 1 , 2 , 3 for percent gives 12.3% as shown and can be writen as 0.123
Using numbers 0 , 5 , 7 for decimal gives 0.57 as shown
Using numbers 4 and 6 for fraction gives $$\frac{4}{6}$$ and can be writen as 0.66
The order from least to greatest is 0.123 , 0.57 , 0.66 .
So , The percent, decimal, and fraction are ordered from least to greatest are 12.3%, 0.57, $$\frac{4}{6}$$

Lesson 6.2 The Percent Proportion

EXPLORATION 1

Using Percent Models
Work with a partner.
a. Complete each model. Explain what each model represents.

b. Use the models in part (a) to answer each question.

• What number is 50% of 30?
• 15 is what percent of 75?
• 96 is 133$$\frac{1}{3}$$% of what number?

c. How can you use ratio tables to check your answers in part(b)? How can you use proportions? Provide examples to support your reasoning.
d. Write a question different from those in part (b) that can be answered using one of the models in part(a). Trade questions with another group and find the solution.
Answer: All the answers are given below the explanation.

Explanation:
a.
This model represents the number 30 which is divided in to two equal halves ,which can be represented in percent and numbers,
that is 50% of 30 is 15.
50 percent × 30 =
(50:100) × 30 =
(50 × 30):100 =
1500:100 = 15

This model represents the number 75 which is divided in to five equal halves, which can be represented in percent and numbers ,That is
20% of 75 is 15,
40% of 75 is 30,
60% of 75 is 45,
80% of 75 is 60.

This model represents the number 96 which is divided in to four equal halves, which can be represented in percent and numbers ,That is
33$$\frac{1}{3}\%$$ of 96 is 24,
66$$\frac{2}{3}\%$$ of 96 is 48,
100% of 96 is 72.

b. As shown in the models in part (a) 15 is the number which is 50% of 30,
From the figure 2 of part (a) we know, 15 is the number which is 20% of 75,
From the figure 3 of part (a) we know, 96 is the number which is 133$$\frac{1}{3}\%$$ of 96.

c. The models provided in the part (a) are representing the each of the number individually which are divided in to equal number of parts in respective of their numbers , Also the divided parts can be represented as percent and number form
Since the divided equal parts are equally represents the whole number given ,So that the ratios of the parts can be easily combined in to the whole number .
Thus, the proportion can be used to calculate the percent of each ratio and have the unchanged output .
For example,

In the picture shown The number 30 is divided into 1:1 ratio equally which is exactly the half of the number and percent that is 15 is 50% of 30.

d. From the figure 2 of part (a), we can assume the question as
What number is 80% of 75?

Answer: 60 is the number which is 80% of 75 , as shown in the model above.

Try It

Write and solve a proportion to answer the question.
Question 1.
What percent of 5 is 3?
Answer: 60%

Explanation:
3 : 5 × 100 =
(3 × 100): 5
300 : 5 = 60
So, 5 is 60% of 3.

Question 2.
24 is what percent of 20?
Answer: 120%

Explanation:
24 : 20 × 100 =
(24 × 100) : 20 =
2400 : 20 = 120
So, 24 is the 120% of 20.

Write and solve a proportion to answer the question.
Question 3.
What number is 80% of 60?
Answer:  48

Explanation:
80 % × 60 =
(80 :100) × 60 =
(80 × 60) :100 =
4800 : 100 = 48
So, 48 is 80% of 60.

Question 4.
10% of 40.5 is what number?
Answer: 4.05

Explanation:
10 % × 40.5 =
(10 : 100) × 40.5 =
(10 × 40.5) : 100 =
4.05 : 100 = 4.05
So, 4.05 is the 10% of 40.5

Write and solve a proportion to answer the question.
Question 5.
0.1% of what number is 4?
Answer: 4,000

Explanation:
Let the number be X
0.1% × X = 4
X = 4 ÷ 0.1%
= 4 ÷ (0.1 ÷ 100)
= (100 × 4 ) ÷ 0.1
= 400 ÷ 0.1
=4,000
So, 4 is the 0.1% of 4,000.

Question 6.
$$\frac{1}{2}$$ is 25% of what number?
Answer:  2

Explanation:
To make calculation easier we can rewrite $$\frac{1}{2}$$ as 0.5
let the number be X
25% × X = 0.5
X = 0.5 ÷ 25%
= 0.5 ÷ (25 ÷ 100)
= (100 × 0.5 ) ÷ 25
= 50 ÷ 25
=2
So, 0.5 is the 25% of 2.

Self-Assessment for Concepts & Skills
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 7.
USING THE PERCENT PROPORTION
Write and solve a proportion to determine what percent of 120 is 54.
Answer:  45%

Explanation:
By using percent proportion, we have
$$\frac{a}{w}$$ = $$\frac{p}{100}$$
$$\frac{54}{120}$$ = $$\frac{p}{100}$$
$$\frac{54 × 100}{120}$$ = p
p = $$\frac{5400}{120}$$
p = 45%
so, 54 is the 45% of 120.

Question 8.
CHOOSE TOOLS
Use a model to find 60% of 30.
Answer: 18

Explanation:
60% × 30
= (60 : 100) × 30
= (60 × 30) : 100
= 1800 : 100
= 18
So, 60% of 30 is 18.

Question 9.
WHICH ONE DOESN’T BELONG?
Which proportion at the left does not belong with the other three? Explain your reasoning.

Answer: second proportion doesn’t belong to other three

Explanation:
second proportion: 15/50 = p /100
p = (15 × 100) ÷50
p = 150 ÷ 5
p = 30
Here we got 30% , as the p values of the other three proportions are 50%
So, second proportion does not fit in to the other three proportions.

Self-Assessment for Problem Solving
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 10.
An arctic woolly-bear caterpillar lives for 7 years and spends 90% of its life frozen. How many days of its life is the arctic woolly-bear frozen?

Answer: The arctic woolly-bear frozen for 6.3 years in their life time.

Explanation:
Given, An arctic woolly-bear caterpillar lives for 7 years and spends 90% of its life frozen
We have, 90% × 7
= (90 : 100 ) × 7
= (90 × 7) : 100
= 630 : 100
= 6.3 years
So, The arctic woolly-bear frozen for 6.3 years in their life time.

Question 11.
DIG DEEPER!
The table shows the numbers of pictures you upload to a social media website for 5 days in a row. How many total pictures do you upload during the week when 32% of the total pictures are uploaded on Saturday and Sunday?

Answer: The total pictures uploaded during the week are 53.

Explanation:
Given,  32% of the total pictures are uploaded on Saturday and Sunday
And adding the photos that we uploaded during the 5 days are 2 + 2 + 4 + 1 + 8 = 17
Let the number we should find be X
32% × X = 17
X = 17 ÷ 32%
= 17 ÷ (32 ÷ 100)
= (100 × 17 ) ÷ 32
= 1700 ÷ 32
= 53.12
So, 17 is the 32% of 53.
Finally we have, The total pictures uploaded during the week are 53.

The Percent Proportion Homework & Practice 6.2

Review & Refresh

Write the fraction as a decimal and a percent.
Question 1.
$$\frac{42}{100}$$
Answer: decimal = 0.42 , percent = 42%

Explanation:
To get the percent or decimal from fraction $$\frac{42}{100}$$ we have to divide 42 by 100 ,
Then, we get 0.42,
To get the percent of 0.42 multiply by 100 , it will be 42%
So, $$\frac{42}{100}$$ can be written as  0.42 or 42%

Question 2.
$$\frac{7}{1000}$$
Answer: decimal = 0.007 , percent = 0.7%

Explanation:
To get the percent or decimal from  fraction $$\frac{7}{1000}$$ we have to divide 7 by 1000 ,
Then, we get 0.007,
To get the percent of 0.007 multiply by 100 , it will be 0.7%
So, $$\frac{7}{1000}$$ can be written as  0.007 or 0.7%

Question 3.
$$\frac{13}{9}$$
Answer: decimal = 1.444 , percent = 144.4%

Explanation:
To get the percent or decimal from $$\frac{13}{9}$$  fraction we have to divide 13 by 9 ,
Then, we get 1.444,
To get the percent of 1.444 multiply by 100 , it will be 144.4%
So, $$\frac{13}{9}$$ can be written as  1.444 or 144.4%

Question 4.
$$\frac{41}{66}$$
Answer: decimal = $$0.62 \overline{12}$$ , percent = $$62. \overline{12} \%$$

Explanation:
To get the percent or decimal from  fraction $$\frac{41}{66}$$ we have to divide 41 by 66 ,
Then, we get $$0.62 \overline{12}$$,
To get the percent of $$0.62 \overline{12}$$ multiply by 100 , it will be $$62. \overline{12} \%$$
So, $$\frac{41}{66}$$ can be written as  $$0.62 \overline{12}$$ or $$62. \overline{12} \%$$

Evaluate the expression when a = – 15 and b = – 5.
Question 5.
a ÷ 5
Answer: -3

Explanation:
Given , a = – 15
Then , – 15 ÷ 5
= $$\frac{-15}{5}$$
= – 3.
so, a ÷ 5 = – 3.

Question 6.
$$\frac{b+14}{a}$$
Answer: $$\frac{9}{-15}$$

Explanation:
Given , a = -15 , b = -5 , by substituting the given values in the expression, we get
= $$\frac{(- 5)+14}{-15}$$
= $$\frac{9}{-15}$$
So, $$\frac{b+14}{a}$$ = $$\frac{9}{-15}$$

Question 7.
$$\frac{b^{2}}{a+5}$$
Answer: $$\frac{25}{-10}$$

Explanation:
Given , a = -15 , b = -5, by substituting the given values in the expression, we get
= $$\frac{(-5)^{2}}{(-15)+5}$$
= $$\frac{25}{-10}$$
So, $$\frac{b^{2}}{a+5}$$ = $$\frac{25}{-10}$$

What is the solution of 9x = 1.8?
A. x = – 5
B. x = – 0.2
C. x = 0.2
D. x = 5
Answer: C . x = 0.2

Explanation:
Given, 9x = 1.8
x = $$\frac{1.8}{9}$$
x = 0.2.

Concepts, Skills, &Problem Solving

CHOOSE TOOLS Use a model to answer the question. Use a proportion to check your answer. (See Exploration 1, p. 241.)
Question 9.
What number is 20% of 80?
Answer: 16

Explanation:
By using proportion , we have
$$\frac{a}{w}$$  = $$\frac{p}{100}$$
$$\frac{a}{80}$$ = $$\frac{20}{100}$$
$$\frac{a}{80}$$ = $$\frac{1}{5}$$
a = $$\frac{80}{5}$$
a = 16.
So, 16 is 20% of 80.

Question 10.
10 is what percent of 40?
Answer: 25%

Explanation:
By using proportion , we have
$$\frac{a}{w}$$ = $$\frac{p}{100}$$
$$\frac{10}{40}$$ = $$\frac{p}{100}$$
$$\frac{1}{4}$$  = $$\frac{p}{100}$$
p = $$\frac{100}{4}$$
p = 25
So, 10 is 25% of 40.

Question 11.
15 is 30% of what number?
Answer: 50

Explanation:
By using proportion , we have
$$\frac{a}{w}$$  = $$\frac{p}{100}$$
$$\frac{15}{w}$$  = $$\frac{30}{100}$$
by cross multiplication we get,
30 × w = 15 × 100
w  = $$\frac{1500}{30}$$
w = $$\frac{150}{30}$$
w = 50.
So, 15 is 30% of 50.

Question 12.
What number is 120% of 70?
Answer: 84

Explanation:
By using proportion , we have
$$\frac{a}{w}$$ = $$\frac{p}{100}$$
$$\frac{a}{70}$$  = $$\frac{120}{100}$$
$$\frac{a}{70}$$  = $$\frac{6}{5}$$
a = $$\frac{70 × 6}{5}$$
a = $$\frac{420}{5}$$
a = 84
So, 84 is 120% of 70.

Question 13.
20 is what percent of 50?
Answer: 40%

Explanation:
By using proportion , we have
$$\frac{a}{w}$$ = $$\frac{p}{100}$$
$$\frac{20}{50}$$  = $$\frac{p}{100}$$
p = $$\frac{20 × 100}{50}$$
p = $$\frac{200}{5}$$
p = 40
So, 20 is 40% of 50.

Question 14.
48 is 75% of what number?
Answer: 64

Explanation:
By using proportion , we have
$$\frac{a}{w}$$  = $$\frac{p}{100}$$
$$\frac{48}{w}$$  = $$\frac{75}{100}$$
w =$$\frac{48 × 100}{75}$$
w = $$\frac{4800}{75}$$
w = 64
So, 48 is 75% of 64.

USING THE PERCENT PROPORTION Write and solve a proportion to answer the question.
Question 15.
What percent of 25 is 12?
Answer: 48%

Explanation:
By using proportion , we have
$$\frac{a}{w}$$  = $$\frac{p}{100}$$
$$\frac{12}{25}$$  = $$\frac{p}{100}$$
p = $$\frac{12 × 100}{25}$$
p = $$\frac{1200}{25}$$
p = 48
So, 12 is 48% of 25.

Question 16.
14 is what percent of 56?
Answer: 25%

Explanation:
By using proportion , we have
$$\frac{a}{w}$$ = $$\frac{p}{100}$$
$$\frac{14}{56}$$  = $$\frac{p}{100}$$
p = $$\frac{14 × 100}{56}$$
p = $$\frac{1400}{56}$$
p = 25
So, 14 is 25% of 56.

Question 17.
25% of what number is 9?
Answer: 36

Explanation:
By using proportion , we have
$$\frac{a}{w}$$ = $$\frac{p}{100}$$
$$\frac{9}{w}$$ = $$\frac{25}{100}$$
$$\frac{9}{w}$$ = $$\frac{1}{4}$$
w = 9 × 4
w = 36.
So, 9 is 25% of 36.

Question 18.
36 is 0.9% of what number?
Answer: 4,000

Explanation:
By using proportion , we have
$$\frac{a}{w}$$  = $$\frac{p}{100}$$
$$\frac{36}{w}$$ = $$\frac{0.9}{100}$$
w = $$\frac{36 × 100}{0.9}$$
w = $$\frac{3600}{0.9}$$
w = $$\frac{36,000}{9}$$
w = 4,000
So, 36 is 0.9% of 4,000.

Question 19.
75% of 124 is what number?
Answer: 93

Explanation:
By using proportion , we have
$$\frac{a}{w}$$ = $$\frac{p}{100}$$
$$\frac{a}{124}$$ = $$\frac{75}{100}$$
a = $$\frac{75 × 124}{100}$$
a  = $$\frac{9,300}{100}$$
a = 93
So, 93 is 75% of 124.

Question 20.
110% of 90 is what number?
Answer: 99

Explanation:
By using proportion , we have
$$\frac{a}{w}$$  = $$\frac{p}{100}$$
$$\frac{a}{90}$$  = $$\frac{110}{100}$$
a = $$\frac{110 × 90}{100}$$
a = $$\frac{9900}{100}$$
a = 99
So, 99 is 110% of 90.

Question 21.
What number is 0.4% of 40?
Answer: 0.16

Explanation:
By using proportion , we have
$$\frac{a}{w}$$  = $$\frac{p}{100}$$
$$\frac{a}{40}$$  = $$\frac{0.4}{100}$$
a = $$\frac{0.4 × 40}{100}$$
a = $$\frac{16}{100}$$
a = 0.16
So, 0.16 is 0.4% of 40.

Question 22.
72 is what percent of 45?
Answer:  160%

Explanation:
By using proportion , we have
$$\frac{a}{w}$$ = $$\frac{p}{100}$$
$$\frac{72}{45}$$ = $$\frac{p}{100}$$
p = $$\frac{72 × 100}{45}$$
p = $$\frac{7200}{45}$$
p = 160
So, 72 is 160% of 45.

Question 23.
YOU BE THE TEACHER
Your friend uses the percent proportion to answer the question below. Is your friend correct? Explain your reasoning.
“40%of what number is 34?”

Answer:  yes, he used the correct percent proportion.

Explanation:
By using proportion , we have
$$\frac{a}{w}$$ = $$\frac{p}{100}$$
$$\frac{34}{w}$$ = $$\frac{40}{100}$$
w = $$\frac{34 × 100}{40}$$
w = $$\frac{3400}{40}$$
w = 85
So, 34 is 40% of 85.

Question 24.
MODELING REAL LIFE
Of 140 seventh-grade students, 15% earn the Presidential Youth Fitness Award. How many students earn the award?

Answer: 21 students earn the Presidential Youth Fitness Award.

Explanation:
Given, Of 140 seventh-grade students, 15% earn the Presidential Youth Fitness Award.
By using percent proportion , we have
$$\frac{a}{w}$$ = $$\frac{p}{100}$$
$$\frac{a}{140}$$ = $$\frac{15}{100}$$
a = $$\frac{15 × 140}{100}$$
a = $$\frac{2100}{100}$$
a = 21
So, 21 is 15% of 140.
Totally ,21 students earn the Presidential Youth Fitness Award.

Question 25.
MODELING REAL LIFE
A salesperson receives a 3% commission on sales. The salesperson receives $180 in commission. What is the amount of sales? Answer: The total amount of sale is$6,000.

Explanation:
Given, A salesperson receives a 3% commission on sales, The salesperson receives $180 in commission. By using proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{180}{w}$$ = $$\frac{3}{100}$$ w = $$\frac{180 × 100}{3}$$ w = $$\frac{18,000}{3}$$ w = 6,000 So,$180 is 3% of $6,000. Total amount of sale is$6,000.

USING THE PERCENT PROPORTION Write and solve a proportion to answer the question.
Question 26.
0.5 is what percent of 20?
Answer: 2.5%

Explanation:
By using proportion , we have
$$\frac{a}{w}$$ = $$\frac{p}{100}$$
$$\frac{0.5}{20}$$ = $$\frac{p}{100}$$
p = $$\frac{0.5 × 100}{20}$$
p = 0.5 × 5
p = 2.5
So, 0.5 is 2.5% of 20.

Question 27.
14.2 is 35.5% of what number?
Answer: 40

Explanation:
By using proportion , we have
$$\frac{a}{w}$$ = $$\frac{p}{100}$$
$$\frac{14.2}{w}$$ = $$\frac{35.5}{100}$$
w = $$\frac{14.2 × 100}{35.5}$$
w = $$\frac{142 × 10}{35.5}$$
w = 40
So, 14.2 is 35.5% of 40.

Question 28.
$$\frac{3}{4}$$ is 60% of what number?
Answer: 1.25

Explanation:
$$\frac{3}{4}$$ can be rewrite as 0.75 in decimal,
By using proportion , we have
$$\frac{a}{w}$$ = $$\frac{p}{100}$$
$$\frac{0.75}{w}$$ = $$\frac{60}{100}$$
w = $$\frac{0.75 × 100}{60}$$
w = $$\frac{75}{60}$$
w = 1.25
So, 0.75 is 60% of 1.25.

Question 29.
What number is 25% of $$\frac{7}{8}$$?
Answer: 0.218

Explanation:
$$\frac{7}{8}$$ can be rewrite as 0.875 in decimal,
By using percent proportion , we have
$$\frac{a}{w}$$ = $$\frac{p}{100}$$
$$\frac{a}{0.875}$$ = $$\frac{25}{100}$$
a = $$\frac{0.875 × 25}{100}$$
a = $$\frac{21.87}{100}$$
a = 0.218
So, 0.218 is 25% of 0.875.

Question 30.
MODELING REAL LIFE
You are assigned 32 math exercises for homework. You complete 75% of the exercises before dinner. How many exercises do you have left to do after dinner?
Answer: 24 exercises are left .

Explanation:
You are assigned 32 math exercises for homework. You complete 75% of the exercises before dinner.
By using percent proportion , we have
$$\frac{a}{w}$$ = $$\frac{p}{100}$$
$$\frac{a}{32}$$ = $$\frac{75}{100}$$
a = $$\frac{32 × 75}{100}$$
a = $$\frac{2,400}{100}$$
a = 24
So, 24 is 75% of 32.
Totally, 24 exercise are left .

Question 31.
MODELING REAL LIFE
Your friend earns $10.50 per hour, which is 125% of her hourly wage last year. How much did your friend earn per hour last year? Answer: Friend earned$8.4 per hour last year

Explanation:
Your friend earns $10.50 per hour, which is 125% of her hourly wage last year, By using proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{10.5}{w}$$ = $$\frac{125}{100}$$ w = $$\frac{10.5 × 100}{125}$$ w = $$\frac{1050}{125}$$ w = 8.4 So, 8.4 is 125% of 10.5. Question 32. MODELING REAL LIFE The bar graph shows the numbers of reserved campsites at a campground for one week. What percent of the reservations were for Friday or Saturday? Answer: The percent of reservations For Friday is 74.2% and for Saturday is 85% Explanation: As per the graph shown, The reservations made for the week are 35 , Friday reservations are 26, so By using proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{26}{35}$$ = $$\frac{p}{100}$$ p = $$\frac{26 × 100}{35}$$ p = $$\frac{2600}{35}$$ p = 74.2 So, 26 is 74.2% of 35. Saturday reservations are 30, so By using proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{30}{35}$$ = $$\frac{p}{100}$$ p = $$\frac{30 × 100}{35}$$ p = $$\frac{3,000}{35}$$ p = 85 So, 30 is 85% of 35. Totally, The percent of reservations For Friday is 74.2% and for Saturday is 85%. Question 33. PROBLEM SOLVING Your friend displays the results of a survey that asks several people to vote on a new school mascot. a. What is missing from the bar graph? b. What percent of the votes does the least popular mascot receive? Explain your reasoning. c. There are 124 votes total. How many votes does tiger receive? Answer: a. The numerical values for the votes are missing from the graph b. In order to calculate the percent of the votes does the least popular mascot received will be halted due to lack of clear mentioning of of the proportion of the votes or the ratio of the votes. c. The votes acquired by the tiger are cannot be determined because of figure which does not contain proper information. Question 34. DIG DEEPER! A quarterback completes 18 of 33 passes during the first three quarters of a football game. He completes every pass in the fourth quarter and 62.5% of his passes for the entire game. How many passes does the quarterback throw in the fourth quarter? Justify your answer. Answer: Quarterback throw 20.6 passes in the fourth quarter Explanation: A quarterback completes 18 of 33 passes during the first three quarters of a football game, He completes every pass in the fourth quarter and 62.5% of his passes for the entire game. By using percent proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{a}{33}$$ = $$\frac{62.5}{100}$$ a = $$\frac{33 × 62.5}{100}$$ a = $$\frac{2,062.5}{100}$$ a = 20.6 So, 20.6 is 62.5% of 33. Hence, Quarterback thrown 20.6 passes in the fourth quarter Question 35. REASONING 20% of a number is x. What is 100% of the number? Assume x > 0. Answer: 5x Explanation: By using proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ substitute a = x and p = 20 $$\frac{x}{w}$$ = $$\frac{20}{100}$$ by cross multiplication , we get w × 20 = x × 100 20w = 100x Divide both sides by 20, $$\frac{20w}{20}$$ = $$\frac{100x}{20}$$ w = 5x So, 100% of the number is 5x. Question 36. STRUCTURE Answer each question. Assume x > 0. a. What percent of 8x is 5x? b. What is 65% of 80x? Answer: a. 62.5% , b. 52x Explanation: a. percent of 8x is 5x By using proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{5x}{8x}$$ = $$\frac{p}{100}$$ $$\frac{5}{8}$$ = $$\frac{p}{100}$$ by cross multiplying 8p = 500 p = $$\frac{500}{8}$$ p = 62.5 So, 5x is 62.5% of 8x. b. By using percent proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{a}{8x}$$ = $$\frac{65}{100}$$ a = $$\frac{8x × 65}{100}$$ a = $$\frac{520x}{100}$$ a = 52x So, 52x is 65% of 80x. Lesson 6.3 The Percent Equation EXPLORATION 1 Using Percent Equations Work with a partner. a. The circle graph shows the number of votes received by each candidate during a school election. So far, only half of the students have voted. Find the percent of students who voted for each candidate. Explain your method. c. The circle graph shows the final results of the election after every student voted. Use the equation you wrote in part(b) to find the number of students who voted for each candidate. d. Use a different method to check your answers in part(c). Which method do you prefer? Explain. Answer: a. The percent of students who voted for each candidate are For Person A = 20% , For Person B = 25% For Person C = 15% For Person D = 40% b. The equation is a = $$\frac{w × p}{100}$$ c. The number of students who voted for each candidate are For Person A = 30 For Person B = 24 For Person C = 24 For Person D = 42 d. ratio proportion is used as another method. Explanation: a. Given, The circle graph shows the number of votes received by each candidate during a school election. So far, only half of the students have voted. The number of votes received till now are 12 + 15 + 9 + 24 = 60, To know the percent of students who voted for each candidate we have , For person A, w = 60 , a = 12 , p = ? By using proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{12}{60}$$ = $$\frac{p}{100}$$ p = $$\frac{12 × 100}{60}$$ p = $$\frac{1200}{60}$$ p = 20 So, 12 is 20% of 60. The percent of students who voted for person A is 20% For person B , w = 60 , a = 15 , p = ? By using proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{15}{60}$$ = $$\frac{p}{100}$$ p = $$\frac{15 × 100}{60}$$ p = $$\frac{1500}{60}$$ p = 25 So, 15 is 25% of 60. The percent of students who voted for person B is 25% For person C , w = 60 , a = 9 , p = ? By using proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{9}{60}$$ = $$\frac{p}{100}$$ p = $$\frac{9 × 100}{60}$$ p = $$\frac{900}{60}$$ p = 15 So, 9 is 15% of 60. The percent of students who voted for person C is 15% For person D , w = 60 , a = 24 , p = ? By using proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{24}{60}$$ = $$\frac{p}{100}$$ p = $$\frac{24 × 100}{60}$$ p = $$\frac{2400}{60}$$ p = 40 So, 24 is 40% of 60. The percent of students who voted for person D is 40% b. we know that percent proportion is $$\frac{a}{w}$$ = $$\frac{p}{100}$$ where as a = part , w= whole , p = percent , To solve for a , we can write it as a = $$\frac{w × p}{100}$$ c. The figure showing the percent of all the candidates individually are after the final results , as shown in part (a) half of the students are 60 , The half of the students voted in all the students are 60 and total strength of students are 60 + 60 = 120 To calculate the number of voting acquired by each candidate we have, For person A , w = 120 , p = 25 , a = ? By using percent proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{a}{120}$$ = $$\frac{25}{100}$$ a = $$\frac{120 × 25}{100}$$ a = $$\frac{120}{4}$$ a = 30 So, 30 is 25% of 120. The number of students who voted for person A after final results are 30 For person B , w = 120 , p = 20 , a = ? By using percent proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{a}{120}$$ = $$\frac{20}{100}$$ a = $$\frac{120 × 20}{100}$$ a = $$\frac{120}{5}$$ a = 24 So, 24 is 20% of 120. The number of students who voted for person B after final results are 24 For person C , w = 120 , p = 20 , a = ? By using percent proportion , we have same as for person B , So, 24 is 20% of 120. The number of students who voted for person C after final results are 24 For person D , w = 120 , p = 35 , a = ? By using percent proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{a}{120}$$ = $$\frac{35}{100}$$ a = $$\frac{120 × 35}{100}$$ a = $$\frac{4,200}{100}$$ a = 42 So, 42 is 35% of 120. The number of students who voted for person D after final results are 42. d. We are using ratio proportion method , To check the answers of part (c), For person A , 25 % × 120 = (25 : 100) × 120 = (25 × 120) : 100 = 3,000 : 100 = 30 So, 30 is the 25% of 120. For person B , 20 % × 120 = (20 : 100) × 120 = (20 × 120) : 100 = 2,400 : 100 = 24 So, 24 is the 20% of 120. For person C , 20% × 120 is same as person B So, 24 is the 20% of 120. For person D , 35 % × 120 = (35 : 100) × 120 = (35 × 120) : 100 = 4,200 : 100 = 42 So, 42 is the 35% of 120. All the answers are verified with ratio proportion method. Try It Write and solve an equation to answer the question. Question 1. What number is 10% of 20? Answer: 2 Explanation: 10 % × 20 = (10 : 100) × 20 = (10 × 20) : 100 = 200 : 100 = 2 So, 2 is the 10% of 20. Question 2. What number is 150% of 40? Answer: 60 Explanation: 150 % × 40 = (150 : 100) × 40 = (150 × 40) : 100 = 6,000 : 100 = 60 So, 60 is the 150% of 40. Write and solve an equation to answer the question. Question 3. 3 is what percent of 600? Answer: 0.5% Explanation: By using proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{3}{600}$$ = $$\frac{p}{100}$$ p = $$\frac{3 × 100}{600}$$ p = $$\frac{300}{600}$$ p = $$\frac{1}{2}$$ p = 0.5 So, 3 is 0.5% of 600. Question 4. 18 is what percent of 20? Answer: By using proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{18}{20}$$ = $$\frac{p}{100}$$ p = $$\frac{18 × 100}{20}$$ p = $$\frac{1800}{20}$$ p = $$\frac{180}{2}$$ p = 90 So, 18 is 90% of 20. Write and solve an equation to answer the question. Question 5. 8 is 80% of what number? Answer: 10 Explanation: By using proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{8}{w}$$ = $$\frac{80}{100}$$ w = $$\frac{8 × 100}{80}$$ w = $$\frac{800}{80}$$ w = 10 So, 8 is 80% of 10. Question 6. 90 is 180% of what number? Answer: 50 Explanation: By using proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{90}{w}$$ = $$\frac{180}{100}$$ w = $$\frac{90 × 100}{180}$$ w = $$\frac{9,000}{180}$$ w = 50 So, 90 is 180% of 50. Self-Assessment for Concepts & Skills Solve each exercise. Then rate your understanding of the success criteria in your journal. Question 7. VOCABULARY Write the percent equation in words. Answer: Percent Equation: In this equation, the whole is the number of which we are taking a percentage and the part is the value that results from taking the percent of the whole. This means that in any percent problem, there are three basic values to be concerned about: the percent, the whole, and the resulting part. we can represent percent = p , whole = w , part = a So, we have the percent equation as, $$\frac{a}{w}$$ = $$\frac{p}{100}$$. USING THE PERCENT EQUATION Write and solve an equation to answer the question. Question 8. 14 is what percent of 70? Answer: 20% Explanation: By using proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{14}{70}$$ = $$\frac{p}{100}$$ p = $$\frac{14 × 100}{70}$$ p = $$\frac{1400}{70}$$ p = 20 So, 14 is 20% of 70. Question 9. What number is 36% of 85? Answer: 30.6 Explanation: By using percent proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{a}{85}$$ = $$\frac{36}{100}$$ a = $$\frac{85 × 36}{100}$$ a = $$\frac{3,060}{100}$$ a = 30.6 So, 30.6 is 36% of 85. Question 10. 9 is 12% of what number? Answer: 75 Explanation: By using proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{9}{w}$$ = $$\frac{12}{100}$$ w = $$\frac{9 × 100}{12}$$ w = $$\frac{900}{12}$$ w = 75 So, 9 is 12% of 75. Question 11. 108 is what percent of 72? Answer: 150% Explanation: By using proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{108}{72}$$ = $$\frac{p}{100}$$ p = $$\frac{108 × 100}{72}$$ p = $$\frac{10,800}{72}$$ p = 150 So, 108 is 150% of 72. Question 12. DIFFERENT WORDS, SAME QUESTION Which is different? Find “both” answers. Answer: 55 is 20% of what number ? , is different from other three questions. Explanation: Given , 20% of 55 , we have to find the part of whole number By using percent proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{a}{55}$$ = $$\frac{20}{100}$$ a = $$\frac{55 × 20}{100}$$ a = $$\frac{1,100}{100}$$ a = 11 So, 11 is 20% of 55. But this 55 is 20% of what number ? is different from other three, because here we have to find out the whole number By using proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{55}{w}$$ = $$\frac{20}{100}$$ w = $$\frac{55 × 100}{20}$$ w = $$\frac{5500}{20}$$ w = 275 So, 55 is 20% of 275. Hence , 55 is 20% of what number ? , is different from other three questions. Self-Assessment for Problem Solving Solve each exercise. Then rate your understanding of the success criteria in your journal. Question 13. DIG DEEPER! A school offers band and chorus classes. The table shows the percents of the 1200 students in the school who are enrolled in band, chorus, or neither class. How many students are enrolled in both classes? Explain. Answer: The total number of students enrolled for both classes are 744. Explanation: Given, The table shows the percent of the 1200 students in the school who are enrolled in band, chorus, or neither class. For Band , w = 1200 , p = 34 , a = ? By using percent proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{a}{1200}$$ = $$\frac{34}{100}$$ a = $$\frac{1200 × 34}{100}$$ a = 12 × 34 a = 408 So, 408 is 34% of 1200. The number of students enrolled for the Band are 408. For Band , w = 1200 , p = 28 , a = ? By using percent proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{a}{1200}$$ = $$\frac{28}{100}$$ a = $$\frac{1200 × 28}{100}$$ a = 12 × 28 a = 336 So, 336 is 28% of 1200. The number of students enrolled for the chorus are 336. The total number of students enrolled for both classes are 408 + 336 = 744. Question 14. Water Tank A has a capacity of 550 gallons and is 66% full. Water Tank B is 53% full. The ratio of the capacity of Water Tank A to Water Tank B is 11:15. a. How much water is in each tank? b. What percent of the total volume of both tanks is filled with water? Answer: a. The water tank A is filled with 363 gallons of water. The water tank B is filled with 397.5 gallons of water. b. The percent of the total volume of both tanks is filled with water is 58.5%. Explanation: a. Given , Water Tank A has a capacity of 550 gallons and is 66% full. w = 550 gallons , p = 66% , a = ? By using percent proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{a}{550}$$ = $$\frac{66}{100}$$ a = $$\frac{550 × 66}{100}$$ a = 550 × 0.66 a = 363 So, 363 is 66% of 550. The water tank A is filled with 363 gallons of water. Given, Water Tank B is 53% full. The ratio of the capacity of Water Tank A to Water Tank B is 11:15. The capacity of Water Tank A is 550 gallons Let the capacity of tank B is x gallons $$\frac{550}{x}$$ = $$\frac{11}{15}$$ x = $$\frac{550 × 15}{11}$$ x = $$\frac{8,250}{11}$$ x = 750. The capacity of the water Tank B is 750 gallons, To know the amount of water filled in the tank we have, By using percent proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{a}{750}$$ = $$\frac{53}{100}$$ a = $$\frac{750 × 53}{100}$$ a = 7.5 × 53 a = 397.5 So, 397.5 is 53% of 750. The water tank B is filled with 397.5 gallons of water. b. To know the percent of the total volume of both tanks is filled with water, we have The total capacity of Water tank A and Water tank B = 550 + 750 = 1,300 gallons The total amount of water filled in both tanks are 363 + 397.5 = 760.5 gallons So, w = 1,300 , a = 760.5 , p = ? By using proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{760.5}{1,300}$$ = $$\frac{p}{100}$$ p = $$\frac{760.5 × 100}{1,300}$$ p = $$\frac{760.5}{13}$$ p = 58.5 So, 760.5 is 58.5% of 1,300. The percent of the total volume of both tanks is filled with water is 58.5%. The Percent Equation Homework & Practice 6.3 Review & Refresh Write and solve a proportion to answer the question. Question 1. 30% of what number is 9? Answer: 30 Explanation: By using proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{9}{w}$$ = $$\frac{30}{100}$$ w = $$\frac{9 × 100}{30}$$ w = $$\frac{900}{30}$$ w = 30 So, 9 is 30% of 30. Question 2. 42 is what percent of 80? Answer: 52.5% Explanation: By using proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{42}{80}$$ = $$\frac{p}{100}$$ p = $$\frac{42 × 100}{80}$$ p = $$\frac{420}{8}$$ p = 52.5 So, 42 is 52.5% of 80. Question 3. What percent of 36 is 20? Answer: Explanation: By using proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{20}{36}$$ = $$\frac{p}{100}$$ p = $$\frac{20 × 100}{36}$$ p = $$\frac{2,000}{36}$$ p = 5.55 So, 20 is55.5% of 36. Question 4. What number is 120% of 80? Answer: 96 Explanation: By using percent proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{a}{80}$$ = $$\frac{120}{100}$$ a = $$\frac{80 × 120}{100}$$ a = 8 × 12 a = 96 So, 96 is 120% of 80. Find the distance between the two numbers on a number line. Question 5. – 4 and 10 Answer: The distance between the two numbers – 4 and 10 on a number line is 14 . Explanation: For the number line above we have divided each segment in to equal parts , So, The distance between the two numbers – 4 and 10 on a number line is 14 . Question 6. –$$\frac{2}{3}$$ and $$\frac{4}{3}$$ Answer: The distance between the two numbers –$$\frac{2}{3}$$ and $$\frac{4}{3}$$ on a number line is 6 . Explanation: For the number line above we have divided each segment in to equal parts , So, The distance between the two numbers –$$\frac{2}{3}$$ and $$\frac{4}{3}$$ on a number line is 6 . Question 7. – 5$$\frac{2}{3}$$ and – 1 $$\frac{3}{10}$$ Answer: The distance between the two numbers – 5$$\frac{2}{3}$$ and – 1 $$\frac{3}{10}$$ on a number line is 6. Explanation: Given , – 5$$\frac{2}{3}$$ and – 1 $$\frac{3}{10}$$ can be written as $$\frac{-13}{3}$$ and $$\frac{-7}{10}$$ converting into decimal form we get , -4.3 and -0.7 For the number line above we have divided each segment in to equal parts , So, The distance between the two numbers – 5$$\frac{2}{3}$$ and – 1 $$\frac{3}{10}$$ on a number line is 6. Question 8. – 4.3 and 7.5 Answer: The distance between the two numbers – 4.3 and 7.5 on a number line is 8 . Explanation: For the number line above we have divided each segment in to equal parts , So, The distance between the two numbers – 4.3 and 7.5 on a number line is 8 . Question 9. There are 160 people in a grade. The ratio of boys to girls is 3 to 5. Which proportion can you use to find the number x of boys? Answer: A. $$\frac{3}{8}$$ = $$\frac{x}{160}$$ Explanation: Given, The ratio of boys to girls is$$\frac{3}{5}$$ The ratio of boys to the grade is $$\frac{3}{8}$$ to find the number of x boys, we have to $$\frac{3}{8}$$ = $$\frac{x}{160}$$ x = $$\frac{3 × 160}{8}$$ x = $$\frac{480}{8}$$ x = 60. So, A = $$\frac{3}{8}$$ = $$\frac{x}{160}$$ is the correct answer. Concepts, Skills, & Problem Solving USING PERCENT EQUATIONS The circle graph shows the number of votes received by each candidate during a school election. Find the percent of students who voted for the indicated candidate. Each Candidate(See Exploration 1, p. 247.) Question 10. Candidate A Answer: The percent of students who voted for the candidate A is 36% Explanation: Given, the circle graph shows the number of votes received by each candidate during a school election. The total number of students voted are 54 + 60 + 36 = 150 For candidate A we have , w = 150 , a = 54 , p = ? By using proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{54}{150}$$ = $$\frac{p}{100}$$ p = $$\frac{54 × 100}{150}$$ p = $$\frac{540}{15}$$ p = 36 So, 54 is 36% of 150. The percent of students who voted for the candidate A is 36%. Question 11. Candidate B Answer: The percent of students who voted for the candidate B is 40%. Explanation: For candidate B we have , w = 150 , a = 60 , p = ? By using proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{60}{150}$$ = $$\frac{p}{100}$$ p = $$\frac{60 × 100}{150}$$ p = $$\frac{600}{15}$$ p = 40 So, 60 is 40% of 150. The percent of students who voted for the candidate B is 40%. Question 12. Candidate C Answer: The percent of students who voted for the candidate C is 24%. Explanation: For candidate C we have , w = 150 , a = 36 , p = ? By using proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{36}{150}$$ = $$\frac{p}{100}$$ p = $$\frac{36 × 100}{150}$$ p = $$\frac{360}{15}$$ p = 24 So, 36 is 24% of 150. The percent of students who voted for the candidate C is 24%. USING THE PERCENT EQUATION Write and solve an equation to answer the question. Question 13. 20% of 150 is what number? Answer: 30 Explanation: By using percent proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{a}{150}$$ = $$\frac{20}{100}$$ a = $$\frac{150 × 20}{100}$$ a = 15 × 2 a = 30 So, 30 is 20% of 150. Question 14. 45 is what percent of 60? Answer: 75% Explanation: By using proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{45}{60}$$ = $$\frac{p}{100}$$ p = $$\frac{45 × 100}{60}$$ p = $$\frac{4,500}{60}$$ p = 75 So, 45 is 75% of 60. Question 15. 35% of what number is 35? Answer: 35 is 35% of 100. Explanation: By using proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{35}{w}$$ = $$\frac{35}{100}$$ w = $$\frac{35 × 100}{35}$$ w = $$\frac{3500}{35}$$ w = 100 So, 35 is 35% of 100. Question 16. 0.8% of 150 is what number? Answer: 1.2 . Explanation: By using percent proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{a}{150}$$ = $$\frac{0.8}{100}$$ a = $$\frac{150 × 0.8}{100}$$ a = $$\frac{120}{100}$$ a = 1.2 So, 1.2 is 0.8% of 150. Question 17. 29 is what percent of 20? Answer: 145% Explanation: By using proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{29}{20}$$ = $$\frac{p}{100}$$ p = $$\frac{29 × 100}{20}$$ p = $$\frac{2,900}{20}$$ p = $$\frac{2,90}{2}$$ p = 145 So, 29 is 145% of 20. Question 18. 0.5% of what number is 12? Answer: Explanation: By using proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{12}{w}$$ = $$\frac{0.5}{100}$$ w = $$\frac{12 × 100}{0.5}$$ w = $$\frac{1200}{0.5}$$ w = 2,400. So, 12 is 0.5% of 2,400. Question 19. What percent of 300 is 51? Answer: 17% Explanation: By using proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{51}{300}$$ = $$\frac{p}{100}$$ p = $$\frac{51 × 100}{300}$$ p = $$\frac{51}{3}$$ p = 17 So, 51 is 17% of 300. Question 20. 120% of what number is 102? Answer: 85 Explanation: By using proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{102}{w}$$ = $$\frac{120}{100}$$ w = $$\frac{102 × 100}{120}$$ w = $$\frac{1020}{12}$$ w = 85 So, 102 is 120% of 85. YOU BE THE TEACHER Your friend uses the percent equation to answer the question. Is your friend correct? Explain your reasoning. Question 21. What number is 35% of 20? Answer: yes , He is correct . Explanation: By using percent proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{a}{20}$$ = $$\frac{35}{100}$$ a = $$\frac{20 × 35}{100}$$ a = $$\frac{700}{100}$$ a = 7 So, 7 is 35% of 20. Question 22. 30 is 60% of what number? Answer: 30 is 60% of 50. Explanation: By using proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{30}{w}$$ = $$\frac{60}{100}$$ w = $$\frac{30 × 100}{60}$$ w = $$\frac{300}{6}$$ w = 50 So, 30 is 60% of 50. Question 23. MODELING REAL LIFE A salesperson receives a 2.5% commission on sales. What commission does the salesperson receive for$8000 in sales?
Answer: He receives $200 for commission of the sale. Explanation: By using percent proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{a}{8000}$$ = $$\frac{2.5}{100}$$ a = $$\frac{8000 × 2.5}{100}$$ a = 80 × 2.5 a =$200
So, $200 is 2.5% of$8000.
He receives $200 for commission of the sale. Question 24. MODELING REAL LIFE Your school raised 125% of its fundraising goal. The school raised$6750. What was the goal?
Answer: The fundraising goal of the school is $5,400. Explanation: By using proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{6750}{w}$$ = $$\frac{125}{100}$$ w = $$\frac{6750 × 100}{125}$$ w =54 × 100 w =$5,400
So, $6750 is 125% of$5,400.
The fundraising goal of the school is $5,400. Question 25. MODELING REAL LIFE The sales tax on the model rocket shown is$1.92. What is the percent of sales tax?

Answer: The percent of sales tax on the rocket model  is 8%.

Explanation:
Given, The sales tax on the model rocket shown is $1.92. The tax on the rocket is$24 , we have ,
w = 24 , a = 1.92 , p = ?
By using proportion , we have
$$\frac{a}{w}$$ = $$\frac{p}{100}$$
$$\frac{1.92}{24}$$ = $$\frac{p}{100}$$
p = $$\frac{1.92 × 100}{24}$$
p = $$\frac{192}{24}$$
p = 8
So, $1.92 is 8% of$24.
The percent of sales tax on the rocket model  is 8%.

PUZZLE There were n signers of the Declaration of Independence. The youngest was Edward Rutledge, who was x years old. The oldest was Benjamin Franklin, who was y years old.

Question 26.
x is 25% of 104. What was Rutledge’s age?
Answer: The age of Rutledge is 26.

Explanation:
By using percent proportion , we have
$$\frac{a}{w}$$ = $$\frac{p}{100}$$
$$\frac{a}{104}$$ = $$\frac{25}{100}$$
a = $$\frac{104 × 25}{100}$$
a = $$\frac{104}{4}$$
a = 26
x = 26.
So, 26 is 25% of 104.

Question 27.
7 is 10% of y. What was Franklin’s age?
Answer: The Franklin’s age is 70.

Explanation:
By using proportion , we have
$$\frac{a}{w}$$ = $$\frac{p}{100}$$
$$\frac{7}{w}$$ = $$\frac{10}{100}$$
w = $$\frac{7 × 100}{10}$$
w = $$\frac{700}{10}$$
w = 70
y = 70.
So, 7 is 10% of 70.
The Franklin’s age is 70.

Question 28.
n is 80% of y. How many signers were there?
Answer: There are n = 56 members signers.

Explanation:
y = 70 , p = 80 ,
By using percent proportion , we have
$$\frac{a}{w}$$ = $$\frac{p}{100}$$
$$\frac{a}{70}$$ = $$\frac{80}{100}$$
a = $$\frac{70 × 80}{100}$$
a = 7 × 8
a =56
n = 56
So, 56 is 80% of 70.
There are n = 56 members signers.

Question 29.
LOGIC
How can you tell whether a percent of a number will be greater than, less than, or equal to the number? Give examples to support your answer.
Answer: The percent of a number is less than 100% , Then the percent of the number will be less than the number.
The percent of a number is 100% , Then the percent of the number will be equal the number.
The percent of a number is greater than 100% , Then the percent of the number will be greater than the number.

Explanation:
If the percent of a number is less than 100% , Then the percent of the number will be less than the number.
For example , 80% of 50
= 0.8 × 50
= 40
80% < 100% ,  so 40 < 50.

If the percent of a number is 100% , Then the percent of the number will be equal the number.
For example , 100% of 50
= 1 × 50
= 50.
100% = 100% , So, 50 = 50.

If the percent of a number is greater than 100% , Then the percent of the number will be greater than the number.
For example , 120% of 50
= 1.2 × 50
= 60.
120% > 100% , So, 60 > 50.

Question 30.
PROBLEM SOLVING
In a survey, a group of students is asked their favorite sport. Eighteen students choose “other” sports.
a. How many students participate in the survey?
b. How many choose football?
Answer:  a. The number of students participated  are 80.
b. The number of students chose football are 30.

Explanation:
a. 18 students chose ” other” sports So, a = 18 ,
The percent of the “other” sport = 100% – (40% + 37.5%) = 22.5%
By using proportion , we have
$$\frac{a}{w}$$ = $$\frac{p}{100}$$
$$\frac{18}{w}$$ = $$\frac{22.5}{100}$$
w = $$\frac{18 × 100}{22.5}$$
w = $$\frac{1800}{22.5}$$
w =  80
So, 18 is 22.5% of 80.
The number of students participated  are 80.

b. 80 students are participated , so w = 80
The percent of the students who chose football is 37.5%
By using percent proportion , we have
$$\frac{a}{w}$$ = $$\frac{p}{100}$$
$$\frac{a}{80}$$ = $$\frac{37.5}{100}$$
a = $$\frac{80 × 37.5}{100}$$
a = $$\frac{3,000}{100}$$
a = 30
So, 30 is 37.5% of 80.
The number of students chose football are 30.

Question 31.
TRUE OR FALSE?
Tell whether the statement is true or false. Explain your reasoning.
If W is 25% of Z, then Z : W is 75 : 25.
Answer: The statement is False.

Explanation:
Given , W is 25% of Z
$$\frac{a}{w}$$ = $$\frac{p}{100}$$
$$\frac{W}{Z}$$ = $$\frac{25}{100}$$
So, $$\frac{Z}{W}$$ = $$\frac{100}{25}$$
But given that Z : W is 75 : 25
So, The statement is False.

Question 32.
DIG DEEPER!
At a restaurant, the amount of your bill before taxes and tip is $19.83. A 6% sales tax is applied to your bill, and you leave a tip equal to 19% of the original amount. Use mental math to estimate the total amount of money you pay. Explain your reasoning. (Hint: Use 10% of the original amount.) Answer: The total amount of the money to be paid is$22.18.

Explanation:
Given ,The amount of your bill before taxes and tip is $19.83. A 6% sales tax is applied to your bill, and you leave a tip equal to 19% of the original amount. bill before the tax =$19.83
sales tax = 6%
So, $19.83 – 6% =$19.83 – 0.06
= $18.64 Tip = 19% So,$18.64 + 19%
= $18.64 + 0.19 =$22.18
The total amount of the money to be paid is $22.18. Question 33 REASONING The table shows your test results in a math class. What score do you need on the last test to earn 90% of the total points on the tests? Answer: Of all the total test points you need 720 points to earn 90% Explanation: Total point value = 100 + 250 + 150 + 300 = 800 Given p = 90% , w = 800 , a= ? By using percent proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{a}{800}$$ = $$\frac{90}{100}$$ a = $$\frac{800 × 90}{100}$$ a = 8 × 90 a = 720 So, 720 is 90% of 800. Finally, of all the total test points you need 720 points to earn 90% Lesson 6.4 Percents of Increase and Decrease EXPLORATION 1 Exploring Percent of Change Work with a partner. Each year in the Columbia River Basin, adult salmon swim upriver to streams to lay eggs. To go up the river, the adult salmon use fish ladders. But to go down the river, the young salmon must pass through several dams. At one time, there were electric turbines at each of the eight dams on the main stem of the Columbia and Snake Rivers. About 88% of the young salmon pass through a single dam unharmed. a. One thousand young salmon pass through a dam. How many pass through unharmed? b. One thousand young salmon pass through the river basin. How many pass through all 8 dams unharmed? c. By what percent does the number of young salmon decrease when passing through a single dam? d. Describe a similar real-life situation in which a quantity increases by a constant percent each time an event occurs. Answer: a. 880 salmon passed through single dam unharmed. b. Totally , 358 salmon pass through all 8 dams unharmed. c. The percent of the number of young salmon decrease when passing through a single dam is 12% d. An example for, real-life situation in which a quantity increases by a constant percent each time an event occurs. is given below in explanation. Explanation: a. Given , One thousand young salmon pass through a dam. 88% of the young salmon pass through a single dam unharmed. To know the number of salmon passed through unharmed we have, 88% of 1,000 = 88% × 1,000 = 0.88 × 1,000 = 880. So, 880 salmon passed through single dam unharmed. b. Given , One thousand young salmon pass through a dam. 88% of the young salmon pass through a single dam unharmed. To calculate number of salmon pass through all 8 dams unharmed are number of salmon passed through dam 1 , unharmed are 880 (as shown in part a) number of salmon passed through dam 2 , unharmed = 88% of 880 = 0.88 × 880 = 774 number of salmon passed through dam 3 , unharmed = 88% of 774 = 0.88 × 774 = 681 number of salmon passed through dam 4 , unharmed = 88% of 681 = 0.88 × 681 = 599 number of salmon passed through dam 5 , unharmed = 88% of 599 = 0.88 × 599 = 527 number of salmon passed through dam 6 , unharmed = 88% of 527 = 0.88 × 527 = 463 number of salmon passed through dam 7 , unharmed = 88% of 463 = 0.88 × 463 = 407 number of salmon passed through dam 8 , unharmed = 88% of 407 = 0.88 × 407 = 358 Totally , 358 salmon pass through all 8 dams unharmed. c. To calculate The percent of the number of young salmon decrease when passing through a single dam is The total percent of salmon is 100%, The percent of salmon pass through a single dam is 88% So, 100% – 88% = 12% Finally , The percent of the number of young salmon decrease when passing through a single dam is 12% d. An Example of real-life situation in which a quantity increases by a constant percent each time an event occurs. is , while we are filling the tank with water , The amount of water ingoing increases constantly with the speed of the motor power running the water , water levels in the tank increases by a constant percent each time until the tank is filled up with the water fully. Try It Find the percent of change. Round to the nearest tenth of a percent if necessary. Question 1. 10 inches to 25 inches Answer: percent of change is 150% Explanation: We know that , formula for percent change = $$\frac{New value – old value}{old value}$$ where New value = 25 and old value = 10 , because a change of 10 to 25 is a positive (increase) change So, percent change = $$\frac{25 – 10}{10}$$ = $$\frac{15}{10}$$ = $$\frac{15}{10}$$ × 100 = 150% So, percent of change is 150% Question 2. 57 people to 65 people Answer: percent of change is 14% Explanation: We know that , formula for percent change = $$\frac{New value – old value}{old value}$$ where New value = 65 and old value = 57 , because a change of 57 to 65 is a positive (increase) change So, percent change = $$\frac{65 – 57}{57}$$ = $$\frac{8}{57}$$ = $$\frac{8}{57}$$ × 100 = 14.03 % Approximately We can write it as 14 % So, percent of change is 14% Question 3. In Example 2, what was the percent of change from 2014 to 2015? Answer: percent of change is – 44% Explanation: In Example 2, change from 2014 to 2015 , that is 18 to 10 We know that , formula for percent change = $$\frac{New value – old value}{old value}$$ where New value = 10 and old value = 18 , because a change of 18 to 10 is a negative (decrease) change So, percent change = $$\frac{10 – 18}{18}$$ = $$\frac{-8}{18}$$ = – 0.444 = – 0.444 × 100 = – 44.4 % Approximately We can write it as – 44 % So, percent of change is – 44% Self-Assessment for Concepts & Skills Solve each exercise. Then rate your understanding of the success criteria in your journal. Question 4. VOCABULARY What does it mean for a quantity to change by n%? Answer: The meaning of a quantity to change is given by, which the amount of quantity increases or decreases of its original value to new value is called percent of change, Then percent of change can be in positive or negative depending on the value of change , if it is for n% then it can be increase or decrease in the quantity of change. Question 5. NUMBER SENSE Without calculating, determine which situation has a greater percent of change. Explain. • 5 bonus points added to 50 points • 5 bonus points added to 100 points Answer: 5 bonus points added to 50 points has the greater percent of change. Explanation: 5 bonus points added to 50 points Then 50 points to 55 points We know that , formula for percent change = $$\frac{New value – old value}{old value}$$ where New value = 55 and old value = 50 , because a change of 50 to 55 is a positive (increase) change So, percent change = $$\frac{55 – 50}{50}$$ = $$\frac{5}{50}$$ = $$\frac{1}{10}$$ × 100 = 10% So, percent of change is 10% 5 bonus points added to 100 points Then 100 points to 105 points We know that , formula for percent change = $$\frac{New value – old value}{old value}$$ where New value = 105 and old value = 100 , because a change of 100 to 105 is a positive (increase) change So, percent change = $$\frac{105 – 100}{100}$$ = $$\frac{5}{100}$$ = $$\frac{1}{20}$$ × 100 = 5% So, percent of change is 5% Percent of change of 5 bonus points added to 50 points is 10% Percent of change of 5 bonus points added to 100 points is 5% So, 5 bonus points added to 50 points has the greater percent of change. FINDING A PERCENT OF CHANGE Identify the percent of change as an increase or a decrease. Then find the percent of change. Question 6. 8 feet to 24 feet Answer: percent of change is increased that is 200% Explanation: We know that , formula for percent change = $$\frac{New value – old value}{old value}$$ where New value = 24 and old value = 8 , because a change of 8 to 24 is a positive (increase) change So, percent change = $$\frac{24 -8}{8}$$ = $$\frac{16}{8}$$ = 2 × 100 = 200% So, percent of change is 200% Question 7. 300 miles to 210 miles Answer: percent of change is decreased that is -30% Explanation: We know that , formula for percent change = $$\frac{New value – old value}{old value}$$ where New value = 210 and old value = 300 , because a change of300 to 210 is a negative (decrease) change So, percent change = $$\frac{210 – 300}{300}$$ = $$\frac{-90}{300}$$ = $$\frac{-9}{30}$$ × 100 = -0.3 × 100 = -30% So, percent of change is -30% Self-Assessment for Problem Solving Solve each exercise. Then rate your understanding of the success criteria in your journal. Question 8. In one round of a game, you are asked how many bones are in a human body. If the percent error of your answer is at most 5%, you earn two points. If the percent error is at most 10%, but greater than 5%, you earn one point. You guess 195 bones. The correct answer is 206 bones. How many points do you earn? Answer: The percent Error we calculated is at most of 5% so points we earned is 2 points Explanation: Given, You guess 195 bones. The correct answer is 206 bones. The amount of error is 206 – 195 = 11 We know that , formula for percent error = $$\frac{Error value}{Actual value}$$ where Error value = 11 and Actual value = 206 , So, percent Error = $$\frac{11}{206}$$ = 0.053 = 0.053 × 100 = 5.3% Approximately we can write as 5% So, percent Error is 5% Then , the percent Error we calculated is at most of 5% so points we earned is 2 points Question 9. DIG DEEPER! The manager of a restaurant offers a 20% decrease in price to tennis teams. A cashier applies a 10% decrease and then another 10% decrease. Is this the same as applying a 20% decrease? Justify your answer. Answer: There is slight difference between these two methods but are approximately equal. Explanation: Given, The manager of a restaurant offers a 20% decrease in price to tennis teams. Let the total price be 100 So, 100 decrease 20% = 100 × (1 – 20%) = 100 × (1 – 0.2) = 80. Given, A cashier applies a 10% decrease and then another 10% decrease. Let the total price be 100 So, 100 decrease 10% = 100 × (1 – 10%) = 100 × (1 – 0.1) = 90. Again applying 10% decrease 90 decrease 10% = 90 × (1 – 10%) = 90 × (1 – 0.1) = 81. So, There is slight difference between these two methods but are approximately equal Percents of Increase and Decrease Homework & Practice 6.4 Review & Refresh Write and solve an equation to answer the question. Question 1. What number is 25% of 64? Answer: 16 Explanation: By using percent proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{a}{64}$$ = $$\frac{25}{100}$$ a = $$\frac{64 × 25}{100}$$ a = $$\frac{64}{4}$$ a = 16 So, 16 is 25% of 64. Question 2. 39.2 is what percent of 112? Answer: 35% Explanation: By using proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{39.2}{112}$$ = $$\frac{p}{100}$$ p = $$\frac{39.2 × 100}{112}$$ p = $$\frac{3,920}{112}$$ p = 35 So, 39.2 is 35% of 112. Find the sum. Write fractions in simplest form. Question 5. $$\frac{4}{7}$$ + (- $$\frac{6}{7}$$) Answer: – $$\frac{2}{7}$$ Explanation: To find the sum of the fractions given, we have to do addition, that is $$\frac{4}{7}$$ + (- $$\frac{6}{7}$$) = $$\frac{4 – 6}{7}$$ = – $$\frac{2}{7}$$ So, The sum of $$\frac{4}{7}$$ + (- $$\frac{6}{7}$$) is – $$\frac{2}{7}$$ Question 6. – 4.621 + 3.925 Answer: – 0.696 Explanation: To find the sum we have to add the given numbers , that is – 4.621 + 3.925 = 3.925 – 4.621 = – 0.696 So, the sum of – 4.621 + 3.925 is – 0.696. Question 7. –$$\frac{5}{12}$$ + $$\frac{3}{4}$$ Answer: $$\frac{1}{3}$$ Explanation: To find the sum of the fractions given, we have to do addition, that is Given , –$$\frac{5}{12}$$ + $$\frac{3}{4}$$ $$\frac{3}{4}$$ – $$\frac{5}{12}$$ Expand the fraction , multilpy the numerator and denominator by 3 We get , $$\frac{3 × 3}{3 × 4}$$ = $$\frac{3 × 3}{3 × 4}$$ – $$\frac{5}{12}$$ = $$\frac{9}{12}$$ – $$\frac{5}{12}$$ = $$\frac{9 – 5}{12}$$ = $$\frac{4}{12}$$ The simplest form of $$\frac{4}{12}$$ is $$\frac{1}{3}$$ So, The sum of –$$\frac{5}{12}$$ + $$\frac{3}{4}$$ is $$\frac{1}{3}$$ Concepts, Skills, & Problem Solving EXPLORING PERCENT CHANGE You are given the percent of salmon that pass through a single dam unharmed. By what percent does the number of salmon decrease when passing through a single dam? (See Exploration 1, p. 253.) Question 8. 75% Answer: The percent of the number of salmon decrease when passing through a single dam is 25% Explanation: The percent of salmon that pass through a single dam unharmed is 75% Let the percent of salmon coming to the dam are 100% The percent of the number of salmon decrease when passing through a single dam is 100% – 75% = 25% So, The percent of the number of salmon decrease when passing through a single dam is 25% Question 9. 80% Answer: The percent of the number of salmon decrease when passing through a single dam is 20% Explanation: The percent of salmon that pass through a single dam unharmed is 80% Let the percent of salmon coming to the dam are 100% The percent of the number of salmon decrease when passing through a single dam is 100% – 80% = 20% So, The percent of the number of salmon decrease when passing through a single dam is 20% Question 10. 62% Answer: The percent of the number of salmon decrease when passing through a single dam is 38% Explanation: The percent of salmon that pass through a single dam unharmed is 62% Let the percent of salmon coming to the dam are 100% The percent of the number of salmon decrease when passing through a single dam is 100% – 62% = 38% So, The percent of the number of salmon decrease when passing through a single dam is 38% Question 11. 94% Answer: The percent of the number of salmon decrease when passing through a single dam is 6% Explanation: The percent of salmon that pass through a single dam unharmed is 94% Let the percent of salmon coming to the dam are 100% The percent of the number of salmon decrease when passing through a single dam is 100% – 94% = 6% So, The percent of the number of salmon decrease when passing through a single dam is 6% FINDING A PERCENT OF CHANGE Identify the percent of change as an increase or a decrease. Then find the percent of change. Round to the nearest tenth of a percent if necessary. Question 12. 12 inches to 36 inches Answer: percent of change is 200% Explanation: We know that , formula for percent change = $$\frac{New value – old value}{old value}$$ where New value = 36 and old value = 12 , because a change of 12 to 36 is a positive (increase) change So, percent change = $$\frac{36 – 12}{12}$$ = $$\frac{24}{12}$$ = 2 × 100 = 200% So, percent of change is 200% Question 13. 75 people to 25 people Answer: percent of change is 66% Explanation: We know that , formula for percent change = $$\frac{New value – old value}{old value}$$ where New value = 25 and old value = 75 , because a change of 75 to 25 is a negative (decrease) change So, percent change = $$\frac{25 – 75}{75}$$ = $$\frac{- 50}{75}$$ = – 0.666 = – 0.666 × 100 = – 66.6% Approximately we can write it as – 66% So, percent of change is – 66% Question 14. 50 pounds to 35 pounds Answer: percent of change is – 30% Explanation: We know that , formula for percent change = $$\frac{New value – old value}{old value}$$ where New value = 35 and old value = 50 , because a change of 35 to 50 is a negative (decrease) change So, percent change = $$\frac{35 – 50}{50}$$ = $$\frac{- 15}{50}$$ = – 0.3 = – 0.3 × 100 = – 30% So, percent of change is – 30% Question 15. 24 songs to 78 songs Answer: percent of change is 225% Explanation: We know that , formula for percent change = $$\frac{New value – old value}{old value}$$ where New value = 78 and old value = 24 , because a change of 24 to 78 is a positive (increase) change So, percent change = $$\frac{78 – 24}{24}$$ = $$\frac{54}{24}$$ = 2.25 × 100 = 225% So, percent of change is 225% Question 16. 10 gallons to 24 gallons Answer: percent of change is 140% Explanation: We know that , formula for percent change = $$\frac{New value – old value}{old value}$$ where New value = 24 and old value = 10 , because a change of 10 to 24 is a positive (increase) change So, percent change = $$\frac{24 – 10}{10}$$ = $$\frac{14}{10}$$ = 1.4 = 1.4 × 100 = 140% So, percent of change is 140% Question 17. 72 paper clips to 63 paper clips Answer: percent of change is – 12.5% Explanation: We know that , formula for percent change = $$\frac{New value – old value}{old value}$$ where New value = 63 and old value = 72 , because a change of 72 to 63 is a negative (decrease) change So, percent change = $$\frac{63 – 72}{72}$$ = $$\frac{- 9}{72}$$ = $$\frac{- 1}{8}$$ = – 0.125 = – 0.125 × 100 = – 12.5% So, percent of change is – 12.5% Question 18. 16 centimeters to 44.2 centimeters Answer: percent of change is 176% Explanation: We know that , formula for percent change = $$\frac{New value – old value}{old value}$$ where New value = 44.2 and old value = 16 , because a change of 16 to 44.2 is a positive (increase) change So, percent change = $$\frac{44.2 – 16}{16}$$ = $$\frac{28.2}{16}$$ = 1.76 = 1.76 × 100 = 176% So, percent of change is 176% Question 19. 68 miles to 42.5 miles Answer: percent of change is – 37.5% Explanation: We know that , formula for percent change = $$\frac{New value – old value}{old value}$$ where New value = 42.5 and old value = 68 , because a change of 68 to 42.5 is a negative (decrease) change So, percent change = $$\frac{42.5 – 68}{68}$$ = $$\frac{- 25.5}{68}$$ = – 0.375 = – 0.375 × 100 = – 37.5% So, percent of change is – 37.5% Question 20. YOU BE THE TEACHER Your friend finds the percent increase from 18 to 26. Is your friend correct? Explain your reasoning. Answer: No , The percent of change of 18 to 26 is positive (increase) that is 44.4% Explanation: We know that , formula for percent change = $$\frac{New value – old value}{old value}$$ where New value = 26 and old value = 18 , because a change of 18 to 26 is a positive (increase) change So, percent change = $$\frac{26 – 18}{18}$$ = $$\frac{8}{18}$$ = 0.444 = 0.444 × 100 = 44.4% So, percent of change is 44.4% Question 21. MODELING REAL LIFE Last week, you finished Level 2 of a video game in 32 minutes. Today, you finish Level 2 in 28 minutes. What is the percent of change? Answer: The percent of change from last week to today is – 12.5% Explanation: Given, you finished Level 2 of a video game in 32 minutes. Today, you finish Level 2 in 28 minutes. We know that , formula for percent change = $$\frac{New value – old value}{old value}$$ where New value = 28 and old value = 32 , because a change of 32 to 28 is a negative (decrease) change So, percent change = $$\frac{28 – 32}{32}$$ = $$\frac{- 4}{32}$$ = $$\frac{- 1}{8}$$ = – 0.125 = – 0.125 × 100 = – 12.5% So, percent of change is – 12.5% Question 22. MODELING REAL LIFE You estimate that a baby pig weighs 20 pounds. The actual weight of the baby pig is 16 pounds. Find the percent error. Answer: The percent Error is 20%. Explanation: Given , You estimate that a baby pig weighs 20 pounds. The actual weight of the baby pig is 16 pounds. The amount of error is 20 – 16 = 4 We know that , formula for percent error = $$\frac{Error value}{Actual value}$$ where Error value = 4 and Actual value = 20 , So, percent Error = $$\frac{4}{20}$$ = $$\frac{1}{5}$$ = 0.2 = 0.2 × 100 = 20% So, percent Error is 20% . Question 23. PRECISION A researcher estimates that a fossil is 3200 years old. Using carbon-14 dating, a procedure used to determine the age of an object, the researcher discovers that the fossil is 3600 years old. a. Find the percent error. b. What other estimate gives the same percent error? Explain your reasoning. Answer: a. The percent Error is 11.1% b. The other estimate that gives the same percent error is 3,199 years old. Explanation: a. Given, A researcher estimates that a fossil is 3200 years old the researcher discovers that the fossil is 3600 years old. The amount of error is 3600 – 3200 = 400 We know that , formula for percent error = $$\frac{Error value}{Actual value}$$ where Error value = 400 and Actual value = 3600 , So, percent Error = $$\frac{400}{3600}$$ = $$\frac{1}{9}$$ = 0.111 = 0.111 × 100 = 11.1% So, percent Error is 11.1% . b. If The other estimate that gives the same percent error is 3,199 years old. The amount of error is 3600 – 3199 = 401 We know that , formula for percent error = $$\frac{Error value}{Actual value}$$ where Error value = 401 and Actual value = 3600 , So, percent Error = $$\frac{401}{3600}$$ = 0.111 = 0.111 × 100 = 11.1% So, percent Error is 11.1% . So , The other estimate that gives the same percent error is 3,199 years old. FINDING A PERCENT OF CHANGE Identify the percent of change as an increase or a decrease. Then find the percent of change. Round to the nearest tenth of a percent if necessary. Question 24. $$\frac{1}{4}$$ to $$\frac{1}{2}$$ Answer: percent of change is 100% Explanation: The given fractions can be written in decimal form then we have, $$\frac{1}{4}$$ as 0.25 $$\frac{1}{2}$$ as 0.5 , so , 0.25 to 0.5 We know that , formula for percent change = $$\frac{New value – old value}{old value}$$ where New value = 0.5 and old value = 0.25 , because a change of 0.25 to 0.5 is a positive (increase) change So, percent change = $$\frac{0.5 – 0.25}{0.25}$$ = $$\frac{0.25}{0.25}$$ = 1 × 100 = 100% So, percent of change is 100% Question 25. $$\frac{4}{5}$$ to $$\frac{3}{5}$$ Answer: percent of change is – 25% Explanation: The given fractions can be written in decimal form then we have, $$\frac{4}{5}$$ as 0.8 $$\frac{3}{5}$$ as 0.6 , so , 0.8 to 0.6 We know that , formula for percent change = $$\frac{New value – old value}{old value}$$ where New value = 0.6 and old value = 0.8 , because a change of 0.8 to 0.6 is a negative (decrease) change So, percent change = $$\frac{0.6 – 0.8}{0.8}$$ = $$\frac{-0.2}{0.8}$$ = – 0.25 = – 0.25 × 100 = – 25% So, percent of change is – 25% Question 26. $$\frac{3}{8}$$ to $$\frac{7}{8}$$ Answer: percent of change is 135% Explanation: The given fractions can be written in decimal form then we have, $$\frac{3}{8}$$ as 0.37 $$\frac{7}{8}$$ as 0.87 , So 0.37 to 0.87 We know that , formula for percent change = $$\frac{New value – old value}{old value}$$ where New value = 0.87 and old value = 0.37 , because a change of 0.37 to 0.87 is a positive (increase) change So, percent change = $$\frac{0.87 – 0.37}{0.37}$$ = $$\frac{0.5}{0.37}$$ = 1.35 = 1.35 × 100 = 135% So, percent of change is 135% Question 27. $$\frac{5}{4}$$ to $$\frac{3}{8}$$ Answer: percent of change is – 70.4% Explanation: The given fractions can be written in decimal form then we have, $$\frac{5}{4}$$ as 1.25 $$\frac{3}{8}$$ as 0.37 , So, 1.25 to 0.37 We know that , formula for percent change = $$\frac{New value – old value}{old value}$$ where New value = 0.37 and old value = 1.25 , because a change of 1.25 to 0.37 is a negative (decrease) change So, percent change = $$\frac{0.37 – 1.25}{1.25}$$ = $$\frac{- 0.88}{1.25}$$ = – 0.704 = – 0.704 × 100 = – 70.4% So, percent of change is – 70.4% Question 28. CRITICAL THINKING Explain why a change from 20 to 40 is a 100% increase, but a change from 40 to 20 is a 50% decrease. Answer: From 20 to 40 is a 100% increase because of increase in number value and from 40 to 20 is a 50% decrease because of decrease in number value. Explanation: Given , 20 to 40 We know that , formula for percent change = $$\frac{New value – old value}{old value}$$ where New value = 40 and old value = 20 , because a change of 20 to 40 is a positive (increase) change So, percent change = $$\frac{40 – 20}{20}$$ = $$\frac{20}{20}$$ = 1 × 100 = 100% Then , percent of change is 100% So, The percent of change from 20 to 40 is a 100% increase Given , 40 to 20 We know that , formula for percent change = $$\frac{New value – old value}{old value}$$ where New value = 20 and old value = 40 , because a change of 40 to 20 is a negative (decrease) change So, percent change = $$\frac{20 – 40}{40}$$ = $$\frac{- 20}{40}$$ = $$\frac{- 1}{2}$$ = – 0.5 = – 0.5 × 100 = – 50% Then , percent of change is – 50% So , The percent of change from 40 to 20 is a 50% decrease. Finally , From 20 to 40 is a 100% increase because of increase in number value and from 40 to 20 is a 50% decrease because of decrease in number value. Question 29. MODELING REAL LIFE The table shows population data for a community. a. What is the percent of change from 2011 to 2017? b. Predict the population in 2023. Explain your reasoning. Answer: a. The percent of change from 2011 to 2017 is 169% b. The estimated population of 2023 will be 158,000. Explanation: a. Given 2011 to 2017 , so from the table we know it as , 118,000 to 138,000 We know that , formula for percent change = $$\frac{New value – old value}{old value}$$ where New value = 138,000 and old value = 118,000 , because a change of 118,000 to 138,000 is a positive (increase) change So, percent change = $$\frac{138,000 – 118,000}{118,000}$$ = $$\frac{20,000}{118,000$$ = 0.169 =0.169 × 100 = 169% So, percent of change is 169% b. The population from 2011 to 2017 increased from 118,000 to 138,000, The difference between 2011 to 2017 is 6 years ,So the population increase in numbers are 138,000 – 118,000 = 20,000. If the population in 6 years is increased by 20,000. Then from 2017 to 2023 is 6 years , So increase in population is 20,000 Then for 2023 The population will be 138,000 + 20,000 = 158,000 finally, The estimated population of 2023 will be 158,000. Question 30. GEOMETRY Suppose the length and the width of the sandbox are doubled. a. Find the percent of change in the perimeter. b. Find the percent of change in the area. Answer: a. the percent of change in the perimeter of sandbox is 100% b. the percent of change in the area of sandbox is Explanation: a. Given , length of sandbox = 10 ft , width of sandbox = 6 ft The sandbox is in the form of a rectangle , So the perimeter of a rectangle is P = 2 ( l + w) , where l = length of the rectangle and w = width of the rectangle, Then P = 2( l + w ) = 2 ( 10 + 6 ) = 2 × 16 =18 So, the perimeter of sandbox is 32 ft Given that , the length and the width of the sandbox are doubled. Then l = 20 ft and w = 12 ft , P = 2( l + w ) = 2 ( 20 + 12 ) = 2 × 32 = 64 So, the perimeter of sandbox after the length and the width are doubled. is 64 ft . The perimeter of sandbox changed from 32 ft to 64 ft , Then We know that , formula for percent change = $$\frac{New value – old value}{old value}$$ where New value = 64 and old value = 32 , because a change of 32 to 64 is a positive (increase) change So, percent change = $$\frac{64 – 32}{32}$$ = $$\frac{32}{32}$$ = 1 × 100 = 100% So, percent of change is 100% Finally , the percent of change in the perimeter of sandbox is 100% b. Given , length of sandbox = 10 ft , width of sandbox = 6 ft The sandbox is in the form of a rectangle , So the area of a rectangle is A = l × w , Then A = l × w = 10 × 6 = 60 So , The Area of Sandbox is 60 ft Given that , the length and the width of the sandbox are doubled. Then , l = 20 ft and w = 12 ft , Then A = l × w = 20 × 12 = 240 So , The Area of Sandbox after the length and the width are doubled is 240 ft The area of sandbox changed from 60 ft to 240 ft We know that , formula for percent change = $$\frac{New value – old value}{old value}$$ where New value = 240 and old value = 60 , because a change of 60 to 240 is a positive (increase) change So, percent change = $$\frac{240 – 60}{60}$$ = $$\frac{180}{60}$$ = 3 = 3 × 100 = 300% So, percent of change is 300% Finally , the percent of change in the area of sandbox is 300%. Question 31. MODELING REAL LIFE A company fills boxes with about 21 ounces of cereal. The acceptable percent error in filling a box is 2.5%. Box A contains 20.4 ounces of cereal and Box B contains 21.5 ounces of cereal. Tell whether each box is an acceptable weight. Answer: The percent error of Box A is greater than acceptable percent error that is 2.8% and The percent error of Box B is less than acceptable percent error that is 2.3%. Explanation: For Box A , Given , A company fills boxes with about 21 ounces of cereal and Box A contains 20.4 ounces of cereal The amount of error is 21 – 20.4 = 0.6 We know that , formula for percent error = $$\frac{Error value}{Actual value}$$ where Error value = 0.6 and Actual value = 21 , So, percent Error = $$\frac{0.6}{21}$$ = 0.028 = 0.028 × 100 = 2.8% So, percent Error is 2.8% The percent error of Box A is greater than acceptable percent error that is 2.8% For Box B A company fills boxes with about 21 ounces of cereal and Box B contains 21.5 ounces of cereal. The amount of error is 21.5 – 21 = 0.5 We know that , formula for percent error = $$\frac{Error value}{Actual value}$$ where Error value = 0.5 and Actual value = 21 , So, percent Error = $$\frac{0.5}{21}$$ = 0.023 = 0.023 × 100 = 2.3% So, percent Error is 2.3% The percent error of Box B is less than acceptable percent error that is 2.3% Question 32. PRECISION Find the percent of change from June to September in the mile-run times shown. Answer: The percent of change from June to September in the mile-run times is -26%. Explanation: Given , change from 7.45 to 5.51 We know that , formula for percent change = $$\frac{New value – old value}{old value}$$ where New value = 5.51 and old value = 7.45 , because a change of 7.45 to 5.51 is a negative (decrease) change So, percent change = $$\frac{5.51 – 7.45}{7.45}$$ = $$\frac{- 1.94}{7.45}$$ = – 0.26 = – 0.26 × 100 = – 26% So, percent of change is – 26% Finally , The percent of change from June to September in the mile-run times is -26%. Question 33. CRITICAL THINKING A number increases by 10% and then decreases by 10%. Will the result be greater than, less than or equal to, the original number? Explain. Answer: A number increases by 10% and then decreases by 10% , the result will be less than the original number. Explanation: Let original number is 100 Given ,The original number increases by 10% The new number = original number + The original number increases by 10% = 100 + ( 100 × 10%) = 100 + ( 100 × 0.1) = 100 + 10 = 110 So, The new number is 110. Let the original number is 110 Given ,The original number decreases by 10% The new number = original number – The original number decreases by 10% = 110 – ( 110 × 10%) = 110 – ( 110 × 0.1) = 110 – 11 = 99 So, The new number is 99 99 < 100 . So , A number increases by 10% and then decreases by 10% , the result will be less than the original number. Question 34. PROBLEM SOLVING You want to reduce your daily calorie consumption by about 9%. You currently consume about 2100 calories per day. Use mental math to estimate the number of calories you should consume in one week to meet your goal. Explain. Answer: The estimated calories you should consume per week is 13,377. Explanation: Given ,You currently consume about 2100 calories per day, reduce your daily calorie consumption by about 9%. so, The calories you have to consume after reduction of 9% is The new number = original number – The original number decreases by 9% = 2100 – ( 2100 × 9%) = 2100 – (2100 × 0.09) = 2100 – 189 = 1,911 . The calories you have to consume after reduction of 9% is 1,911 The calories you should consume per day is 1,911. The calories you should consume per week (7 days) = 1,911 × 7 = 13,377. Finally , The estimated calories you should consume per week is 13,377. Question 35. DIG DEEPER! Donations to an annual fundraiser are 15% greater this year than last year. Last year, donations were10% greater than the year before. The amount raised this year is$10,120. How much was raised two years ago?
Answer: The amount raised 2 years ago is $7,741.8. Explanation: Given , The amount raised this year is$10,120.
Let the amount raised last year = x
Donations are 15% greater than last year
The amount raised last year = The amount raised this year – ((The amount raised this year .15%)
x = 10,120 – ( 10,120 × 0.15)
x = 10,120 – 1,518
x = 8,602
The amount raised last year = $8,602. We know that , The amount raised last year =$8,602.
Let the amount raised the year before = x
Donations are 10% greater than the year before
The amount raised the year before = The amount raised last year – ((The amount raised last year .10%)
x = 8,602 – ( 8,602 × 0.1)
x = 8,602 – 860.2
x = 7,741.8
The amount raised the year before = $7,741.8. So, The amount raised 2 years ago is$7,741.8.

Question 36.
REASONING
Forty students are in the science club. Of those, 45% are girls. This percent increases to 56% after more girls join the club. How many more girls join?
Answer:  The number of new girls join the club is 10.

Explanation:
Let the number of new girls = x
The number of girls = x + 18
The number of students = x + 40
So, the number of girls = 56% The number of students
x + 18 = 0.56( x + 40 )
x + 18  = 0.56x + 22.4
0.56x – x = 22.4 – 18
0.44x = 4.4
x = $$\frac{4.4}{0.44}$$
x = 10.
So , The number of new girls is 10.

Lesson 6.5 Discounts and Markups

EXPLORATION 1

Comparing Discounts
Work with a partner.
a. The same pair of earrings is on sale at three stores. Which store has the best price? Use the percent models to justify your answer.

b. You buy the earrings on sale for 30% off at a different store. You pay $22.40. What was the original price of the earrings? Use the percent model to justify your answer. c. You sell the earrings in part(b) to a friend for 60% more than what you paid. What is the selling price? Use a percent model to justify your answer. Answer: a. For store A ,The sales price is$27
For store B  ,The sales price is $24.5 For store C ,The sales price is$31.2

b. the original price  of the earrings is $32 c. the selling price is$35.84.

Explanation:
a. Given store A = $45 with 40% off We know , The sales price be 100% – 40% = 60% of the original price sales price = 60% of 45 = 0.6 × 45 = 27 So, The sales price is$27

For store B = $49 with 50% off We know , The sales price be 100% – 50% = 50% of the original price sales price = 50% of 49 = 0.5 × 49 = 24.5 So, The sales price is$24.5

For store c = $39 with 20% off We know , The sales price be 100% – 20% = 80% of the original price sales price = 80% of 39 = 0.8 × 39 = 31.2 So, The sales price is$31.2

b. Given , You buy the earrings on sale for 30% off at a different store. You pay $22.40 The saples price = 100% – 30% = 70% we know a = 22.4 , p = 70% , w = ? a = p% × w 22.4 = 0.7 × w w = $$\frac{22.4}{0.7}$$ w = 32. So , the original price of the earrings is$32

c. Given , You sell the earrings in part(b) to a friend for 60% more than what you paid.
If the selling price is more than the buying price then it is called markup

Here , the markup is 60% of $22.4 a = p% × w a = 0.6 × 22.4 a = 13.44 So, the markup is$13.44
We know that selling price = cost of buying + markup
= 22.4 + 13.44
= 35.84
So , the selling price is $35.84. Try It Question 1. The original price of a skateboard is$50. The skateboard is on sale for 20% off. What is the sale price?
Answer: The sales price of skateboard is $40 Explanation: Given , skateboard is$50 with 20% off
We know , The sales price be 100% – 20% = 80% of the original price
sales price = 80% of 50
= 0.8 × 50 = 40
So, The sales price is $40 Question 2. The discount on a DVD is 50%. It is on sale for$10. What is the original price of the DVD?
Answer: The original price of DVD is $20. Explanation: Given , discount on a DVD is 50%. , It is on sale for$10
We know , The sales price be 100% – 50% = 50%
a = p% × w
10 = 0.5 × w
w = $$\frac{10}{0.5}$$
w = 20
So, The original price is $20. Self-Assessment for Concepts & Skills Solve each exercise. Then rate your understanding of the success criteria in your journal. Question 3. WRITING Describe how to find the sale price of an item that has a 15% discount. Answer: To get the sales price , The discount percent must be cleared from the total percent of sales price , it gives the sales percentage of the original price , which is used to find the sales price. So, The sales price = original price minus discount. Explanation: Let the original price be$50 with 15% off
We know , The sales price be 100% – 15% = 85% of the original price
sales price = 85% of 50
= 0.85 × 50 = 42.5
So, The sales price is $42.5 FINDING A SALE PRICE Find the sale price. Use a percent model to check your answer. Question 4. A portable table tennis set costs$30 before a 30% discount.
Answer: The sales price of portable table tennis set is $21. Explanation: Given , tennis set is$30 with 30% off
We know , The sales price be 100% – 30% = 70% of the original price
sales price = 70% of 30
= 0.7 × 30 = 21
So, The sales price is $21. Question 5. The original price of an easel is$70. The easel is on sale for 20% off.
Answer: The sales price  of an easel is $56. Explanation: Given , easel is$70 with 20% off
We know , The sales price be 100% – 20% = 80% of the original price
sales price = 80% of 70
= 0.8 × 70 = 56
So, The sales price is $56. FINDING AN ORIGINAL PRICE Find the original price. Use a percent model to check your answer. Question 6. A bracelet costs$36 after a 25% discount.

Answer: The original price of bracelet is $48. Explanation: Given , discount on a bracelet is 25%. , It is cost$36  after discount
We know , The sales price be 100% – 25% = 75%
a = p% × w
36 = 0.75 × w
w = $$\frac{36}{0.75}$$
w = 48
So, The original price is $48. Question 7. The discount on a toy robot is 40%. The toy robot is on sale for$54.
Answer: The sale price toy robot is $32.4 . Explanation: Given , discount on a toy robot is 40%. The toy robot is on sale for$54.
We know , The sales price be 100% – 40% = 60%
a = p% × w
a = 0.6 × 54
a = 32.4
So, The sale price is $32.4 . Self-Assessment for Problem Solving Solve each exercise. Then rate your understanding of the success criteria in your journal. Question 8. DIG DEEPER! You have two coupons for a store. The first coupon applies a$15 discount to a single purchase, and the second coupon applies a 10% discount to a single purchase. You can only use one coupon on a purchase. When should you use each coupon? Explain.
Answer: The first coupon of $15 discount is to be used on your highest cost of purchase and the second coupon with 10% off is to be used on your least cost of purchase . Explanation: Given , The first coupon applies a$15 discount to a single purchase,
Let the purchase be $50 Then the first coupon applies =$50 – $15 =$35.
So, when the first coupon applies price will be $35. Given , the second coupon applies a 10% discount to a single purchase Let the original price be$50 and with 10% off
We know , The sales price be 100% – 10% = 90% of the original price
sales price = 90% of 50
= 0.9 × 50 = 45
So, The sales price is $45. Here , we can see that$15 discount is offering the reduction of the original price and  discount with 10% of is offering to pay the 90% of its original price.

Finally , The first coupon of $15 discount is to be used on your highest cost of purchase and the second coupon with 10% off is to be used on your least cost of purchase . Question 9. A store sells memory cards for$25 each.

a. The markup for each memory card is 25%. How much did the store pay for 50 memory cards?
b. The store offers a discount when a customer buys two or more memory cards. A customer pays $47.50 for two memory cards. What is the percent of discount? c. How much does a customer pay for three memory cards if the store increases the percent of discount in part (b) by 2%? Answer: a. The store pay$937.5 for 50 memory cards
b. The discount offered by the store is 5%
c. The customer paid $69.75 for 3 memory cards. Explanation: a. The markup is 25% of$25
a = p% × w
= 25% × 25
= 0.25 × 25
= 6.25
So , the markup is $6.25. To , find the cost to store , we have cost to store = selling price – markup =$25 – $6.25 =$18.75.
The cost to store for each memory card is $18.75. Then for 50 memory cards = 50 ×$18.75 = $937.5 b. A customer pays$47.50 for two memory cards.
Then for one memory card $$\frac{47.5}{2}$$ = $23.75 we have , a =$23.75 , w = $25 , p = ? By using proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{23.75}{25}$$ = $$\frac{p}{100}$$ p = $$\frac{23.75 × 100}{25}$$ p = $$\frac{2375}{25}$$ p = 95 So, 23.75 is 95% of 25. To find the discount , = The percent of original price – the percent of selling price = 100% – 95% = 5% So, The discount offered by the store is 5% c. If the discount is increased by 2% , Then the discount offered by store is 5% + 2% = 7% The amount of selling price = 100% – 7% = 93% So , 93% of$25
By using percent proportion , we have
$$\frac{a}{w}$$ = $$\frac{p}{100}$$
$$\frac{a}{25}$$ = $$\frac{93}{100}$$
a = $$\frac{25 × 93}{100}$$
a = $$\frac{2325}{100}$$
a = 23.25
So, The amount after  the discount is $23.25 Then for 3 memory cards =$23.25 × 3 = $69.75 The customer paid$69.75 for 3 memory cards.

Discounts and Markups Homework & Practice 6.5

Review & Refresh

Identify the percent of change as an increase or decrease. Then find the percent of change. Round to the nearest tenth of a percent if necessary.
Question 1.
16 meters to 20 meters
Answer: The percent of change is 25%

Explanation:
We know that , formula for percent change = $$\frac{New value – old value}{old value}$$
where New value = 20 and old value = 16 , because a change of 16 to 20 is a positive (increase) change
So, percent change = $$\frac{20 – 16}{16}$$
= $$\frac{4}{16}$$
= $$\frac{1}{4}$$
= 0.25
= 0.25 × 100
= 25%
So, percent of change is 25%

Question 2.
9 points to 4 points
Answer: The percent of change is – 55.5%

Explanation:
We know that , formula for percent change = $$\frac{New value – old value}{old value}$$
where New value = 4 and old value = 9 , because a change of 9 to 4 is a negative (decrease) change
So, percent change = $$\frac{4 – 9}{9}$$
= $$\frac{- 5}{9}$$
= – 0.555
= – 0.555 × 100
= – 55.5%
So, percent of change is – 55.5%

Question 3.
15 ounces to 5 ounces
Answer: The percent of change is – 66.6%

Explanation:
We know that , formula for percent change = $$\frac{New value – old value}{old value}$$
where New value = 5 and old value = 15 , because a change of 15 to 5 is a negative (decrease) change
So, percent change = $$\frac{5 – 15}{15}$$
= $$\frac{- 10}{15}$$
= – 0.666
= – 0.666 × 100
= – 66.6%
So, percent of change is – 66.6%

Question 4.
38 staples to 55 staples
Answer:

Explanation:
We know that , formula for percent change = $$\frac{New value – old value}{old value}$$
where New value = 55 and old value = 38 , because a change of 38 to 55 is a positive (increase) change
So, percent change = $$\frac{55 – 38}{38}$$
= $$\frac{17}{38}$$
= 0.447
= 0.447 × 100
= 44.7%
So, percent of change is 44.7%

Find the product. Write fractions in simplest form.
Question 5.
$$\frac{4}{7}\left(-\frac{1}{6}\right)$$
Answer: The product of  $$\frac{4}{7}\left(-\frac{1}{6}\right)$$ is 0.094.

Explanation:
The given fractions can be written in decimal form then we have,
$$\frac{4}{7}\left(-\frac{1}{6}\right)$$ as 0.571 ( – 0.166)
To find the product we have to multiply the two numbers
= 0.571 × ( – 0.166)
= 0.094
So, The product of  $$\frac{4}{7}\left(-\frac{1}{6}\right)$$ is 0.094.

Question 6.
– 1.58(6.02)
Answer: The product of – 1.58(6.02) is  – 9.51.

Explanation:
To find the product we have to multiply the two numbers
= – 1.58 × (6.02)
=  – 9.51
So, The product of – 1.58(6.02) is  – 9.51.

Question 7.
– 3(- 2$$\frac{1}{8}$$)
Answer:

Explanation:
The given fractions can be written in decimal form then we have,
– 3(- 2$$\frac{1}{8}$$) as – 3 ( – 2.12)
To find the product we have to multiply the two numbers
= – 3 × ( – 2.12)
=  6.36
So, The product of  – 3 ( – 2.12) is 6.36

Concepts, Skills, & Problem Solving

COMPARING DISCOUNTS The same item is on sale at two stores. Which one is the better price? Use percent models to justify your answer. (See Exploration 1, p. 259.)
Question 8.
60% off $60 or 55% off$50
Answer: The item has better price at 55% off $50 . Explanation: a. Given , 60% off$60.
We know , The sales price be 100% – 60% = 40% of the original price
sales price = 40% of 60
= 0.4 × 60 = 24
So, The sales price is $24 b. Given , 55% off$50

We know , The sales price be 100% – 55% = 45% of the original price
sales price = 45% of 50
= 0.45 × 50 = 22.5
So, The sales price is $22.5 Question 9. 85% off$90 or 70% off $65 Answer: The item has better price at 85% off$90 .

Explanation:

a. Given , 85% off $90 We know , The sales price be 100% –85% = 15% of the original price sales price = 15% of 90 = 0.15 × 90 = 13.5 So, The sales price is$13.5 .

b. Given , 70% off $65 We know , The sales price be 100% –70% = 30% of the original price sales price = 30% of 65 = 0.3 × 65 = 19.5 So, The sales price is$19.5.

USING TOOLS Copy and complete the table.

Answer:
10. Given , Original price of the item is $80 , percent of discount is 20% , Find sales price ? Answer: The sales price of the is$64.

Explanation:
We know , The sales price be 100% – 20% = 80% of the original price
sales price = 80% of 80
= 0.8 × 80 = 64
So, The sales price is $64. 11. Given , Original price of the item is$42 , percent of discount is 15% , Find sales price ?
Answer: The sales price of the is $35.7. Explanation: We know , The sales price be 100% – 15% = 85% of the original price sales price = 85% of 42 = 0.85 × 42 = 35.7 So, The sales price is$35.7.

12. Given , Original price of the item is $120 , percent of discount is 80% , Find sales price ? Answer: The sales price of the is$24.

Explanation:
We know , The sales price be 100% – 80% = 20% of the original price
sales price = 20% of 120
= 0.2 × 120 = 24
So, The sales price is $24. 13. Given , Original price of the item is$112 , percent of discount is 32% , Find sales price ?
Answer: The sales price of the is $76.16. Explanation: We know , The sales price be 100% – 32% = 68% of the original price sales price = 68% of 112 = 0.68 × 112 = 76.16 So, The sales price is$76.16.

14. Given , Original price of the item is $69.8 , percent of discount is 60% , Find sales price ? Answer: The sales price of the is$27.92.

Explanation:
We know , The sales price be 100% – 60% = 40% of the original price
sales price = 40% of 69.8
= 0.4 × 69.8 = 27.92
So, The sales price is $27.92. 15. Given , sales price of the item is$40 , percent of discount is 25% , Find original price ?
Answer: The Original price of the is $53. Explanation: The sales price be 100% – 25% = 75% we know a = 40 , p = 75% , w = ? a = p% × w a = 75% × w 40 = 0.75 × w w = $$\frac{40}{0.75}$$ w = 53 So , the original price of the earrings is$53.

16. Given , sales price of the item is $57 , percent of discount is 5% , Find original price ? Answer: The Original price of the is$60.

Explanation:
The sales price be 100% – 5% = 95%
we know  a = 57 , p = 95% , w = ?
a  =  p% × w
a = 95% × w
57 = 0.95 × w
w = $$\frac{57}{0.95}$$
w = 60
So , the original price of the earrings is $60. 17. Given , sales price of the item is$90 , percent of discount is 80% , Find original price ?
Answer: The Original price of the is $450. Explanation: The sales price be 100% – 80% = 20% we know a = 90 , p = 20% , w = ? a = p% × w a = 20% × w 90 = 0.2 × w w = $$\frac{90}{0.2}$$ w = 450 So , the original price of the earrings is$450.

18. Given , sales price of the item is $72 , percent of discount is 64% , Find original price ? Answer: The Original price of the is$200.

Explanation:
The sales price be 100% – 64% = 36%
we know  a = 72 , p = 36% , w = ?
a  =  p% × w
a = 36% × w
72 = 0.36 × w
w = $$\frac{72}{0.36}$$
w = 200
So , the original price of the earrings is $200. 19. Given , sales price of the item is$146.54 , percent of discount is 15% , Find original price ?
Answer: The Original price of the is $172.4. Explanation: The sales price be 100% – 15% = 85% we know a = 146.54 , p = 85% , w = ? a = p% × w a = 85% × w 146.54 = 0.85 × w w = $$\frac{146.54}{0.85}$$ w = 172.4 So , the original price of the earrings is$172.4.

20. Given , original price  of the item is $60 , sales price of the item is$45 , Find percent of discount ?
Answer: The percent of discount is 25% .

Explanation:
We have a = 45 , w = 60 , p = ?
By using proportion , we have
$$\frac{a}{w}$$ = $$\frac{p}{100}$$
$$\frac{45}{60}$$ = $$\frac{p}{100}$$
p = $$\frac{45 × 100}{60}$$
p = $$\frac{4,500}{60}$$
p = 75
So, 45 is 75% of 60.
To get the percent of discount we have ,
The percent of discount = The percent of original price – the percent of sales price
= 100% – 75%
= 25%.
So, The percent of discount is 25% .

21. Given , original price  of the item is $82 , sales price of the item is$65.6 , Find percent of discount ?
Answer: The percent of discount is 20% .

Explanation:
We have a = 65.6 , w = 82 , p = ?
By using proportion , we have
$$\frac{a}{w}$$ = $$\frac{p}{100}$$
$$\frac{65.6}{82}$$ = $$\frac{p}{100}$$
p = $$\frac{65.6 × 100}{82}$$
p = $$\frac{6,560}{82}$$
p = 80
So, 65.6 is 80% of 82.
To get the percent of discount we have ,
The percent of discount = The percent of original price – the percent of sales price
= 100% – 80%
= 20%.
So, The percent of discount is 20% .

22. Given , original price  of the item is $95 , sales price of the item is$61.75 , Find percent of discount ?
Answer: The percent of discount is 35% .

Explanation:
We have a = 61.75 , w = 95 , p = ?
By using proportion , we have
$$\frac{a}{w}$$ = $$\frac{p}{100}$$
$$\frac{61.75}{95}$$ = $$\frac{p}{100}$$
p = $$\frac{61.75 × 100}{95}$$
p = $$\frac{6,175}{95}$$
p = 65
So, 61.75 is 65% of 90.
To get the percent of discount we have ,
The percent of discount = The percent of original price – the percent of sales price
= 100% – 65%
= 35%.
So, The percent of discount is 35% .

FINDING A SELLING PRICE .

Question 23.
Cost to store: $50 Markup: 10% Answer: The selling price is$55.

Explanation:
The markup is 10% of $50 a = p% × w = 10% × 50 = 0.1 × 50 = 5 So , the markup is$5.
To , find the selling price , we have
selling price = cost to store + markup
= $50 +$5
= $55. The selling price is$55.

Question 24.
Cost to store: $80 Markup: 60% Answer: The selling price is$128.

Explanation:
The markup is 60% of $80 a = p% × w = 60% × 80 = 0.6 × 80 = 48 So , the markup is$48.
To , find the selling price , we have
selling price = cost to store + markup
= $80 +$48
= $128. The selling price is$128.

Question 25.
Cost to store: $140 Markup: 25% Answer: The selling price is$175.

Explanation:
The markup is 25% of $140 a = p% × w = 25% × 140 = 0.25 × 140 = 35 So , the markup is$35.
To , find the selling price , we have
selling price = cost to store + markup
= $140 +$35
= $175. The selling price is$175.

Question 26.
YOU BE THE TEACHER
A store pays $60 for an item. Your friend finds the selling price when the markup is 20%. Is your friend correct? Explain your reasoning. Answer: No , The selling price is$72.

Explanation:
Given , The markup is 20% of $60 a = p% × w = 20% × 60 = 0.2 × 60 = 12 So , the markup is$12.
To , find the selling price , we have
selling price = cost to store + markup
= $60 +$12
= $72. The selling price is$72.

Question 27.
STRUCTURE
The scooter is being sold at a 10% discount. The original price is shown. Which methods can you use to find the new sale price? Which method do you prefer? Explain.

Answer: The sales price is $37.8. Used the method of Multiplying$42 by 0.9.

Explanation:
By Using the method of Multiplying $42 by 0.9 We know , The sales price be 100% – 10% = 90% of the original price sales price = 90% of 42 = 0.9 × 42 = 37.8 So, The sales price is$37.8.

Question 28.
NUMBER SENSE
The original price of an item is P dollars. Is the price of the item with an 18% markup the same as multiplying the original price by 1.18? Use two expressions to justify your answer.
Answer: The selling price is $1.18P and it is same as multiplying the original price by 1.18 . Explanation: Given , The original price of an item is P dollars. The markup is 18% of$P
a = p% × w
= 18% × P
= 0.18 × P
= $0.18P So , the markup is$0.18P.
To , find the selling price , we have
selling price = cost to store + markup
= $P +$0.18P
= $P ( 1 + 0.18 ) =$1.18P.
The selling price is $1.18P. The given method is multiplying the original price by 1.18, The original price is$P = $P × 1018. So , The selling price is$1.18P.

Finally , The selling price is $1.18P and it is same as multiplying the original price by 1.18 . Question 29. PROBLEM SOLVING You are shopping for a video game system. a. At which store should you buy the system? b. Store A has a weekend sale. What discount must Store A offer for you to buy the system there? Answer: a. you should buy the system at store C has The selling price is$200.
b. To buy the system at store A  , it should have the discount of  28.6%.

a. Given , For store A cost to store is $162 , Markup is 40% Then , The markup is 40% of$162
a = p% × w
= 40% × 162
= 0.4 × 162
= 64.8
So , the markup is $64.8. To , find the selling price , we have selling price = cost to store + markup =$162 + $64.8 =$226.8.
So , For store A , The selling price is $226.8. Given , For store B cost to store is$155 , Markup is 30%
Then , The markup is 30% of $155 a = p% × w = 30% × 155 = 0.3 × 155 = 46.5 So , the markup is$46.5.
To , find the selling price , we have
selling price = cost to store + markup
= $155 +$46.5
= $201.5. So , For store B , The selling price is$201.5.

Given , For store C cost to store is $160 , Markup is 25% Then , The markup is 25% of$160
a = p% × w
= 25% × 160
= 0.25 × 160
= 40
So , the markup is $40. To , find the selling price , we have selling price = cost to store + markup =$160 + $40 =$200.
So , For store C , The selling price is $200. you should buy the system at store C has The selling price is$200.

b. Given , Store A has a weekend sale, to buy the system there , it should have the discount of ,
For store A , The selling price is $226.8 , cost to store is$162
We know a = $162 , w =$226.8
By using proportion , we have
$$\frac{a}{w}$$ = $$\frac{p}{100}$$
$$\frac{162}{226.8}$$ = $$\frac{p}{100}$$
p = $$\frac{162 × 100}{226.8}$$
p = $$\frac{16,200}{226.8}$$
p = 71.4%
So, 162 is 71.4% of 226.8.
To get the percent of discount we have ,
The percent of discount = The percent of original price – the percent of sales price
= 100% – 71.4%
= 28.6%.
So, The percent of discount is 28.6% .
Hence , To buy the system at store A  , it should have the discount of  28.6%.

Question 30.
DIG DEEPER!
A pool manager balances the pH level of a pool. The price of a bucket of chlorine tablets is $90, and the price of a pH test kit is$11. The manager uses a coupon that applies a 40% discount to the total cost of the two items. How much money does the pool manager pay for each item?
Answer: The pool manager pays for each item $60.6. Explanation: Now, to find the money the pool manager pay for each item The price of a bucket of chlorine tablets is =$90.
The price of a pH test kit = $11. So, to get the total price of both items we add both prices:$90 + $11 =$101.
Total cost of two items = $101. As, given a coupon that applies a 40% discount to the total cost of the two items. Now, to get the total cost by applying discount: 101 – (40% of 101) = 101 – (0.4 × 101) = 101 – 40.4 = 60.6 So, The pool manager pays for each item$60.6

Question 31.
PRECISION
You buy a pair of jeans at a department store.

a. What is the percent of discount to the nearest percent?
b. What is the percent of sales tax to the nearest tenth of a percent?
c. The price of the jeans includes a 60% markup. After the discount, what is the percent of markup to the nearest percent?
Answer: a. The percent of discount is to the nearest percent 25% .
b. The percent of sales tax to the nearest percent is 7%.
c. The percent of markup to the nearest percent is 54%

Explanation:
a. To find the percent of discount to the nearest percent , we have
a = 29.99 , w = 39.99 , p = ?
By using proportion , we have
$$\frac{a}{w}$$ = $$\frac{p}{100}$$
$$\frac{29.99}{39.99}$$ = $$\frac{p}{100}$$
p = $$\frac{29.99 × 100}{39.99}$$
p = 0.749 × 100
p = 74.9
So, 29.99 is 74.9% of 39.99.
To get the percent of discount we have ,
The percent of discount = The percent of original price – the percent of sales price
= 100% – 74.9%
= 24.9%.
So, The percent of discount is 25% .

b. To find the percent of sales tax to the nearest tenth of a percent we have
sales tax = 1.95 , price = 29.99 , so , a = 1.95 , w = 29.99 , p = ?
By using proportion , we have
$$\frac{a}{w}$$ = $$\frac{p}{100}$$
$$\frac{1.95}{29.99}$$ = $$\frac{p}{100}$$
p = $$\frac{1.95 × 100}{29.99}$$
p = 0.065 × 100
p = 6.5
So, 1.95 is 6.5% of 29.99.
Thus , The percent of sales tax is 7%

c. Given , The price of the jeans includes a 60% markup.
The markup is 60% of $39.99 a = p% × w = 60% × 39.99 = 0.6 × 39.99 = 23.99 So , the markup is$23.99.
To , find the original price , we have
selling price = cost to store + markup
cost to store = selling price – markup
= $39.99 –$23.99
= $15.99. The selling price is$15.99.
After the discount, The amount is $29.99 To know the mark up we have , a = 15.99 , w = 29.99 a = p% × w p% = $$\frac{a}{w}$$ p% = $$\frac{15.99}{29.99}$$ p% = 0.533 p = 0.533 × 100 p = 53.3 .% So , The percent of markup to the nearest percent is 54% Question 32. CRITICAL THINKING You buy a bicycle helmet for$22.26, which includes 6% sales tax. The helmet is discounted 30% off the selling price. What is the original price?
Answer: The original price of the helmet is $30. Explanation: Given , You buy a bicycle helmet for$22.26, which includes 6% sales tax.
Then , 6% of $22.26 is$1.33 , by decreasing the tax amount from the buying price  we get $20.93 The helmet is discounted 30% off the selling price. The sales price be 100% – 30% = 70% we know a = 20.93 , p = 70% , w = ? a = p% × w a = 70% × w 20.93 = 0.7 × w w = $$\frac{20.93}{0.7}$$ w = 29.9 , approximately equal to 30 So , the original price of the helmet is$30.

Question 33.
REASONING
A drone that costs $129.50 is discounted 40%. The next month, the sale price is discounted an additional 60%. Is the drone now “free”? If so, explain. If not, find the sale price. Answer: The sales price of the next month is$31.8 , The drone not for free.

Explanation:
Given , A drone that costs $129.50 is discounted 40%. We know , The sales price be 100% – 40% = 60% of the original price sales price = 60% of 129.50 = 0.6 × 129.5 =$77.7
So, The sales price is $77.7. Given , The next month, the sale price is discounted an additional 60%. The original price this time is$77.7
We know , The sales price be 100% – 60% = 40% of the original price
sales price = 40% of 77.7
= 0.4 × 77.7 = $31.8 So, The sales price of the next month is$31.8 .

Thus , The drone not for free.

Lesson 6.6 Simple Interest

EXPLORATION 1

Understanding Simple Interest
Work with a partner. You deposit $150 in an account that earns 6% simple interest per year. You do not make any other deposits or withdrawals. The table shows the balance of the account at the end of each year. a. Describe any patterns you see in the account balance. b. How is the amount of interest determined each year? c. How can you find the amount of simple interest earned when you are given an initial amount, an interest rate, and a period of time? d. You deposit$150 in a different account that earns simple interest. The table shows the balance of the account each year. What is the interest rate of the account? What is the balance after 10 years?

Answer: a. we observe the pattern of $9 increment in each year of the deposited amount. b. the amount of interest each year is , For 0 years =$0
For 1 years  I = $9.54 For 2 years I =$20.16
For 3 years  I = $31.86 For 4 years I =$44.64
For 5 years  I = $58.5 For 6 years I =$73.44
c. Formula for simple interest is I = Prt
d. The annual rate of interest is $2.60 . Explanation: a. As the table shows , we have a series of pattern among the years of amount deposited. that is , For 0 years =$150
For 1 years = $150 +$9 = $159 For 2 years =$159 + $9 =$168
For 3 years = $168 +$9 = $177 For 4 years =$177 + $9 =$186
For 5 years = $186 +$9 = $195 For 6 years =$195 + $9 =$204 ,
Here, we observe the pattern of $9 increment in each year of the deposited amount. b. From the table, we have , simple interest for each year as, Principal = P , t = time in years , r = Annual interest rate That is P =$150 , t = 0 years , r = 0.06
To find the simple interest  I  we know , I = Prt ,
For 0 years  I = 150 × 0 × 0.06 = $0 . For 1 years , P =$159 , t = 1 year , r = 0.06
I = 159 × 1 × 0.06 = $9.54 For 2 years , P =$168 , t = 2 year , r = 0.06
I = 168 × 2 × 0.06 = $20.16 For 3 years , P =$177 , t = 3 year , r = 0.06
I = 177 × 3 × 0.06 = $31.86 For 4 years , P =$186 , t = 4 year , r = 0.06
I = 186 × 4 × 0.06 = $44.64 For 5 years , P =$195 , t = 5 year , r = 0.06
I = 195 × 5 × 0.06 = $58.5 For 6 years , P =$204 , t = 6 year , r = 0.06
I = 204 × 6 × 0.06 = $73.44 C. To find the simple interest , we have Principal = P , t = time in years , r = Annual interest rate , Formula for simple interest is I = Prt , So by using this formula we can find the simple interest of the deposit . d. From the table given , we have the pattern of increment of$5 in each year ,
in this case the deposit in the 10 year will be $230 . To find the interest rate of the account , r = $$\frac{I}{pt}$$ r = $$\frac{0.06}{230 × 10}$$ r = 2.60 So , the annual rate of interest is$ 2.60 .

Interest principal is money paid or earned for using or lending money. The is the amount of money borrowed or deposited.

Try It

Question 1.
What is the balance of the account after 9 months?
Answer: The balance after 9 months = $500 +$1.66 = $501.66 . Explanation: From the example given , we know P =$500 , r = 0.03 , t =  9 months
I = prt
I = $$\frac{500 × 0.03}{9}$$
I = $$\frac{15}{9}$$
I = 1.66
The interest earned is $1.66 after 9 months , The balance after 9 months =$500 + $1.66 =$501.66 .

Question 2.
You deposit $350 in an account. The account earns$17.50 simple interest in 2.5 years. What is the annual interest rate?
Answer: The annual interest rate is 2%

Explanation:
Given , P = $350 , I =$17.5 , t = 2.5 ,
We know that  I = Prt
So , r = $$\frac{I}{Pt}$$
r = $$\frac{17.5}{350 × 2.5}$$
= $$\frac{17.5}{875}$$
= 0.02
we can write in percent as 2%
So , The annual interest rate is 2% .

Question 3.
In Example 3, how long does it take an account with a principal of $10,000 to earn$750 in interest?
Answer: It takes 3.75years to an account with a principal of $10,000 to earn$750 in interest .

Explanation:
Given , P = $10,000 , I =$750 , r = 0.02 , t = ? ,
We know that  I = Prt
t = $$\frac{I}{Pr}$$
= $$\frac{750}{10,000 × 0.02}$$
= $$\frac{750}{200}$$
= 3.75
So, It takes 3.75years to an account with a principal of $10,000 to earn$750 in interest .

Self-Assessment for Concepts & Skills
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 4.
VOCABULARY
Explain the meaning of simple interest.
Answer:
Simple interest is interest calculated on the principal portion of a loan or the original contribution to a savings account. Simple interest does not compound, meaning that an account holder will only gain interest on the principal, and a borrower will never have to pay interest on interest already accrued.
So , To put in simplest form we can say that ,  simple interest money paid or earned only on the principal .

USING THE SIMPLE INTEREST FORMULA Use the simple interest formula.
Question 5.
You deposit $20 in a savings account. The account earns 4% simple interest per year. What is the balance after 4 years? Answer: The balance of after 4 years is$23.2 .

Explanation:
To fin the principal , Given , P = $20 , r = 0.04 , t = 4 , We know that I = Prt I = 20 × 0.04 × 4 = 3.2 Simple interest I =$3.2
To find the balance after 4 years we have to add the simple interest to principal amount = $20 +$3.2 = $23.2 . The balance of after 4 years is$23.2 .

Question 6.
You deposit $800 in an account. The account earns$360 simple interest in 3 years. What is the annual interest rate?
Answer: The annual interest rate is 15% .

Explanation:
Given , P = $800 , I =$360 , t = 3 years ,
We know that  I = Prt
So , r = $$\frac{I}{Pt}$$
r = $$\frac{360}{800 × 3}$$
= $$\frac{360}{2400}$$
= 0.15
we can write in percent as 15%
So , The annual interest rate is 15% .

Question 7.
You deposit $650 in a savings account. How long does it take an account with an annual interest rate of 5% to earn$178.25 in interest?
Answer: It takes 7.13 years an account with an annual interest rate of 5% to earn $178.25 in interest . Explanation: Given , P =$650 , I = $178.25 , r = 0.05 , t = ? , We know that I = Prt t = $$\frac{I}{Pr}$$ = $$\frac{178.25}{650 × 0.05}$$ = $$\frac{178.25}{32.5}$$ = 7.13 So, It takes 7.13 years an account with an annual interest rate of 5% to earn$178.25 in interest .

Self-Assessment for Problem Solving
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 8.
You want to deposit $1000 in a savings account for 3 years. One bank adds a$100 bonus to your principal and offers a 2% simple annual interest rate. Another bank does not add a bonus, but offers 6% simple interest per year. Which bank should you choose? Explain.
Answer: Second bank offers the best deal .

Explanation:
Given , we know P = $1000 , t = 3 years , And one bank adds$100 bonus and offers  a 2% simple annual interest rate.
Then p = $1000 +$100 = $1100 , r = 0.02 , I = Prt I = 1100 × 0.02 × 3 I = 11 × 2 × 3 I = 66 The simple interest of one bank =$66 .
The balance of the account will be = $1100 +$66 = $1166 . Another bank offers no bonus, but 6% simple interest per year , r = 0.06 , p =$1000 , t = 3 years
I = prt
I = 1000 × 0.06 × 3
I = 10 × 6 × 3
I = 180
The simple interest of Another bank = $180 . The balance of the account will be =$1000 + $180 =$1180 .

So, Second bank offers the best deal .

Question 9.
Your cousin borrows $1125 to repair her car. The simple annual interest rate is 10%. She makes equal monthly payments of$25. How many years will it take to pay off the loan?
Answer: It takes 2.2 years to pay off the loan.

Explanation:
Given , P = $1125 , r = 0.01 , I =$25 , t = ?
We know that  I = Prt
t = $$\frac{I}{Pr}$$
= $$\frac{25}{1125 × 0.01}$$
= $$\frac{25}{11.25}$$
= 2.2
So, It takes 2.2 years to pay off the loan.

Question 10.
DIG DEEPER!
You borrow$900 to buy a laptop. You plan to pay off the loan after 5 years of equal monthly payments. After 10 payments, you have$1200 left to pay. What is the simple annual interest rate of your loan?
Answer: The annual interest rate is 15% .

Explanation:
Given , P = $900 , I =$1200 , t = 5 years ,
We know that  I = Prt
So , r = $$\frac{I}{Pt}$$
r = $$\frac{900}{1200 × 5}$$
= $$\frac{900}{6000}$$
= 0.15
we can write in percent as 15%
So , The annual interest rate is 15% .

Simple Interest Homework & Practice 6.6

Review & Refresh

Find the selling price.
Question 1.
A store pays $8 for a pool noodle. The markup is 20%. Answer: The selling price is$9.6

Explanation:
Then , The markup is 20% of $8 a = p% × w = 20% × 8 = 0.2 × 8 = 1.6 So , the markup is$1.6 .
To , find the selling price , we have
selling price = cost to store + markup
= $8 +$1.6
= $9.6. So ,The selling price is$9.6 .

Question 2.
A store pays $3 for a magazine. The markup is 5%. Answer: The selling price is$3.15

Explanation:
Then , The markup is 5% of $3 a = p% × w = 5% × 3 = 0.05 × 3 = 0.15 So , the markup is$0.15 .
To , find the selling price , we have
selling price = cost to store + markup
= $3 +$0.15
= $3.15. So ,The selling price is$3.15 .

Solve the inequality. Graph the solution.
Question 3.
x + 5 < 2
Answer:  x  < -3

Explanation:
Given , x + 5 < 2
add  – 5 on both sides ,
x + 5 – 5 < 2 – 5
x  < -3

Question 4.
b – 2 ≥ – 1
Answer: b  ≥ 1

Explanation:
Given , b – 2 ≥ – 1
add  2 on both sides
b – 2 + 2  ≥ – 1 + 2
b  ≥ 1

Question 5.
w + 6 ≤ – 3
Answer: w  ≤ – 9

Explanation:
Given , w + 6 ≤ – 3
add -6 on both sides
w + 6 – 6  ≤ – 3 – 6
w  ≤ – 9

Concepts, Skills, & Problem Solving

UNDERSTANDING SIMPLE INTEREST The table shows the balance of an account each year. What is the interest rate of the account? What is the balance after 10 years? (See Exploration 1, p. 265.)
Question 6.

Answer: The annual interest rate is 3% .

Explanation:
Given , There is an increment of $2 each year , for 10 years it will be$60 .
Now we have , P = $60 , t = 10 years , I =$2 , r = ?
We know that  I = Prt
So , r = $$\frac{I}{Pt}$$
r = $$\frac{2}{60 × 10}$$
= $$\frac{2}{60}$$
= 0.03
we can write in percent as 3%
So , The annual interest rate is 3% .

Question 7.

Answer: The annual interest rate is 0.4% .

Explanation:
Given , There is an increment of $14 each year , for 10 years it will be$315 .
Now we have , P = $315 , t = 10 years , I =$14 , r = ?
We know that  I = Prt
So , r = $$\frac{I}{Pt}$$
r = $$\frac{14}{315 × 10}$$
= 0.004
we can write in percent as 0.4%
So , The annual interest rate is 0.4% .

FINDING INTEREST EARNED An account earns simple annual interest. (a) Find the interest earned. (b) Find the balance of the account.
Question 8.
$600 at 5% for 2 years Answer: The simple interest of bank is$60 .
The balance of the account will be $660 . Explanation: Given , P =$600 , t =  2 years , r = 0.05
I = prt
I = 600 × 0.05 × 2
I = 6 × 5 × 2
I = 60
The simple interest of bank is $60 . The balance of the account will be =$600 + $60 =$660 .

Question 9.
$1500 at 4% for 5 years Answer: The simple interest is$300 .
The balance of the account will be  $1800 . Explanation: Given , P =$1500 , t =  5 years , r = 0.04
I = prt
I = 1500 × 0.04 × 5
I = 15 × 4 × 5
I = 300
The simple interest of  bank = $300 . The balance of the account will be$1500 + $300 =$1800 .

Question 10.
$350 at 3 % for 10 years Answer: The simple interest of bank is$105 .
The balance of the account will be $455 . Explanation: Given , P =$350 , t =  10 years , r = 0.03
I = prt
I = 350 × 0.03 × 10
I = 35 × 3
I = 105
The simple interest of  bank is $105 . The balance of the account will be$350 + $105 =$455 .

Question 11.
$1800 at 6.5% for 30 months Answer: The interest earned is$3.9 ,
The balance of the account will be $1,803.9 . Explanation: Given , P =$1800 , t =  30 months , r = 0.065
I = prt
I = $$\frac{1800 × 0.065}{30}$$
I = $$\frac{117}{30}$$
I = 3.9
The interest earned is $3.9 , The balance of the account will be$1800 + $3.9 =$1,803.9 .

Question 12.
$925 at2.3% for 2.4 years Answer: The interest earned is$638.1 ,
The balance of the account will be $1,563.1 . Explanation: Given , P =$925 , t =  28 months , r = 0.023
I = prt
I = $$\frac{925 × 0.023}{28}$$
I = $$\frac{21.27}{30}$$
I = 638.1
The interest earned is $638.1 , The balance of the account will be$925 + $638.1 =$1,563.1 .

Question 13.
$5200 at 7.36% for 54 months Answer: The interest earned is$7.08 ,
The balance of the account will be $5,207. Explanation: Given , P =$5200 , t = 54 months , r = 0.0736
I = prt
I = $$\frac{5200 × 0.0736}{54}$$
I = $$\frac{382.72}{54}$$
I = 7.08
The interest earned is $7.08 , The balance of the account will be$5200 + $7.08 =$5,207.

Question 14.
YOU BE THE TEACHER
Your friend finds the simple interest earned on $500 at 6% for 18 months. Is your friend correct? Explain your reasoning. Answer: No , The simple interest is$1.6 .

Explanation:
Given , P = $500 , t = 18 months , r = 0.06 I = prt I = $$\frac{500 × 0.06}{18}$$ I = $$\frac{30}{18}$$ I = 1.6 The simple interest is$1.6 .

FINDING AN ANNUAL INTEREST RATE Find the annual interest rate.
Question 15.
I = $24, P =$400, t = 2 years
Answer: The annual interest rate is 3% .

Explanation:
We know that  I = Prt
So , r = $$\frac{I}{Pt}$$
r = $$\frac{24}{400 × 2}$$
= $$\frac{24}{800}$$
= 0.03
we can write in percent as 3%
So , The annual interest rate is 3% .

Question 16.
I = $562.50, P =$1500, t = 5 years
Answer: The annual interest rate is 7.5% .

Explanation:
We know that  I = Prt
So , r = $$\frac{I}{Pt}$$
r = $$\frac{562.5}{1500 × 5}$$
= $$\frac{562.5}{7500}$$
= 0.075
we can write in percent as 7.5%
So , The annual interest rate is 7.5% .

Question 17.
I = $54, P =$900, t = 18 months
Answer: The annual interest rate is 108% .

Explanation:
We know that  I = Prt
So , r = $$\frac{I}{Pt}$$
r = $$\frac{54 × 18 }{900 }$$
= $$\frac{972}{900}$$
= 1.08
we can write in percent as 108%
So , The annual interest rate is 108% .

Question 18.
I = $160, P =$2000, t = 8 months
Answer: The annual interest rate is 64% .

Explanation:
We know that  I = Prt
So , r = $$\frac{I}{Pt}$$
r = $$\frac{160 × 8 }{2000 }$$
= $$\frac{1,280}{2,000}$$
= 0.64
we can write in percent as 64%
So , The annual interest rate is 64% .

FINDING AN AMOUNT OF TIME Find the amount of time.
Question 19.
I $30, P =$500, r = 3%
Answer: The amount of time is 1.6 years .

Explanation:
We know that  I = Prt
t = $$\frac{I}{Pr}$$
= $$\frac30}{500 × 0.03}$$
= $$\frac{30}{18}$$
=1.6
So , The amount of time is 1.6 years .

Question 20.
I = $720, P =$1000, r = 9%
Answer: The amount of time is 8 years .

Explanation:
We know that  I = Prt
t = $$\frac{I}{Pr}$$
= $$\frac{720}{1000 × 0.09}$$
= $$\frac{720}{90}$$
= 8
So , The amount of time is 8 years .

Question 21.
I = $54, P =$800, r = 4.5%
Answer: The amount of time is 1.5 years .

Explanation:
We know that  I = Prt
t = $$\frac{I}{Pr}$$
= $$\frac{54}{800 × 0.045}$$
= $$\frac{54}{36}$$
= 1.5
So , The amount of time is 1.5 years .

Question 22.
I = $450, P =$2400, r = 7.5%
Answer:  The amount of time is 2.5 years .

Explanation:
We know that  I = Prt
t = $$\frac{I}{Pr}$$
= $$\frac{450}{2400 × 0.075}$$
= $$\frac{450}{180}$$
= 2.5
So , The amount of time is 2.5 years .

Question 23.
FINDING AN ACCOUNT BALANCE
A savings account earns 5% simple interest per year. The principal is $1200. What is the balance after 4 years? Answer: The bank balance is$1440 .

Explanation:
Given , P = $1200 , t = 4 years , r = 0.05 I = prt I = 1200 × 0.05 × 4 I = 12 × 5 × 4 I = 240 The simple interest of bank is$240 .
The bank balance is $1200 +$240 = $1440 . Question 24. FINDING AN ANNUAL INTEREST RATE You deposit$400 in an account. The account earns $18 simple interest in 9 months. What is the annual interest rate? Answer: The annual interest rate is 40.5% . Explanation: Given , P =$400 , I = $18 , t = 9 months , We know that I = Prt So , r = $$\frac{I}{Pt}$$ r = $$\frac{18 × 9}{400}$$ = $$\frac{162}{400}$$ = 0.405 we can write in percent as 40.5% So , The annual interest rate is 40.5% . Question 25. FINDING AN AMOUNT OF TIME You deposit$3000 in a CD (certificate of deposit) that earns 5.6% simple annual interest. How long will it take to earn $336 in interest? Answer: It takes 2 years to earn$336 in interest .

Explanation:
Given , P = $3000 , r = 0.056 , I =$336
We know that  I = Prt
t = $$\frac{I}{Pr}$$
= $$\frac{336}{3000 × 0.056}$$
= $$\frac{336}{168}$$
= 2
So , It takes 2 years to earn $336 in interest . FINDING AN AMOUNT PAID Find the amount paid for the loan. Question 26.$1500 at 9% for 2 years
Answer: The amount paid for the loan is $270 . Explanation: Given , P =$1500 , t =  2 years , r = 0.09
I = prt
I = 1500 × 0.09 × 2
I = 15 × 9 × 2
I = 270
So , The amount paid for the loan is $270 . Question 27.$2000 at 12% for 3 years
Answer: The amount paid for the loan is $72 . Explanation: Given , P =$2000 , t =  3 years , r = 0.012
I = prt
I = 2000 × 0.012 × 3
I = 2 × 12 × 3
I = 72
So , The amount paid for the loan is $72 . Question 28.$2400 at 10.5% for 5 years
Answer: The amount paid for the loan is $1,260 . Explanation: Given , P =$2400 , t =  5 years , r = 0.105
I = prt
I = 2400 × 0.105 × 5
I = 252 × 5
I = 1,260
So , The amount paid for the loan is $1,260 . Question 29.$4800 at 9.9% for 4 years
Answer: The amount paid for the loan is $1,900 . Explanation: Given , P =$4800 , t =  4 years , r = 0.099
I = prt
I = 4800 × 0.099 × 4
I = 475.2 × 4
I = 1,900
So , The amount paid for the loan is $1,900 . USING THE SIMPLE INTEREST FORMULA Copy and complete the table. Answer: 30. The simple interest is$1,530 .
31. The principal amount is $2,29,491 . 32. The time required is 4 years . 33. The annual interest rate is 1,275% . 30. Explanation: Given , P =$12,000 , t =  5 years , r = 0.0425
I = prt
I = 12,000 × 0.0425 × 5
I = 510 ×
I = 1,530
So , The simple interest  is $1,530 . 31. Explanation: Given , I =$828.75 , t =  18 months , r = 0.065
I = Prt
P = $$\frac{I}{rt}$$
= $$\frac{828.75 × 18}{0.065}$$
= $$\frac{14,916.96}{0.065}$$
= 2,29,491
So , The principal amount is $2,29,491 . 32. Explanation: Given , P =$15,500 , I = $5425 , r = 0.0875 We know that I = Prt t = $$\frac{I}{Pr}$$ = $$\frac{5425}{15,500 × 0.0875}$$ = $$\frac{5425}{1,356.25}$$ = 4 So , The time required is 4 years . 33. Explanation: Given , P =$18,000 , I = $4252.5 , t = 54 months , We know that I = Prt So , r = $$\frac{I}{Pt}$$ r = $$\frac{4252.5 × 54}{18,000}$$ = $$\frac{2,29,635}{18,000}$$ = 12.75 we can write in percent as 1,275% So , The annual interest rate is 1,275% . Question 34. MODELING REAL LIFE A family borrows money for a rainforest tour. The simple annual interest rate is 12%. The loan is paid after 3 months. What is the total amount paid for the tour? Answer: The total amount paid for the tour will be$1,230 + $49.2 =$1,279.2 .

Explanation:
Given , P = $940 +$170 + $120 = 1,230 P =$1,230 , t =  3 months , r = 0.12
I = prt
I = $$\frac{1,230 × 0.12}{3}$$
I = $$\frac{147.6}{3}$$
I = 49.2
The simple interest  is $49.2 . So , The total amount paid for the tour will be$1,230 + $49.2 =$1,279.2 .

Question 35.
MODELING REAL LIFE
You deposit $5000 in an account earning 7.5% simple interest per year. How long will it take for the balance of the account to be$6500?
Answer: It takes 4 years for the balance of the account to be $6500. Explanation: Given , the balance of the account be$6500 ,  You deposit $5000 We know that . The total balance = deposit + simple interest So, To get simple interest we have = The total balance – deposit =$6500 – $5000 =$1500 ,
P = $5000 , r = 0.075 , I =$1500 , t = ?
We know that  I = Prt
t = $$\frac{I}{Pr}$$
= $$\frac{1500}{5000 × 0.075}$$
= $$\frac{1500}{375}$$
= 4
So , It takes 4 years for the balance of the account to be $6500. Question 36. MODELING REAL LIFE You borrow$1300 to buy a telescope. What is the monthly payment?

Answer: The monthly payment is $66.95 . Explanation: Given , P =$1300 , t =  2 years , r = 0.118
I = prt
I = 1300 × 0.118 × 2
I = 153.4 × 2
I = 306.8
The simple interest of  bank is $306.8 . The balance of the account will be$1300 + $306.8 =$1,606.8 .
To get the monthly payment ,  convert 2 years into 24 months then , $$\frac{1,606.8}{24}$$ = 66.95
So , The monthly payment is $66.95 . Question 37. REASONING How many years will it take for$2000 to double at a simple annual interest rate of 8%? Explain how you found your answer.
Answer: It will take 12.5 years to make the principal amount double .

Explanation:
Given , it take for $2000 to double , so the total balance will be$2000 + $2000 =$4000
We know that . The total balance = deposit + simple interest
So, To get simple interest we have = The total balance – deposit = $4000 –$2000 = $2000 P =$2000 , r = 0.08 , I = $2000 , t = ? We know that I = Prt t = $$\frac{I}{Pr}$$ = $$\frac{2000}{2000 × 0.08}$$ = $$\frac{2000}{160}$$ = 12.5 So , It will take 12.5 years to make the principal amount double . Question 38. DIG DEEPER! You take out two loans. After 2 years, the total interest for the loans is$138. On the first loan, you pay 7.5% simple annual interest on a principal of $800. On the second loan, you pay 3% simple annual interest. What is the principal for the second loan? Answer: The principal amount of the second loan is$300 .

Explanation:
Given , For first loan , P = $800 , t = 2 years , r = 0.075 I = prt I = 800 × 0.075 × 2 I = 120 The simple interest of bank is$120 .

For Second loan , Given , the total interest for the loans is $138 So , I =$138 – $120 =$18 ,
Given r = 0.03 , t = 2 years , I = $18 I = Prt p = $$\frac{I}{rt}$$ = $$\frac{18}{0.03 × 2}$$ = $$\frac{18}{0.06}$$ = 300 So, The principal amount of the second loan is$300 .

Question 39.
REPEATED REASONING
You deposit $500 in an account that earns 4% simple annual interest. The interest earned each year is added to the principal to create a new principal. Find the total amount in your account after each year for 3 years. Answer: The balance of the account for the first year will be$500 + $20 =$520
The balance of the account for the 2 year will be $520 +$41.6 = $561.6 . The balance of the account for the 3 year will be$561.6 + $67.3 =$5683.9.

Explanation:
Given , P = $500 , t = 1 years , r = 0.04 I = prt I = 500 × 0.04 × 1 I = 5 × 4 I = 20 The simple interest of bank is$20 .
The balance of the account for the first year will be $500 +$20 = $520 . For 2 years , p =$520 , t = 2 , r = 0.04
I = prt
I = 520 × 0.04 × 2
I = 20.8 × 2
I = 41.6
The simple interest of  bank is $41.6 . The balance of the account for the 2 year will be$520 + $41.6 =$561.6 .

For 3 years , p = $561.6 , t = 3 , r = 0.04 I = prt I = 561.6 × 0.04 × 3 I = 22.46× 3 I = 67.3 The simple interest of bank is$67.3 .
The balance of the account for the 3 year will be $561.6 +$67.3 = $5683.9. Question 40. NUMBER SENSE An account earns r% simple interest per year. Does doubling the initial principal have the same effect on the total interest earned as doubling the amount of time? Justify your answer. Answer: Yes ,Doubling the initial principal have the same effect on the total interest earned as doubling the amount of time Explanation: Given , An account earns r% simple interest per year. Does doubling the initial principal Let us say P = 2P , t = t years , r = r% we know I = Prt , I = 2P × t × r I = 2Prt The simple interest of the doubling initial principal is 2Prt . Now , doubling the amount of time , t = 2t years we know I = Prt , I = P × 2t × r I = 2Prt The simple interest of the doubling the amount is 2Prt . So, Doubling the initial principal have the same effect on the total interest earned as doubling the amount of time Percents Connecting Concepts Using the Problem-Solving Plan Question 1. The table shows the percent of successful shots for each team in a hockey game. A total of 55 shots are taken in the game. The ratio of shots taken by the Blazers to shots taken by the Hawks is 6 : 5. How many goals does each team score? Understand the problem. You know that 55 shots are taken in a hockey game and that the Blazers take 6 shots for every 5 shots taken by the Hawks. You also know the percent of successful shots for each team. Make a plan. Use a ratio table to determine the number of shots taken by each team. Then use the percent equation to determine the number of successful shots for each team. Solve and check. Use the plan to solve the problem. Then check your solution. Answer: The goals made by Hawks are 50. and The goals made by Blazers are 60 Explanation: Given , The ratio of shots taken by the Blazers to shots taken by the Hawks is 6 : 5 the percent of successful shots for each team in a hockey game is Blazers is 10% , Hawks is 16% , The goals made by Blazers are w = ? , a = 6 , p = 10% By using proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{6}{w}$$ = $$\frac{10}{100}$$ w = $$\frac{6 × 100}{10}$$ w = $$\frac{600}{10}$$ w = 60 So, The goals made by Blazers are 60 The goals made by Hawks are w= ? , a = 5 , p = 16% By using proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{5}{w}$$ = $$\frac{10}{100}$$ w = $$\frac{5 × 100}{10}$$ w = $$\frac{500}{10}$$ w = 50 So , The goals made by Hawks are 50. Question 2. Fill in the blanks with positive numbers so that the sum of the fractions is 37.5% of the first fraction. Justify your answer. Answer: $$\frac{2}{5}$$ + (- $$\frac{0.1}{4}$$) = 37.5% Explanation: Let us say that missing values be x and y we have ,$$\frac{x}{5}$$ + (- $$\frac{y}{4}$$) To get the simplified answer cross check the values of x and y If x = 2 and y = 0.1 ,then $$\frac{2}{5}$$ + (- $$\frac{0.1}{4}$$) 0.4 – 0.025 = 0.375 0.375 can be written as 37.5% So , $$\frac{2}{5}$$ + (- $$\frac{0.1}{4}$$) = 37.5% Question 3. The graph shows the distance traveled by a motorcycle on a dirt road. After turning onto a paved road, the motorcycle travels $$\frac{1}{5}$$ mile every $$\frac{1}{4}$$ minute. Find the percent of change in the speed of the motorcycle. Round to the nearest tenth of a percent if necessary. Answer: The percent of change in the speed of the bike is 38% Explanation: speed of the bike on the dirty road is $$\frac{2.33}{4}$$ = 0.58 speed of the bike on the paved road is $$\frac{0.2}{0.25}$$ = 0.8 old value = 0.58 , new value = 0.8 , The amount of change = 0.8 – 0.58 = 0.22 So , percent of change is $$\frac{amount of change }{original amount}$$ = $$\frac{ 0.22}{0.58}$$ = 0.379 = 0.379 × 100 = 37.9 = 38% So, the percent of change is 38% Performance Task Tornado Alley At the beginning of this chapter, you watched a STEAM Video called “Tornado!” You are now ready to complete the performance task related to this video, available at BigIdeasMath.com. Be sure to use the problem-solving plan as you work through the performance task. Answer: Percents Chapter Review Review Vocabulary Write the definition and give an example of each vocabulary term. percent of change : It is the percent that a quantity changes from the original amount = $$\frac{amount of change }{original amount}$$ Example: 2 feet to 6 feet We know that , formula for percent change = $$\frac{New value – old value}{old value}$$ where New value = 6 and old value = 2 , because a change of 2 to 6 is a positive (increase) change So, percent change = $$\frac{6 – 2}{2}$$ = $$\frac{4}{2}$$ = 2 × 100 = 200% So, percent of change is 200% percent of increase: When the original amount increases then the percent of change is called percent of increase = $$\frac{New value – old value}{old value}$$ Example: 5 cups to 10 cups We know that , formula for percent change = $$\frac{New value – old value}{old value}$$ where New value = 10 and old value = 5 , because a change of 5 to 10 is a positive (increase) change So, percent change = $$\frac{10 – 5}{5}$$ = $$\frac{5}{5}$$ = 1 × 100 = 100% So, percent of increase is 100% percent of decrease: When the original amount decrease then the percent of change is called percent of decrease =$$\frac{old value – New value}{old value}$$ Example: 15 inches to 12 inches We know that , formula for percent change = $$\frac{old value – new value}{old value}$$ where New value = 12 and old value = 15 , because a change of 15 to 12 is a negative (decrease) change So, percent change = $$\frac{15 – 12}{15}$$ = $$\frac{3}{15}$$ = 0.2 × 100 = 20% So, percent of decrease is 20% percent error: It is the percent that estimated value differs from Actual value $$\frac{Error value}{Actual value}$$ Example: Estimated value = 40 , Actual value = 30 The amount of error is 40 – 30 = 10 We know that , formula for percent error = $$\frac{Error value}{Actual value}$$ where Error value = 10 and Actual value = 30 , So, percent Error = $$\frac{10}{30}$$ = 0.333 = 0.333 × 100 = 33.3% So, percent Error is 33.3% Discount: It is a decrease in amount of original price of an item Example: A box is$5 with 20% off
We know , The sales price be 100% – 20% = 80% of the original price
sales price = 80% of 5
= 0.8 × 5 = 4
So, The sales price is $4. Markup: The increase from what the stores pays to the selling price is called as Markup Example: The markup is 15% of$40
a = p% × w
= 15% × 40
= 0.15 × 40
= 6
So , the markup is $6. To , find the selling price , we have selling price = cost to store + markup =$40 + $6 =$46.
The selling price is $46. Interest: A money paid or earned for using or lending money is called interest Example: A bank offers a loan of$500 with an interest of $10 . Here$10 is the interest amount to be paid for the loan

Principal: The amount of money borrowed or deposited
Example:
A bank offers a loan of $500 with an interest of$10 .
here the loan amount is the principal amount

Simple Interest: It is the money paid or earned only on the principal I = Prt , where I = simple interest , P = principal , r = rate of interest , t = time in years
Example:
A bank offers a loan of $500 with an interest rate of 3% for 2 years . the simple interest is p =$500 , r  = 0.03 , t = 2
I = 500 × 0.03 × 2
I = 30
So , The simple interest is $15 . Graphic Organizers You can use a Summary Triangle to explain a concept. Here is an example of writing a percent as a decimal a Summary Triangle for Writing a percent as a decimal. Answer: Summary triangle for writing a decimal into percent is Choose and complete a graphic organizer to help you study the concept. 1. writing a decimal as a percent 2. comparing and ordering fractions, decimals, and percents 3. the percent proportion 4. the percent equation 5. percent of change 6. discount 7. markup Writing a decimal as a percent Chapter Self-Assessment As you complete the exercises, use the scale below to rate your understanding of the success criteria in your journal. 6.1 Fractions, Decimals, and Percents (pp. 235–240) Learning Target: Rewrite fractions, decimals, and percents using different representations. Write the percent as a decimal or the decimal as a percent. Use a model to represent the number. Question 1. 74% Answer: decimal form is 0.74 Explanation: To write percent into decimal we have to remove percent symbol and divide by 100, which moves the decimal point two places to the left. So, 74% in decimal form is 0.74 Question 2. 2% Answer: decimal form is 0.02 Explanation: To write percent into decimal we have to remove percent symbol and divide by 100, which moves the decimal point two places to the left. So, 2% in decimal form is 0.02 Question 3. 221% Answer: decimal form is 2.21 Explanation: To write percent into decimal we have to remove percent symbol and divide by 100, which moves the decimal point two places to the left. So, 221% in decimal form is 2.21 Question 4. 0.17 Answer: 17% Explanation: To write decimal into percent we should multiply the number by 100, which moves the decimal point two places to the right and add a percent symbol. So, 0.17 can be rewrite as 17% Question 5. $$4 . \overline{3}$$ Answer: $$433. \overline{3} \%$$ Explanation: To write decimal into percent we should multiply the number by 100, which moves the decimal point two places to the right and add a percent symbol. So, $$4 . \overline{3}$$ can be rewrite as $$433. \overline{3} \%$$ Question 6. 0.079 Answer: 7.9% Explanation: To write decimal into percent we should multiply the number by 100, which moves the decimal point two places to the right and add a percent symbol. So, 0.079 can be rewrite as 7.9% Write the fraction as a decimal and a percent. Question 7. $$\frac{17}{20}$$ Answer: decimal = 0.85, percent = 85% Explanation: By using the method of converting fraction into decimal and comparing , We can rewrite $$\frac{17}{20}$$ as 0.85 in decimal form, To write decimal into percent we should multiply the number by 100, which moves the decimal point two places to the right and add a percent symbol , then 0.85 can be rewrite as 85% So,$$\frac{17}{20}$$ in decimal = 0.85, percent = 85% Question 8. $$\frac{3}{8}$$ Answer: decimal = 0.375, percent = 37.5% Explanation: By using the method of converting fraction into decimal and comparing , We can rewrite $$\frac{3}{8}$$ as 0.375 in decimal form, To write decimal into percent we should multiply the number by 100, which moves the decimal point two places to the right and add a percent symbol , then 0.375 can be rewrite as 37.5% So, $$\frac{3}{8}$$ in decimal = 0.375, percent = 37.5% Question 9. $$\frac{14}{9}$$ Answer: decimal = $$1 . \overline{5}$$, percent = $$155. \overline{5} \%$$ Explanation: By using the method of converting fraction into decimal and comparing , We can rewrite $$\frac{14}{9}$$ as $$1. \overline{5}$$ in decimal form, To write decimal into percent we should multiply the number by 100, which moves the decimal point two places to the right and add a percent symbol , then $$1 . \overline{5}$$can be rewrite as $$155. \overline{5} \%$$ So, $$\frac{14}{9}$$ in decimal = $$1 . \overline{5}$$, percent = $$155. \overline{5} \%$$ Question 10. For school spirit day, 11.875% of your class wears orange shirts, $$\frac{5}{8}$$ of your class wears blue shirts, 0.15625 of your class wears white shirts, and the rest of your class wears gold shirts. Order the portions of shirts of each color from least to greatest. Justify your answer. Answer: In ascending order we have 10.1% , 15.6% , 62.5% , 11.875% Explanation: Given , 11.875% of your class wears orange shirts, $$\frac{5}{8}$$ of your class wears blue shirts, It can be written as 0.625 in decimals and 62.5% in percent 0.15625 of your class wears white shirts, it can be written as 15.6% And the rest of your class wears gold shirts. Let the total be 100% , The percent in all colors is 11.875% + 62.5% + 15.6% = 89.9% So , the rest of the girls wears 100% – 89.9% = 10.1% , In ascending order we have 10.1% , 15.6% , 62.5% , 11.875% 6.2 The Percent Proportion (pp. 241–246) Learning Target: Use the percent proportion to find missing quantities. Write and solve a proportion to answer the question. Question 11. What percent of 60 is 18? Answer: 18 is 30% of 60. Explanation: Given , a = 18 , w = 60 , p = ? By using proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{18}{60}$$ = $$\frac{p}{100}$$ p = $$\frac{18 × 100}{60}$$ p = $$\frac{1800}{60}$$ p = 30 So, 18 is 30% of 60. Question 12. 40 is what percent of 32? Answer: 40 is 125% of 32. Explanation: Given , a = 40 , w = 32 , p = ? By using proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{40}{32}$$ = $$\frac{p}{100}$$ p = $$\frac{40 × 100}{32}$$ p = $$\frac{4,000}{32}$$ p = 125 So, 40 is 125% of 32. Question 13. What number is 70% of 70? Answer: 49 is 70% of 70. Explanation: Given , a = ? , w = 70 , p = 70% By using percent proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{a}{70}$$ = $$\frac{70}{100}$$ a = $$\frac{70 × 70}{100}$$ a = $$\frac{4900}{100}$$ a = 49 So, 49 is 70% of 70. Question 14. $$\frac{3}{4}$$ is 75% of what number? Answer: 0.75 is 755% of 1. Explanation: we can write $$\frac{3}{4}$$ as 0.75 in decimal Given , a = 0.75 , w = ? , p = 75% By using proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{0.75}{w}$$ = $$\frac{75}{100}$$ w = $$\frac{0.75 × 100}{75}$$ w = $$\frac{75}{755}$$ w = 1 So, 0.75 is 755% of 1. Question 15. About 29% of the Earth’s surface is covered by land. The total surface area of the Earth is about 510 million square kilometers. What is the area of the Earth’s surface covered by land? Answer: The area of the Earth’s surface covered by land is 147.9 million square kilometers . Explanation: Given , p = 29% , w = 510 , a = ? By using percent proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{a}{510}$$ = $$\frac{29}{100}$$ a = $$\frac{510 × 29}{100}$$ a = $$\frac{14,790}{100}$$ a = 147.9 So, 147.9 is 29% of 510. The area of the Earth’s surface covered by land is 147.9 million square kilometers . 6.3 The Percent Equation (pp. 247–252) Learning Target: Use the percent equation to find missing quantities. Write and solve an equation to answer the question. Question 16. What number is 24% of 25? Answer: 6 is 24% of 25. Explanation: Given p = 24% , w = 25 , a = ? By using percent proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{a}{25}$$ = $$\frac{24}{100}$$ a = $$\frac{25 × 24}{100}$$ a = $$\frac{600}{100}$$ a = 6 So, 6 is 24% of 25. Question 17. 9 is what percent of 20? Answer: 9 is 45% of 20. Explanation: Given p = ? , w = 20 , a = 9 By using proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{9}{20}$$ = $$\frac{p}{100}$$ p = $$\frac{9 × 100}{20}$$ p = $$\frac{900}{20}$$ p = 45 So, 9 is 45% of 20. Question 18. 60.8 is what percent of 32? Answer: 60.8 is 190% of 32. Explanation: Given p = ? , w = 32, a = 60.8 By using proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{60.8}{32}$$ = $$\frac{p}{100}$$ p = $$\frac{60.8 × 100}{32}$$ p = $$\frac{6080}{32}$$ p = 190 So, 60.8 is 190% of 32. Question 19. 91 is 130% of what number? Answer: 91 is 130% of 70. Explanation: Given p = 130% , w = ? , a = 91 By using proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{91}{w}$$ = $$\frac{130}{100}$$ w = $$\frac{91 × 100}{130}$$ w = $$\frac{9100{130}$$ w = 70 So, 91 is 130% of 70. Question 20. 85% of what number is 10.2? Answer: 10.2 is 85% of 12. Explanation: Given p = 85% , w = ? , a = 10.2 By using proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{10.2}{w}$$ = $$\frac{85}{100}$$ w = $$\frac{10.2 × 100}{85}$$ w = $$\frac{1020}{85}$$ w = 12 So, 10.2 is 85% of 12. Question 21. 83% of 20 is what number? Answer: 16.6 is 83% of 20. Explanation: Given p = 83% , w = 20 , a = ? By using percent proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{a}{20}$$ = $$\frac{83}{100}$$ a = $$\frac{20 × 83}{100}$$ a = $$\frac{1,660}{100}$$ a = 16.6 So, 16.6 is 83% of 20. Question 22. 15% of the parking spaces at a school are handicap spaces. The school has 18 handicap spaces. How many parking spaces are there in total? Answer: Totally , there are 120 parking spaces . Explanation: Given , p = 15% , a = 18 , w = ? By using proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{18}{w}$$ = $$\frac{15}{100}$$ w = $$\frac{18 × 100}{15}$$ w = $$\frac{1800}{15}$$ w = 120 So, 18 is 15% of 120. Totally , there are 120 parking spaces . Question 23. Of the 25 students on a field trip, 16 bring cameras. What percent of the students bring cameras? Answer: 64% of students brought the cameras . Explanation: Given , a = 16 , w = 25 , p = ? By using proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{16}{25}$$ = $$\frac{p}{100}$$ p = $$\frac{16 × 100}{25}$$ p = $$\frac{1600}{25}$$ p = 64 So, 16 is 64% of 25. Thus, 64% of students brought the cameras . 6.4 Percents of Increase and Decrease (pp. 253–258) Learning Target: Find percents of change in quantities. Identify the percent of change as an increase or a decrease. Then find the percent of change. Round to the nearest tenth of a percent if necessary. Question 24. 6 yards to 36 yards Answer: percent of change is 500% Explanation: We know that , formula for percent change = $$\frac{New value – old value}{old value}$$ where New value = 36 and old value = 6 , because a change of 6 to 36 is a positive (increase) change So, percent change = $$\frac{36 – 6}{6}$$ = $$\frac{30}{6}$$ = 5× 100 = 500% So, percent of change is 500% Question 25. 120 meals to 52 meals Answer: percent of change is – 56.6% Explanation: We know that , formula for percent change = $$\frac{New value – old value}{old value}$$ where New value = 52 and old value = 120 , because a change of 120 to 52 is a negative (decrease) change So, percent change = $$\frac{52 – 120}{120}$$ = $$\frac{- 68}{120}$$ = – 0.566 = – 0.566 × 100 = – 56.6% So, percent of change is – 56.6% Question 26. You estimate that a jar contains 68 marbles. The actual number of marbles is 60. Find the percent error. Answer: percent Error is 13% Explanation: The amount of error is 68 – 60 = 8 We know that , formula for percent error = $$\frac{Error value}{Actual value}$$ where Error value = 8 and Actual value = 60 , So, percent Error = $$\frac{8}{60}$$ = 0.133 = 0.133 × 100 = 13.3% Approximately we can write as 13% So, percent Error is 13% Question 27. The table shows the numbers of skim boarders at a beach on Saturday and Sunday. What was the percent of change in boarders from Saturday to Sunday? Answer: percent of change is – 12.5% Explanation: Given , 12 to 9 We know that , formula for percent change = $$\frac{New value – old value}{old value}$$ where New value = 63 and old value = 72 , because a change of 72 to 63 is a negative (decrease) change So, percent change = $$\frac{63 – 72}{72}$$ = $$\frac{- 9}{72}$$ = $$\frac{- 1}{8}$$ = – 0.125 = – 0.125 × 100 = – 12.5% So, percent of change is – 12.5% 6.5 Discounts and Markups (pp. 259–264) Learning Target: Solve percent problems involving discounts and markups. Find the sale price or original price. Question 28. Original price:$50
Discount: 15%
Sale price: ?
Answer: The sales price is $42.5 Explanation: We know , The sales price be 100% – 15% = 85% of the original price sales price = 85% of 50 = 0.85 × 50 = 42.5 So, The sales price is$42.5

Question 29.
Original price: ?
Discount: 20%
Sale price: $75 Answer: The original price is$93.75.

Explanation:
We know , The sales price be 100% – 20% = 80%
a = p% × w
75 = 0.8 × w
w = $$\frac{75}{0.8}$$
w = 93.75
So, The original price is $93.75. Question 30. What is the original price of the tennis racquet? Answer: The original price is$30.

Explanation:
We know , The sales price be 100% – 30% = 70%
a = p% × w
21 = 0.7 × w
w = $$\frac{21}{0.7}$$
w = 30
So, The original price is $30. Question 31. A store pays$50 for a pair of shoes. The markup is 25%.
a. What is the selling price for the shoes?
b. What is the total cost for a person to buy the shoes including a 6% sales tax?
Answer: a. The selling price is $62.5. b. Total cost for a person is$66.25 .

Explanation:
The markup is 25% of $50 a = p% × w = 25% × 50 = 0.25 × 50 = 12.5 So , the markup is$12.5.
To , find the selling price , we have
selling price = cost to store + markup
= $50 +$12.5
= $62.5. The selling price is$62.5.

b. with 6% sales tax we have ,
6% of $62.5 = 0.06 × 62.5 = 3.75 , it is the tax rate So , total cost for a person is$62.5 + $3.75 =$66.25

6.6 Simple Interest (pp. 265–270)
Learning Target: Understand and apply the simple interest formula.

An account earns simple interest. (a) Find the interest earned. (b) Find the balance of the account.
Question 32.
$300 at 4% for 3 years Answer: a. The interest earned is$36
b. The balance of the account is $336 . Explanation: we know P =$300 , r = 0.04 , t = 3 years
I = prt
I = 300 × 0.04 × 3
I = 3 × 4 × 3
I = $36 The interest earned is$36
The balance of the account  = $300 +$36 = $336 . Question 33.$2000 at 3.5% for 4 years
Answer: a. The interest earned is $280 b. The balance of the account is$2,280 .

Explanation:
we know P = $2000 , r = 0.035 , t = 4 years I = prt I = 2000 × 0.035 × 4 I = 2 × 35 × 4 I =$280
The interest earned is $280 The balance of the account =$2000 + $280 =$2,280 .

Find the annual interest rate.
Question 34.
I = $17, P =$500, t = 2 years
Answer: The annual interest rate is 1.7% .

Explanation:
We know that  I = Prt
So , r = $$\frac{I}{Pt}$$
r = $$\frac{17}{500 × 2}$$
= $$\frac{17}{1000}$$
= 0.017
we can write in percent as 1.7%
So , The annual interest rate is 1.7% .

Question 35.
I = $426, P =$1200, t = 5 years
Answer: The annual interest rate is 7.1% .

Explanation:
We know that  I = Prt
So , r = $$\frac{I}{Pt}$$
r = $$\frac{426}{1200 × 5}$$
= $$\frac{426}{6000}$$
= 0.071
we can write in percent as 7.1%
So , The annual interest rate is 7.1% .

Find the amount of time.
Question 36.
I = $60, P =$400, r = 5%
Answer: The amount of time is 3 years .

Explanation:
We know that  I = Prt
t = $$\frac{I}{Pr}$$
= $$\frac{60}{400 × 0.05}$$
= $$\frac{60}{20}$$
= 3
So, The amount of time is 3 years .

Question 37.
I = $237.90, P =$1525, r = 2.6%
Answer: The amount of time is 6 years .

Explanation:
We know that  I = Prt
t = $$\frac{I}{Pr}$$
= $$\frac{237.9}{1525 × 0.026}$$
= $$\frac{237.9}{39.65}$$
= 6
So, The amount of time is 6 years .

Question 38.
You deposit $100 in an account. The account earns$2 simple interest in 6 months. What is the annual interest rate?
Answer: The annual interest rate is 12%

Explanation:
Given , p = $100 , I =$2 , t = 6 months ,
We know that  I = Prt
So , r = $$\frac{I}{Pt}$$
r = $$\frac{2 × 6}{100}$$
= $$\frac{12}{100}$$
= 0.12
we can write in percent as 12%
So , The annual interest rate is 12%

Question 39.
Bank A is offering a loan with a simple interest rate of 8% for 2 years. Bank B is offering a loan with a simple interest rate of 6.5% for 3 years.

a. Assuming the monthly payments are equal, what is the monthly payment for the four wheeler from Bank A? from Bank B?
b. Give reasons for why a person might choose Bank A and why a person might choose Bank B for a loan to buy the four wheeler. Explain your reasoning.
Answer: a. The monthly payment for bank A is $261 and for Bank B is$179.25 .
b. If a person chooses Bank A , it will be high in monthly payment but within 2 years loan will be cleared .
If a person chooses Bank B , it will be low monthly payment and can pay the loan a year extra in small amounts.

Explanation:
For bank A , Then p = $5400 , r = 0.08 , t = 2 years I = Prt I = 5400 × 0.08 × 2 I = 54 × 8 × 2 I = 864 The simple interest of one bank =$864 .
The balance of the account will be = $5400 +$864 = $6,264 . The monthly payment = $$\frac{6,264}{24}$$ =$261 .

For bank B,  Then p = $5400 , r = 0.065 , t = 3 years I = Prt I = 5400 × 0.065 × 3 I = 54 × 6.5 × 3 I = 1,053 The simple interest of one bank =$1,053 .
The balance of the account will be = $5400 +$1,053 = $6,453 . The monthly payment = $$\frac{6,453}{36}$$ =$179.25 .

b. If a person chooses Bank A , it will be high in monthly payment but within 2 years loan will be cleared .
If a person chooses Bank B , it will be low monthly payment and can pay the loan a year extra in small amounts.

Percents Practice Test

Write the percent as a decimal, or the decimal as a percent. Use a model to represent the number.
Question 1.
0.96%
Answer: 0.0096

Explanation:
To write percent into decimal we have to remove percent symbol and divide by 100, which moves the decimal point two places to the left.
So, 0.96% in decimal form is 0.0096

Question 2.
3%
Answer: 0.03

Explanation:
To write percent into decimal we have to remove percent symbol and divide by 100, which moves the decimal point two places to the left.
So, 3% in decimal form is 0.03

Question 3.
$$25 . \overline{5} \%$$
Answer: $$0 .25 \overline{5}$$

Explanation:
To write percent into decimal we have to remove percent symbol and divide by 100, which moves the decimal point two places to the left.
So, $$25 . \overline{5} \%$$ in decimal form is $$0 .25 \overline{5}$$

Question 4.
$$0 . \overline{6} \%$$
Answer: $$0 .006\overline{6}$$

Explanation:
To write percent into decimal we have to remove percent symbol and divide by 100, which moves the decimal point two places to the left.
So, $$0 . \overline{6} \%$$  in decimal form is$$0 .006 \overline{6}$$

Question 5.
7.88
Answer: 788%

Explanation:
To write decimal into percent we should multiply the number by 100, which moves the decimal point two places to the right and add a percent symbol.
So, 7.88 can be rewrite as 788%

Question 6.
0.58
Answer: 58%

Explanation:
To write decimal into percent we should multiply the number by 100, which moves the decimal point two places to the right and add a percent symbol.
So, 0.58 can be rewrite as 58%

Order the numbers from least to greatest.
Question 7.
86%, $$\frac{15}{18}$$, 0.84, $$\frac{8}{9}$$, $$0 . \overline{86} \%$$
Answer: The numbers from least to greatest are  $$0 .00 \overline{86}$$ , $$0 .8 \overline{3}$$ , 0.84 , 0.86 , $$0 .\overline{8}$$

Explanation:
86%,  in decimals as 0.86
$$\frac{15}{18}$$, can be $$0 .8 \overline{3}$$
$$\frac{8}{9}$$,  can be $$0 .\overline{8}$$
$$0 . \overline{86} \%$$ can be written as $$0 .00 \overline{86}$$
So, The numbers from least to greatest are  $$0 .00 \overline{86}$$ , $$0 .8 \overline{3}$$ , 0.84 , 0.86 , $$0 .\overline{8}$$

Question 8.
91.6%, 0.91, $$\frac{11}{12}$$, 0.917, 9.2%
Answer:  The numbers from least to greatest are 0.092 , 0.91 , 0.916 , $$0 .91\overline{6}$$ , 0.917 .

Explanation:
91.6%, can be written as 0.916
$$\frac{11}{12}$$, ca be written as $$0 .91\overline{6}$$
9.2% can be written as 0.092
So, The numbers from least to greatest are 0.092 , 0.91 , 0.916 , $$0 .91\overline{6}$$ , 0.917 .

Write and solve a proportion or equation to answer the question.
Question 9.
What percent of 28 is 21?
Answer:  21 is 75% of 28.

Explanation:
Given , a = 21 , w = 28 , p = ?
By using proportion , we have
$$\frac{a}{w}$$ = $$\frac{p}{100}$$
$$\frac{21}{28}$$ = $$\frac{p}{100}$$
p = $$\frac{21 × 100}{28}$$
p = $$\frac{2100}{28}$$
p = 75
So, 21 is 75% of 28.

Question 10.
64 is what percent of 40?
Answer: 64 is 160% of 40.

Explanation:
Given , a = 64 , w = 40 , p = ?
By using proportion , we have
$$\frac{a}{w}$$ = $$\frac{p}{100}$$
$$\frac{64}{40}$$ = $$\frac{p}{100}$$
p = $$\frac{64 × 100}{40}$$
p = $$\frac{6400}{40}$$
p = 160
So, 64 is 160% of 40.

Question 11.
What number is 80% of 45?
Answer: 36 is 80% of 45.

Explanation:
Given , a = ? , w = 45 , p = 80%
By using percent proportion , we have
$$\frac{a}{w}$$ = $$\frac{p}{100}$$
$$\frac{a}{45}$$ = $$\frac{80}{100}$$
a = $$\frac{45 × 80}{100}$$
a = $$\frac{3600}{100}$$
a = 36
So, 36 is 80% of 45.

Question 12.
0.8% of what number is 6?
Answer: 6 is 0.8% of 750.

Explanation:
Given , a = 6 , w = ? , p = 0.8%
By using proportion , we have
$$\frac{a}{w}$$ = $$\frac{p}{100}$$
$$\frac{6}{w}$$ = $$\frac{0.8}{100}$$
w = $$\frac{6 × 100}{0.8}$$
w = $$\frac{600}{0.8}$$
w = 750
So, 6 is 0.8% of 750.

Identify the percent of change as an increase or a decrease. Then find the percent of change. Round to the nearest tenth of a percent if necessary.
Question 13.
4 strikeouts to 10 strikeouts
Answer: percent of change is 150%

Explanation:
We know that , formula for percent change = $$\frac{New value – old value}{old value}$$
where New value = 10 and old value = 4 , because a change of 4 to 10 is a positive (increase) change
So, percent change = $$\frac{10 – 4}{4}$$
= $$\frac{6}{4}$$
= 1.5 × 100
= 150%
So, percent of change is 150%

Question 14.
$24 to$18
Answer: percent of change is – 25%

Explanation:
We know that , formula for percent change = $$\frac{New value – old value}{old value}$$
where New value = 18 and old value = 24 , because a change of 24 to 18 is a negative (decrease) change
So, percent change = $$\frac{18 – 24}{24}$$
= $$\frac{- 6}{24}$$
= – 0.25
= – 0.25 × 100
= – 25%
So, percent of change is – 25%

Find the sale price or selling price.
Question 15.
Original price: $15 Discount: 5% Sale price: ? Answer: The sales price is$14.25

Explanation:
We know , The sales price be 100% – 5% = 95% of the original price
sales price = 95% of 15
= 0.95 × 15 = 14.25
So, The sales price is $14.25 Question 16. Cost to store:$5.50
Markup: 75%
Selling price: ?
Answer: The selling price is $9.625. Explanation: The markup is 75% of$5.50
a = p% × w
= 75% × 5.5
= 0.75 × 5.5
= 4.125
So , the markup is $4.125. To , find the selling price , we have selling price = cost to store + markup =$5.5 + $4.125 =$9.625.
The selling price is $9.625. An account earns simple interest. Find the interest earned or the principal. Question 17. Interest earned: ? Principal:$450
Interest rate: 6%
Time: 8 years
Answer: The interest earned is $216 . Explanation: we know P =$450 , r = 0.06 , t = 8 years
I = prt
I = 450 × 0.06 × 8
I = 27 × 8
I = $216 The interest earned is$216 .

Question 18.
Interest earned: $27 Principal: ? Interest rate: 1.5% Time: 2 years Answer: The principal is$900 .

Explanation:
Given , P = ? , I = $27 , t = 2 years , r = 0.015 We know that I = Prt So , p = $$\frac{I}{rt}$$ p = $$\frac{27}{0.015 × 2}$$ = $$\frac{27}{0.03}$$ = 900 So, The principal is$900 .

Question 19.
You spend 8 hours each weekday at school. (a) Write the portion of a weekday spent at school as a fraction, a decimal, and a percent. (b) What percent of a week is spent at school if you go to school 4 days that week? Round to the nearest tenth.
Answer: a. the portion of a weekday spent at school is 33% , It can be written as $$\frac{100}{3}$$ and in decimal as 0.333
b . The 9% of a week is spent at school if you go to school 4 days that week.

Explanation:
Given , You spend 8 hours each weekday at school.
a. In a day we have 24 hours so , The percent of 8 hours in 24 hours is
a = 8 , w = 24 , p = ?
By using proportion , we have
$$\frac{a}{w}$$ = $$\frac{p}{100}$$
$$\frac{8}{24}$$ = $$\frac{p}{100}$$
p = $$\frac{8 × 100}{24}$$
p = $$\frac{800}{24}$$
p = 33.3 %
It can be written as $$\frac{100}{3}$$ and in decimal as 0.333
So, 8 is 33% of 24.

b. The number of hours in a week  , that is 7 days are 168 hours
The number of hours in 4 days to school are 32
So , we have a = 32 , w = 168 , p = ?
By using proportion , we have
$$\frac{a}{w}$$ = $$\frac{p}{100}$$
$$\frac{32}{168}$$ = $$\frac{p}{100}$$
p = $$\frac{32 × 100}{168}$$
p = $$\frac{3200}{168}$$
p = 9.04 %
p = 9%
So , The 9% of a week is spent at school if you go to school 4 days that week.

Question 20.
Research indicates that90% of the volume of an iceberg is below water. The volume of the iceberg above the water is 160,000 cubic feet. What is the volume of the iceberg below water?

Answer: The volume of the iceberg below water is  1,77,777 cubic feet

Explanation:
Given , p = 90% , a = 160,000 , w = ?
By using proportion , we have
$$\frac{a}{w}$$ = $$\frac{p}{100}$$
$$\frac{160,000}{w}$$ = $$\frac{90}{100}$$
w = $$\frac{160,000 × 100}{90}$$
w = $$\frac{1050}{125}$$
w = 1,77,777.7
So, the volume of the iceberg below water is  1,77,777 cubic feet

Question 21.
You estimate that there are 66 cars in a parking lot. The actual number of cars is 75.
a. Find the percent error.
b. What other estimate gives the same percent error? Explain your reasoning.
Answer: a. percent Error is 12%
b, If the estimated number is 66.1 then percent error will be 12%

Explanation:
The amount of error is 75 – 66 = 9
We know that , formula for percent error = $$\frac{Error value}{Actual value}$$
where Error value = 9 and Actual value = 75 ,
So, percent Error = $$\frac{9}{75}$$
= 0.12
= 0.12 × 100
=  12%
So, percent Error is 12%

b. if the estimation is 66.1
The amount of error is 75 – 66.1 = 8.9
We know that , formula for percent error = $$\frac{Error value}{Actual value}$$
where Error value = 8.9 and Actual value = 75 ,
So, percent Error = $$\frac{8.9}{75}$$
= 0.1198
= 0.1198 × 100
=  11.9%
Apporximately 12%
So, percent Error is 12%

Percents Cumulative Practice

Question 1.
A movie theater offers 30% a movie ticket to students from your school. The regular price of a movie ticket is $8.50. What is the discounted price that you pay for a ticket? A.$2.55
B. $5.50 C.$5.95
D. $8.20 Answer: C.$5.95

Explanation:
Given , original price = $8.5 , with 30% off We know , The sales price be 100% – 30% = 70% of the original price sales price = 70% of 8.5 = 0.7 × 8.5 = 5.95 So, The sales price is$5.95.

Question 2.
What is the least value of x for which the inequality is true?

16 ≥ – 2x
Answer: x = 1

Explanation:
Let us say X = 1
16 ≥ – 2(1)
16 ≥ – 2
So , x = 1 , for 16 ≥ – 2x

Question 3.
You are building a scale model of a park that is planned for a city. The model uses the scale 1 centimeter = 2 meters. The park will have a rectangular reflecting pool with a length of 20 meters and a width of 12 meters. In your scale model, what will be the area of the reflecting pool?
F. 60 cm2
G. 120 cm2
H. 480 cm2
I. 960 cm2
Answer:  F .  60 cm2

Explanation:
Given ,actual  l = 20 meters , The model uses the scale 1 centimeter = 2 meters.  let the model length be x
The model length x will be , $$\frac{1cm}{2m}$$ = $$\frac{x cm}{20m}$$
2x = 20
x = 10 ,
Let the model width be y , The model width y will be $$\frac{1 cm}{2m}$$ = $$\frac{y}{12}$$
y= 6 ,
So The model area of the reflecting pool , Area of rectangle = l × w
l = 10 , y = 6 , then A = 10 × 6 = 60 cm2
The model area of the reflecting pool is 60 cm2

Question 4.
Which proportion represents the problem?

Answer: D . $$\frac{43}{n}$$ = $$\frac{17}{100}$$

Explanation:
Given ,  p = 17% , a = 43
By using proportion , we have
$$\frac{a}{w}$$ = $$\frac{p}{100}$$
$$\frac{43}{w}$$ = $$\frac{17}{100}$$
w = $$\frac{43 × 100}{17}$$
w = $$\frac{4300}{17}$$
w = 252.9
So, 43 is 17% of 252.9.

Question 5.
Which list of numbers is in order from least to greatest?
F. 0.8, $$\frac{5}{8}$$, 70%, 0.09
G. 0.09, $$\frac{5}{8}$$, 0.8, 70%
H. $$\frac{5}{8}$$, 70%, 0.8, 0.09
I. 0.09, $$\frac{5}{8}$$, 70%, 0.8
Answer: H. $$\frac{5}{8}$$, 70%, 0.8, 0.09

Explanation:
Given,
$$\frac{5}{8}$$ can be written as 0.625 ,
70% can be written as 0.7
The order from least to greatest  is  $$\frac{5}{8}$$ , 70% , 0.8 , 0.9

Question 6.
What is the value of $$\frac{9}{8}$$ ÷ (-$$\frac{11}{4}$$)?

Answer: the value of $$\frac{9}{8}$$ ÷ (-$$\frac{11}{4}$$) is – 0.413

Explanation:
$$\frac{9}{8}$$  can be written as  1.125
–$$\frac{11}{4}$$ can be written as  – 2.75
then , $$\frac{1.125}{-2.75}$$ = – 0.413

Question 7.
The number of calories you burn by playing basketball is proportional to the number of minutes you play. Which of the following is a valid interpretation of the graph?

A. The unit rate is $$\frac{1}{9}$$ calorie per minute.
B. You burn 5 calories by playing basketball for 45 minutes.
C. You do not burn any calories if you do not play basketball for at least 1 minute.
D. You burn an additional 9 calories for each minute of basketball you play.
Answer: C. You do not burn any calories if you do not play basketball for at least 1 minute.

Explanation:
As , the graph shows the coordinates of the minutes to calories by (0,0) at the initial stage of playing basketball for at least a minute .
So , the graph represents , You do not burn any calories if you do not play basketball for at least 1 minute.

Question 8.
A softball team is ordering uniforms. Each player receives one of each of the items shown in the table.

Which expression represents the total cost (in dollars) when there are 15 players on the team?
F. x + 24.86
G. 15x + 372.90
H. x + 372.90
I. x + 387.90
Answer: G. 15x + 372.90

Explanation:
Given , jersey = x
So , the only option in the options with an x for 15 members is 15x ,
So, G. 15x + 372.90 is the correct option

Question 9.
Your friend solves the equation. What should your friend do to correct the error that he made?

A. Multiply – 45 by – 3.
B. Add 3 to – 45.
C. Add 2 to – 15.
D. Divide – 45 by – 3.
Answer: A. Multiply – 45 by – 3.

Explanation:
Given , – 3(2 + w) = -45
By , Multiply – 45 by – 3.
we get 2 + w = 15
w = 15 – 2
w = 13.

Question 10.
You are comparing the costs of a certain model of ladder at a hardware store and at an online store.

Part A What is the total cost of buying the ladder at each of the stores? Show your work and explain your reasoning.
Part B Suppose that the hardware store is offering 10% off the price of the ladder and that the online store is offering free shipping and handling. Which store offers the lower total cost for the ladder? by how much? Show your work and explain your reasoning.
Answer: Part A , The total cost at hardware store is $371 and The total cost at online store is$355.2
Part B ,The hardware store offers the best price $336 by$3.2 less than the online store $339.2 Explanation: Part A , The ladder cost at hardware store is$350 , with 6% tax ,
So, 6% of 350
By using percent proportion , we have
$$\frac{a}{w}$$ = $$\frac{p}{100}$$
$$\frac{a}{350}$$ = $$\frac{6}{100}$$
a = $$\frac{350 × 6}{100}$$
a = $$\frac{2100}{100}$$
a = 21
So, 21 is 6% of 350.
The total cost at hardware store is $350 +$ 21 = $371 The ladder cost at online store is$320 , with 6% tax ,
So, 6% of 320
By using percent proportion , we have
$$\frac{a}{w}$$ = $$\frac{p}{100}$$
$$\frac{a}{320}$$ = $$\frac{6}{100}$$
a = $$\frac{320 × 6}{100}$$
a = $$\frac{1920}{100}$$
a = 19.2
So,19.2 is 6% of 320.
Additionally shipping cost is 5% of $320 By using percent proportion , we have $$\frac{a}{w}$$ = $$\frac{p}{100}$$ $$\frac{a}{320}$$ = $$\frac{5}{100}$$ a = $$\frac{320 × 5}{100}$$ a = $$\frac{1600}{100}$$ a = 16 So,16 is 5% of 320. The total cost at online store is$320 + $19.2 +$16 = $355.2 Part B, If the hardware store offers 10% off Then , We know , The sales price be 100% – 10% = 90% of the original price sales price = 90% of 350 = 0.9 × 350 = 315 So, The sales price is$315 .
The 6% of 320  is  $21 The total cost in hardware store after 10% off is$315 + $21 =$336

The online store offers free shipping and handling then the price will be $320 and 6% tax So , 6% of 320 is$19.2
The total cost online store is $320 +$ 19.2 = $339.2 The hardware store offers the best price$336 by $3.2 less than the online store$339.2

Question 11.
Which graph represents the inequality – 5 – 3x ≥ – 11.

Answer:

Explanation:11
Option F represents the x value from -4 to 2
So , if x = -4 ,
Then – 5 – 3(-4) ≥ – 11
-5 + 12 ≥ – 11
7 ≥ – 11
if x = 2
– 5 – 3(2) ≥ – 11
-5 – 6 ≥ – 11
-11 ≥ – 11
So, option F represents the inequality – 5 – 3x ≥ – 11.

Final Conclusion:

We have given Big Ideas Math Answers Grade 7 Chapter 6 Percents Pdf free links here. Download BIM Grade 7 Chapter 6 Pdf from the direct links given in the above section. We wish you make use of the given information and score better marks in the exam. If you have any doubts you can easily clear them within the below-given comment section. All the best to all the candidates!!

Big Ideas Math Answers Grade 3 Chapter 14 Represent and Interpret Data

Follow the free study guide i.e Big Ideas Math Book Grade 3 Chapter 14 Represent and Interpret Data Answers available here. It has various topics that every student must know before going for the next chapter. This BIM Answers 3rd Grade 14th Chapter Represent and Interpret Data is useful to finish the homework in time and prepare for the exams. Hence, download Big Ideas Math Answers Grade 3 Chapter 14 Represent and Interpret Data pdf for free of cost and begin the preparation.

Big Ideas Math Book Grade 3 Chapter 14 Represent and Interpret Data Answers

To ease your learning, we have given Big Ideas Math 3rd Grade 14th Chapter Represent and Interpret Data Solution Key which is designed by the subject experts. So, students and teachers are no need to worry if you have any doubts in questions. Make use of this Big Ideas Math Solutions Grade 3 Chapter 14 Represent and Interpret Data to score good marks in the exams.

The various topics covered in Represent and Interpret Data are Read and Interpret Picture Graphs, Make Picture Graphs, Read and Interpret Bar Graphs, Make Bar Graphs, Make Line Plots, Measure Lengths: Half Inch, and Measure Lengths: Quarter Inc. Practice questions from all these topics and attend performance task at the end to cross-check your learning skills. Students can also tap on the quick links provided below to get the solutions.

Lesson 1 Read and Interpret Picture Graphs

Lesson 2 Make Picture Graphs

Lesson 3 Read and Interpret Bar Graphs

Lesson 4 Make Bar Graphs

Lesson 5 Make Line Plots

Lesson 6 Measure Lengths: Half Inch

Lesson 7 Measure Lengths: Quarter Inch

Performance Task

Lesson 14.1 Read and Interpret Picture Graphs

Explore and Grow
You survey 14 students about their favorite type of party. The results are shown on the left picture graph. Use the key to represent the same data on the right picture graph.

Answer:
In left side
the bounce house has 6 students
the costume has 2 students
the pool has 4 students
and skating has 2 students.
And on the right side
the bounce house was represented with 6 students as 1×6= 6
the costume was represented with 2 students as 1×2= 2
the pool was represented with 4 students as 1×4= 4
and skating was represented with 2 students as 1×2= 2.

Explanation:

Here, we can see that the left side was given each emoji was equal to 2 students
so bounce house has three emojis which means 3×2= 6 students
the costume has one emoji which means 1×2= 2 students
the pool has two emojis which means 2×2= 4 students
and skating has one emoji which means 1×2= 2 students.
So to represent the same data on the right picture graph
here, we can see that in the right picture graph,
one emoji is equal to 1 student, so
the bounce house has  6 students which means 6×1= 6 emojis will be placed
the costume has 2 students which means 2×1= 2 emojis will be placed
the pool has 4 students which means 4×1= 4 emojis will be placed
and skating has 2 students which means 2×1= 2 emojis will be placed.

Structure
You ask one more student to name his favorite type of party. He chooses a pool party. How can you represent this on each graph? Explain.
Answer:
On the left side, we will add half emoji, and on the right side, we will add one emoji.

Explanation:
On the left side, we will represent the pool party by half emoji as on the left side one emoji is equal to 2 students, so to add one student we will place half emoji. And on the right side, we will represent the pool party by one emoji as on the right side one emoji is equal to 1 student, so to add one student we will place one emoji.

Think and Grow : Read and Interpret Picture Graphs
A picture graph shows data using pictures or symbols. The key of a picture graph gives the value of one picture or symbol. The value of one picture, or symbol, can be greater than 1.
Example
Use the graph to answer the questions.

Answer:
There are 6 national forests in Arizona.
There are 5 national forests in Colorado.

Explanation:
Given that one full tree is equal to two forests, and the half tree is equal to 1 forest.
In Arizona, there are 3 full trees which means 3×2= 6 national forests
and in Colorado, there are 2 full trees which means 2×2= 4 national forests, and a half tree which means 1×1= 1 national forest. So the total number of national forests is 4+1= 5 national forests.

Show and Grow

Question 1.
Use the graph to answer the questions. How many students chose dog? How many students chose fish?

Answer:
The number of students who chose the dog is 30 students
The number of students who chose the fish is 15 students.

Explanation:
Given one emoji is equal to 10 students,
as we can see three emojis for the dog
so the number of students who chose the dog is
3×10= 30 students.
And we can see one emoji and a half emoji for the fish
so the number of students who chose the fish is
1×10= 10 students and 1×5= 5 students
so the total number of the students who chose fish are
10+5= 15 students.

Question 2.
Use the graph to answer the questions. What does the symbol represent?

How many more students did not choose sledding?
Answer:
The total number of students who didn’t choose sledding is 7+4+6= 17 students.

Explanation:
Here, each emoji represents 2 students, and to find the number of students who didn’t choose sledding
we will add all the students of the other three activities rather than sledding
so in the skiing activity, there are three emojis and half emoji
which is 3×2= 6 students and 1×1= 1 students,
so the total number of students who choose skiing is 6+1= 7 students.
and in the snowboarding activity, there are two emojis which are 2×2= 4 students.
so the total number of students who choose snowboarding is 4 students.
in the sledding activity, there are four emojis and half emoji
which is 4×2= 8 students and 1×1= 1 students,
so the total number of students who choose skiing is 8+1= 9 students.
And in the ice skating activity, there are three emojis,
which is 3×2= 6 students.
So the total number of students who choose skiing is 6 students.
And the total number of students who didn’t choose sledding are 7+4+6= 17 students.

Question 3.
Use the graph to answer the questions. How many mangoes were eaten in June?

How many total mangoes were eaten in the months shown? Were more mangoes eaten in July or in June and August combined?
Answer:
The number of mangoes eaten in June is 21 mangoes.
The total number of mangoes eaten is 21+42+18= 81 mangoes.
Yes in the month of July more mangoes are eaten.

Explanation:
Given each emoji represents 6 mangoes,
so in the month of June, there are three emojis,
which means 3×6= 18 and a half emoji means 6/2= 3.
So the total number of mangoes eaten in June is 18+3= 21 mangoes.
in the month of July, there are seven emojis
which means 7×6= 42, so the total number of mangoes eaten in July is 42 mangoes.
in the month of August, there are three emojis,
which means 3×6= 18, so the total number of mangoes eaten in august is 18 mangoes.
The total number of mangoes eaten is 21+42+18= 81 mangoes.

Question 4.
on a picture graph, then what value does represent? Explain.

Answer:
10.

Explanation:
Let the single ball be 10 and then the half ball will be 5,
so 10+10+5= 25.

Think and Grow: Modeling Real Life

During which two weeks were a total of 52 cans recycled?

During ___ and ___ 52 cans were recycled.
Answer:
During week 3 and week 4 52 cans were recycled.

Explanation:
In week 1 the number of cans recycled is 8+8+8+4= 28 cans
In week 2 the number of cans recycled is 8+8+8+8= 32 cans
In week 3 the number of cans recycled is 8+4= 12 cans
In week 4 the number of cans recycled is 8+8+8+8+8= 40 cans
so during week 3 and week 4, 52 cans were recycled.

Show and Grow

Question 5.
Which two origami animals did a total of 32 students choose?

How many more students chose a frog or penguin than swan or butterfly?
Answer:
The two origami animals that did a total of 32 students are Swan and Penguin.

Explanation:
Given each emoji represents 4 students
The two origami animals did a total of 32 students are
Swan has three emojis and half emoji
which means 3×4= 12 and a half emoji means 4/2= 2
so the total number of students is 12+2= 14 students.
and Penguin has four emojis and half emoji
which means 4×4= 16 and a half emoji means 4/2= 2
so the total number of students is 16+2= 18 students.
The two origami animals that did a total of 32 students are Swan and Penguin.
How many more students chose frog or penguin than swan or butterfly
Butterfly has two emojis and half emoji
which means 2×4= 8 and a half emoji means 4/2= 2
so the total number of students is 8+2= 10 students.
and Frog has six emojis, which means 6×4= 24 students.
so the total number of students is 10+24= 34 students.
so there are 34-32= 2 more students who choose frog or penguin than Swan or Butterfly.

Read and Interpret Picture Graphs Homework & Practice 14.1

Question 1.
Use the graph to answer the questions. What value does the symbol represent?
How many students chose pterodactyl?

How many students chose stegosaurus or velociraptor?
How many students did choose tyrannosaur?
Answer:
The value of half emoji represents 5 students.
The total number of students who choose pterodactyl is 45 students.
The total number of students who chose stegosaurus or velociraptor is 60 students.
the total number of students who choose tyrannosaur is 20 students.

Explanation:
Given each emoji represent 10 students, and
In pterodactyl, there are four emojis and half emoji which means
4×10= 40 and a half emoji represents 10/2= 5
the total number of students who choose pterodactyl is 40+5= 45 students.
In stegosaurus, there are two emojis and half emoji which means
2×10= 20 and a half emoji represents 10/2= 5
the total number of students who choose stegosaurus is 20+5= 25 students.
In velociraptor, there are three emojis and half emoji which means
3×10= 30 and a half emoji represents 10/2= 5
the total number of students who choose velociraptor is 30+5= 35 students.
The total number of students who chose stegosaurus or velociraptor is 60 students.
In tyrannosaur, there are five emojis which means
2×10= 20 and the total number of students who choose tyrannosaur is 20 students.

Question 2.
Use the graph to answer the questions.
How many dogs participated in the survey?

Which dog treat has more votes than biscuits, but fewer votes than peanut butter? How many dogs chose this treat?
Answer:
The total number of dogs that participated in the survey are 30 dogs.
Dog bone dog treat has more votes than biscuits and fewer votes than peanut butter. And 8 dogs choose this treat.

Explanation:
Given each emoji represent two dogs
so dog bone has four emojis which means 4×2= 8 dogs
Peanut butter has five emojis which means 5×2= 10 dogs
Cheese has two emojis and half emoji which means
2×2= 4 and a half emoji represents 1 dog
so the total number of dogs is 4+1= 5 dogs.
Biscuits have three emojis and half emoji which means
3×2= 6 and a half emoji represents 1 dog
so the total number of dogs is 6+1= 7 dogs.
So the total number of dogs who participated in the survey are
8+10+5+7= 30 dogs.
Dog bone dog treat has more votes than biscuits and fewer votes than peanut butter.
And 8 dogs choose this treat.

DIG DEEPER!
Why would it be difficult to use a key where the value of one symbol represents an odd number of dogs?
Answer:
It would be difficult to use a key where the value of one symbol represents an odd number of dogs because if the symbol is half then the value will be in decimals and we cannot divide the dog into decimals, so it’s difficult to represent an odd number.

YOU BE THE TEACHER
Newton says that one more dog likes peanut butter than dog bones. Is he correct? Explain.
Answer:
Yes, Newton is correct

Explanation:
Yes, Newton is correct. We can see in the table that the number of dogs who likes peanut butter is more than the dogs who like a dog bone. So Newton is correct.

Question 3.
Modeling Real Life
Which creature has 3 more eyes than the squid?

Answer:
The creature which has 3 more eye images than the squid is the spider which has four eye images.

Explanation:
Given each eye image represents 2 eyes
as we can see squid contains one eye image, which means 2 eyes
the spider has four eye images, which means 4×2= 8 eyes
Praying mantis has three eye images and half eye image, which means
3×2= 6 eyes and half eye image represents 1 eye
so the total number of eyes is 6+1= 7 eyes.
Starfish has three eye images, which means
3×2= 6 eyes.
So the creature which has 3 more eye images than the squid is the spider which has four eye images
and the creature has 3 more eyes than the squid is the spid=r, praying mantis, and starfish.

Review & Refresh

Find the area of the shape.

Question 4.

Area = ___
Answer:
The area of the shape is 20 square centimeters.

Explanation:
To find the area of the shape, we will divide the shape into parts, and then we will find the area of the shape.
So we will divide the shape into two rectangles,
the length of rectangle 1 is 7 cm, and
the breadth of rectangle 1 is 2 cm
so the area of rectangle 1 is
area= length×breadth
= 7×2
= 14 square centimeters.
the length of rectangle 2 is 3 cm, and
the breadth of rectangle 2 is 2 cm
so the area of rectangle 2 is
area= length×breadth
= 3×2
= 6 square centimeters.
So the total area of the shape is
14+6= 20 square centimeters.

Question 5.

Area = ___
Answer:
The area of the rectangle is 15 square meters.

Explanation:
The length of the rectangle is 5 m, and
the breadth of the rectangle is 3 m
so the area of the rectangle is
area= length×breadth
= 5×3
= 15 square meters.
The area of the rectangle is 15 square meters.

Lesson 14.2 Make Picture Graphs

Explore and Grow

Flip a two-color counter 10 times. Record the results. Then complete the picture graph.

Reasoning
Why might you change the key if you ﬂip the counter 100 times?
Answer:

Think and Grow: Make Picture Graphs

A frequency table is a table that gives the number of times something occurs.
Example
You survey students about their favorite planet. The frequency table shows the results. Use the table to complete the picture graph.

Step 1: Write the title at the top of the picture graph. Label a row for each category.

Step 2: Look at the numbers Saturn in the table. Choose a value for the key.
Step 3: Use the key to decide how many symbols you need for each planet. Then draw the symbols.
Answer:
Earth needs five symbols,
Mars needs three symbols,
Saturn needs six symbols, and
Jupiter needs four symbols.

Explanation:
The number of students who chooses Saturn is 30 students and the value for the key is six symbols.
By using the key the planet Earth needs five symbols,
Mars needs three symbols,
Saturn needs six symbols, and
Jupiter needs four symbols.

Show and Grow

Question 1.
Use the frequency table to complete the picture graph.

Answer:
Each circle represents three books

Explanation:

Each circle represents three books
To represent in a picture graph
June will be represented with two circles,
July will be represented with one circle,
August will be represented in three circles.

Apply and Grow: Practice

Question 2.
Use the frequency table to complete the picture graph.

How many symbols did you draw to represent 10 inches of snowfall in March?
How many inches do you think April would receive?
Answer:
Each circle represents 5 inches

Explanation:

Each circle represents 5 inches
To represent in a picture graph
January will be represented with six circles,
February will be represented with five circles,
March will be represented with two circles.

Question 3.
Use the frequency table to complete the picture graph.

Structure
Choose a different value for the key. How would the picture graph change?
Answer:
Each emoji represents 4 students.

Explanation:

Each emoji represents 4 students
To represent in a picture graph
Flying will be represented with eight emojis,
Time travel will be represented with eight emojis,
Super strength will be represented with three emojis,
Invisibility will be represented with six emojis,
Super speed will be represented with seven emojis.

Think and Grow: Modeling Real Life

You survey 90 students about their favorite type of ﬁeld trip. 35 students choose science center, 10 choose theater, and 20 choose zoo. The rest of the students choose museum. Complete the picture graph.

Answer:
Each emoji represents 5 students.
The total number of students who choose the Museum is 25 students.

Explanation:
Let’s take each emoji represents 5 students,
As the Science center was chosen by 35 students, which means we can represent with 35/5= 7 emojis,
The theater was chosen by 10 students, which means we can represent with 10/5= 2 emojis,
Zoo was chosen by 20 students, which means we can represent with 20/5= 4 emojis,
so to know how many students choose Museum, we will add all the three field students and then subtract from the total number of the students, so
The number of students from the three fields is 35+10+20= 65 students
and the number of students who choose the Museum is 90-65= 25 students.
And the Museum was chosen by 25 students, which means we can represent with 25/5= 5 emojis.

Show and Grow

Question 4.
You survey 48 students about their favorite type of movie. 8 students choose cartoon, 12 choose action, and 24 choose comedy. The rest of the students choose musical. Complete the picture graph.

All of the students who chose the musical go to see a movie. Each ticket costs $9.The students use two$20 bills to pay for all of the tickets. What is the change?
Answer:
Each emoji represent 4 students.
The remaining change will be $4. Explanation: Let’s take that each emoji represent 4 students. The total number of students who participated in the survey are 48 students, and in that eight students chooses cartoon, which means we can represent with 8/4= 2 emojis, and 12 students choose an action, which means we can represent with 12/4= 3 emojis, and 24 students choose comedy, which means we can represent with 24/4= 6 emojis, so to know how many students choose Musical, we will add all the three type of movie students and then subtract from the total number of the students, so The number of students from three types of movies is 8+12+24= 44 students and the number of students who choose the Musical are 48-44= 4 students. And the Musical was chosen by 4 students, which means we can represent with 4/4= 1 emoji. The total number of students who choose a musical type of movie is 4 students and each ticket costs$9. So for a total of 4 students, it will cost 4×$9=$36. As two students use $20 bills and four students, it will take$40 bill for all of the tickets. So the remaining change will be $40-$36= $4. Make Picture Graphs Homework & Practice 14.2 Question 1. Use the frequency table to complete the picture graph. Which type of art has more votes than painting, but fewer votes than crafts? How many students chose that type of art? Answer: Ceramic has more votes than painting and fewer votes than Crafts. So the total number of students who choose Ceramic is 25 students. Explanation: Each emoji represents 5 students To represent in a picture graph Drawing will be represented with two emojis, Ceramics will be represented with five emojis, The painting will be represented with three emojis, Crafts will be represented with six emojis. Ceramic has more votes than painting and fewer votes than Crafts. So the total number of students who choose Ceramic is 25 students. Question 2. Use the frequency table to complete the picture graph. Answer: Each circle represents 3 insects. Explanation: Each circle represents 3 insects. To represent in a picture graph Ant will be represented with eight circles, Bee will be represented with one circle, Ladybug will be represented with four circles. DIG DEEPER! You see 1 more bee and 4 more ladybugs. How might you change the key? Answer: We might change the key by each circle with 4 insects Explanation: In the above, we can see Bee has three insects and if we add one more bee then the number of insects will be 3+1= 4 insects. And in the above, we can see ladybugs have 12 insects and if we add four more ladybugs then the number of insects will be 12+4= 16 insects. So we might change the key by each circle with 4 insects, so that bee will be represented with one circle, and the ladybug will be represented with 16/4= 4 circles. Question 3. Modeling Real Life You survey 72 students about their favorite carnival ride. 12 choose Ferris wheel, 24 choose swings, and 6 choose bumper cars. The rest of the students choose roller coaster. Complete the picture graph. Answer: Let’s take that each emoji represent 6 students. Explanation: Let’s take that each emoji represent 6 students. The total number of students who participated in the favorite carnival ride survey are 72 students, and in that twelve students chooses Ferris wheel, which means we can represent with 12/6= 2 emojis, and 24 students choose swings, which means we can represent with 24/6= 4 emojis, and 6 students choose bumper cars, which means we can represent with 6/6= 1 emoji, so to know how many students choose roller coaster, we will add all the three types of carnival ride students and then subtract from the total number of the students, so The number of students from three types of carnival rides is 12+24+6= 42 students and the number of students who choose the roller coaster is 72-42= 30 students. And the roller coaster was chosen by 30 students, which means we can represent with 30/6= 5 emojis. Modeling Real Life All of the students who chose roller coaster want to ride together. Each ride ticket costs$2. The students have three $10 bills and four$5 bills. Will they have enough to ride the roller coaster together?
Answer:
No. they don’t have enough money.

Explanation:
As there are a total number of students who choose roller coaster ride are 30 students, so ride ticket costs for 30 students is 30×$2=$60. As three students have a $10 bill which means$10×3= $30 and four students have a$5 bill which means $5×4=$20. So together the students have $30+$20= $50, and they need$60. So they don’t have enough money.

Review & Refresh

Question 4.
532 + 54 = ___
Answer:
532 + 54 = 586.

Explanation:
On adding 532 and 54 we will get 586.

Question 5.
718 + 226 = ___
Answer:
718+226= 944.

Explanation:
On adding 718 and 226 we will get 944.

Question 6.
81 + 647 = ___
Answer:
81+647= 728.

Explanation:
On adding 81 and 647 we will get 728.

Lesson 14.3 Name Read and Interpret Graphs

Explore and Grow

You survey 12 students about their favorite school club. The results are shown on the picture graph. Represent the same data on the bar graph.

Answer:
The drama was chosen by 4 students,
Math was chosen by 2 students,
Robotics was chosen by 6 students.

Explanation:
Given each emoji represents two students,
To represent in a bar graph
Drama is represented with two emojis, which is 2×2= 4 students
Math is represented with one emoji, which is 1×2= 2 students
Robotics is represented with three emojis, which is 3×2= 6 students.
and represented in the graph as shown below.

Structure
How would you change the scale of the bar graph to match the picture graph?
Answer:
The number of students will be represented with 2,4,6,8, etc., and change the scale of the bar graph to match the picture graph.

Think and Grow:
A bar graph shows data using bars. The scale of a bar graph is the group of labels that shows the values at equally spaced grid lines.
Example
Use the graph to answer the questions.

How many gold medals did Jamaica win?
Jamaica won ___ gold medals.
Which country won the fewest gold medals?
__ won the fewest gold medals.
Answer:
Jamaica won 6 gold medals.
Sweden won the fewest gold medals.

Explanation:
In the above graph, we can see Jamaica won 6 gold medals and Sweden won the fewest gold medals.

Show and Grow

Question 1.
Use the graph to answer the questions.

How many students chose grapes? Which fruit is the most favorite?
Answer:
Grapes were chosen by 10 students and the most favorite fruit is Apples.

Explanation:
In the above bar graph, we can see the number of students who chosen grapes are 10 students, and the most favorite fruit apples and the number of students who choose are 45 students.

Apply and Grow: Practice

Question 2.
Use the graph to answer the questions.

How many students does each grid line represent? How many students chose fall? Which season is the least favorite?
Answer:
Each grid line represents four students and the fall was chosen by 18 students and the least favorite season is winter.

Explanation:
In the above bar graph, we can see that each grid line represents four students and we can see that the fall was chosen by 18 students. And the least favorite season is winter because winter was chosen by 10 students.

Question 3.
Use the graph to answer the questions.

How many vegetable seeds did the farmer plant in all? The farmer wants to plant green bean seeds. She plants more green bean seeds than zucchini seeds, but fewer than carrot seeds. How many green bean seeds could the farmer have planted? The farmer plants 30 more potato seeds. Will the farmer have more potatoes or more corn?
Answer:
The number of vegetable seeds did the farmer plant in all is four vegetables.
The number of green bean seeds that the farmer can plant is between 61 to 79 seeds.
The farmer will have more corn seeds than potato seeds.

Explanation:
As we can see in the bar graph the number of vegetable seeds is four. As we can see the number of zucchini seeds planted is 60 seeds and the number of carrot seeds planted is 80 seeds. As she planted more green bean seeds than zucchini seeds and fewer than carrot seeds, so the green bean seeds can be in between 61 to 79 seeds.
As we can see in the bar graph, the corn seeds are 95 seeds and the potato seeds are 45 seeds, so if 30 more seeds are added then the total potato seeds will be 45+30= 75 seeds. So farmers will have more corn seeds than potato seeds.

Question 4.
Writing
Do you think a bar graph or a picture graph is easier to read? Explain.
Answer:
A picture graph is easier to read than a Bar graph.

Explanation:
A picture graph was represented using pictures or symbols and a bar graph was represented using bars and compares the data in each category using bars. So the picture graph is easier than a bar graph because in a picture graph we can easily calculate the values than in the bar graph.

Think and Grow: Modeling Real Life

How many more students need to choose the strategy app so that strategy is the most favorite?

Understand the problem:
Make a plan:
Solve:
___ more students need to choose the strategy app so it is the most favorite.
Answer:
22 more students need to choose the strategy app so it is the most favorite.

Explanation:
As we can see in the above bar graph the strategy app was chosen by 20 numbers and if this strategy app needs to be the most favorite, then 22 more students needed to be chosen.

Show and Grow

Question 5.
Each grade needs to plant 20 trees. How many more trees does second grade need to plant to complete the goal?

How many more trees did fourth-grade plant than first grade and third grade combined?
Answer:
Second grade needs 6 more trees to plant to complete the goal.
The number of more trees that fourth-grade plant than first grade and third grade are 2 trees.

Explanation:
As we can see in the above bar graph, the second-grade students planted 14 trees and each grade needs to plant 20 trees. So second-grade needs 20-14= 6 more trees to plant to complete the goal.
The fourth grade planted 20 trees and first grade planted 8 trees and the third grade planted 10 plants,
so the total combined trees of first grade and third grade are 8+10= 18 trees. So the number of more trees that fourth-grade plant than first grade and third grade are 20-18= 2 trees.

Name Read and Interpret Graphs Homework & Practice 14.3

Question 1.
Use the graph to answer the questions.

How many students does each grid line represent? Which type of exercise is the least favorite? How many fewer students chose running than swimming? How many students chose walking or biking?
Answer:
Each grid line represents 10 students.
The 5 fewer students choose running than swimming.
Walking was chosen by 15 students and biking was chosen by 45 students.

Explanation:
As we can see in the bar graph that each grid line represents 10 students. And in the bar graph, we can see the skating contains 10 number of students which is the least favorite. The number of students who choose running is 30 students and the number of students who choose swimming is 35 students. So the 5 fewer students choose running than swimming. In the bar graph, we can see that walking was chosen by the 15 number of students, and biking was chosen by 45 students.

Question 2.
Use the graph to answer the questions.

Which sunﬂowers are taller than 11 feet? Sunﬂower E has a height of 15 feet. How much taller is Sunﬂower A than Sunﬂower E?
Writing
Which sunﬂower is the shortest? Explain.
Answer:
The sunflowers which are taller than 11 feet are Sunflower A and Sunflower D.
Sunflower A is 11 feet taller than Sunflower E.
Sunflower C is the shortest.

Explanation:
As we can see in the bar graph that the Sunflower A is 26 feet and Sunflower D is 16 feet, and Sunflower B is 10 feet, Sunflower C is 10 feet. So Sunflower A and Sunflower D are taller than 11 feet. Sunflower A is 26 feet and Sunflower E is 15 feet. So the Sunflower A is 26-15= 11 feet taller than Sunflower E. Sunflower C is the shortest, because Sunflower C has the least number of feet.

Question 3.
Modeling Real Life
Use the graph to answer the questions.

How many more students need to choose chameleon so that chameleon is the most favorite? You survey 10 more students and they all choose snake. What is the new total number of students who chose snake? How many more students chose turtle than bearded dragon and lizard combined?
Answer:
There must be 9 more number of students needed to choose Chameleon to be the most favorite.
The new total number of students who choose the snake is 19 students.
There are 3 students more students who choose turtle than bearded dragon and lizard.

Explanation:
As we can see in the bar graph that the total number of students is 30 and in that Chameleon was chosen by 21 number of students. So to choose chameleon as a most favorite then the total number should choose are 30-21=9 students. As we can see in the bar graph the total number of students who choose a snake is 9 students and if 10 more students choose a snake, then the total number of students who choose a snake will be 9+10=19 students. As we can see in the bar graph that the turtle was chosen by 27 students and the bearded dragon was chosen by 18 students and the lizard was chosen by 6 students. So the total number of students who choose bearded dragon and lizard are 18+6= 24 students and the turtle was chosen by 27 students, so there are 27-24= 3 students more students chooses turtle than bearded dragon and lizard.

Review & Refresh

Estimate the difference.

Question 4.
96 – 47 = __
Answer:
96-47= 49.
Explanation:
The difference between 96 and 47 is 49.

Question 5.
678 – 142 = ___
Answer:
678-142= 536.
Explanation:
The difference between 678 and 142 is 536.

Lesson 14.4 Make Bar Graphs

Explore and Grow

Spin the Color Spinner 10 times. Record the results. Then complete the bar graph.

Answer:
Reasoning
Explain how you would change the scale if you spin the spinner 100 times.
Answer:

Think and Grow: Make Bar Graphs

Example
You record the number of times each baseball team wins. The frequency table shows the results. Use the table to complete the bar graph.

Step 1: Write the title at the top of the bar graph. Label a row for each category. Label the categories.
Step 2: Look at the numbers in the table. Use a scale so that most of the bars end on a grid line. Label the scale.
Step 3: Draw and shade a bar for each team.
Answer:
Each grid line represents three wins.

Explanation:

As given the values of the number of wins are the multiples of 3, so we will represent the number of wins with the multiples of 3. To represent these values in a bar graph, we will represent those data with rectangular bars with proper heights and lengths by the given values. And those bars are plotted vertically or horizontally. In the above bar graph, we can see each grid line represents three wins.

Show and Grow

Question 1.
Use the frequency table to complete the bar graph.

Answer:
Each grid line represents two number of students.

Explanation:

In the given frequency table, we can see most of the values are multiplies of 2. So we represent the number of students by the multiplies of 2. To represent these values in a bar graph, we will represent those data with rectangular bars with proper heights and lengths by the given values. And those bars are plotted vertically or horizontally. In the above bar graph, we can see each grid line represents two number of students.

Apply and Grow: Practice

Question 2.
Use the frequency table to complete the bar graph.

How would you use the graph to decide which type of ﬁngerprint is the most common?
Answer:
To decide which type of fingerprint is used most commonly, we will compare the heights of the bars, and then we will choose the most common fingerprint.

Explanation:

In the given frequency table, we can see most of the values are multiplied by 25. So we represent the number of students by the multiplies of 25. To represent these values in a bar graph, we will represent those data with rectangular bars with proper heights and lengths by the given values. And those bars are plotted vertically or horizontally. In the above bar graph, we can see each grid line represents twenty-five number of students.

Question 3.
Use the frequency table to complete the bar graph.

How many fewer students chose the least favorite type of food than the most favorite type of food?
Answer:
The least favorite type of food chosen by vegetarians and the most favorite type of food was chosen by Mexican.
And each grid line represents four number of students.

Explanation:

In the given frequency table, we can see most of the values are multiplies of 4. So we represent the number of students by the multiplies of 4. To represent these values in a bar graph, we will represent those data with rectangular bars with proper heights and lengths by the given values. And those bars are plotted vertically or horizontally. In the above bar graph, we can see each grid line represents four number of students.

Think and Grow: Modeling Real Life

You survey 27 students about their favorite subject. Nine students choose science. Six fewer students choose English than science. The rest of the students choose math. Complete the bar graph.

Answer:
Each grid line represents three number of students.
The total number of students who choose math is 15 students.

Explanation:

In the given frequency table, we can see most of the values are multiplies of 3. So we represent the number of students by the multiplies of 3. To represent these values in a bar graph, we will represent those data with rectangular bars with proper heights and lengths by the given values. And those bars are plotted vertically or horizontally. In the above bar graph, we can see each grid line represents three number of students.
As the total number of students for the survey is 27 students in that nine students choose science, and six fewer students choose English than Science, which means 9-6= 3. So 3 students choose Science. And the rest of the students chooses math, which means we add the two subjects and then subtract from the total number of students, then we can get the value of the students who choose math. So the total number of students who choose English and Science are 9+3= 12 and then subtract with the total number of students, which means 27-12= 15. So the total number of students who choose math is 15 students.

Show and Grow

Question 4.
You survey 22 students about their favorite camp activity. Eight students choose archery. Four more students choose swimming than archery. The rest of the students choose hiking. Complete the bar graph.

How many fewer students chose archery than swimming and hiking combined?
Answer:
Six fewer students chose archery than swimming and hiking combined.

Explanation:

In the given frequency table, we can see most of the values are multiplies of 2. So we represent the number of students by the multiplies of 2. To represent these values in a bar graph, we will represent those data with rectangular bars with proper heights and lengths by the given values. And those bars are plotted vertically or horizontally. In the above bar graph, we can see each grid line represents three number of students.
The number of students who choose archery is eight students and in that four more students choose swimming than archery, which means 8+4= 12 students choose swimming. And the rest choose hiking, which means we will add the number of students who choose archery and swimming and then subtract the value with the total number of students. So the number of students who choose archery and swimming is 8+12= 20, and the number of students who choose hiking is 22-20= 2. The total number of students who choose swimming and hiking combined is 12+2= 14 students, and archery was chosen by 8 students. So 14-8= 6 fewer students choose archery than swimming and hiking combined.

Make Bar Graphs Homework & Practice 14.4

Question 1.
Use the frequency table to complete the bar graph.

How many students does each grid line represent? How would you use the graph to find the most favorite type of music?
Answer:
Each grid line represents five number of students. By using the graph, we will compare the heights of the graphs and pick the most favorite music.

Explanation:

As given the values of the number of students are the multiples of 5, so we will represent the number of students with the multiples of 5. To represent these values in a bar graph, we will represent those data with rectangular bars with proper heights and lengths by the given values. And those bars are plotted vertically or horizontally. In the above bar graph, we can see each grid line represents five number of students. By using the graph, we will compare the heights of the graphs and pick the most favorite music.

Question 2.
Use the frequency table to complete the bar graph.

Structure
On the last day of school, each backpack weighs less than 5 pounds. How could the scale of the bar graph change?
Answer:
We will subtract the given values by 5 pounds and then we will represent those values in a graph.

Explanation:

As given the values of the number of pounds are the multiples of 4, so we will represent the number of pounds with the multiples of 4. To represent these values in a bar graph, we will represent those data with rectangular bars with proper heights and lengths by the given values. And those bars are plotted vertically or horizontally. In the above bar graph, we can see each grid line represents four pounds. As on the last day, each backpack weighs less than 5 pounds, which means we will subtract the given values by 5, and then we will represent those values in a graph.

Question 3.
Modeling Real Life
You survey 26 teachers about their favorite vacation spot. Six teachers choose amusement park. Two more teachers choose camping than amusement park. The rest of the teachers choose beach. Complete the bar graph.
How many fewer teachers chose camping than amusement park and beach combined?

Answer:
The beaches were chosen by 12 teachers.

Explanation:

As given the values of the number of teachers are most of them are multiples of 2, so we will represent the number of teachers with the multiples of 2. To represent these values in a bar graph, we will represent those data with rectangular bars with proper heights and lengths by the given values. And those bars are plotted vertically or horizontally. In the above bar graph, we can see each grid line represents five number of students. By using the graph, we will compare the heights of the graphs and pick the most favorite music. As amusement park was chosen by six teachers and two more teachers choose camping than the amusement park, so camping was chosen by 6+2= 8 teachers. And the rest of the teachers choose the beach, which means we will add both the vacation amusement park and camping and then subtract by the total number of teachers. So the total number of teachers who choose both amusement parks and camping is 6+8= 14 teachers, so the teachers who choose the beach are 26-14= 12 teachers.

Review & Refresh

Find the difference.

Question 4.
474 – 19 = ___
Answer:
474 – 19 = 455.

Explanation:
The difference between 474 and 19 is 455.

Question 5.
615 – 204 = ___
Answer:
615 – 204 = 411.

Explanation:
The difference between 615 and 204 is 411.

Question 6.
232 – 53 = ___
Answer:
232 – 53 =179.

Explanation:
The difference between 232 and 53 is 179.

Lesson 14.5 Make Line Plots

Explore and Grow

A teacher asks students to line up according to the number of siblings they have. The results are shown. Create a Line Plot for the number of siblings the students in your class have.

Answer:

Explanation:
The students that have 0 siblings in the line plot 1 is 2,
The students that have 1 sibling in the line plot 1 is 11,
The students that have 2 siblings in the line plot 1 is 5,
The students that have 3 siblings in the line plot 1 is 2,
The students that have 0 siblings in the line plot 2 is 1,
The students that have 1 sibling in the line plot 2 is 6,
The students that have 2 siblings in the line plot 2 is 3,
The students that have 3 siblings in the line plot 2 is 2.
Structure
Compare the two line plots. How are the line plots the same? How are they different?
Answer:
The line plots are the same in that each class has a maximum of 3 siblings and varies in the count of siblings.

Explanation:
Comparing the two-line plots we can see the count of siblings varies and the two number line plots are the same that the class has the maximum number of siblings is 3.

Think and Grow: Make Line Plots
A lion plot uses marks above a number line to show data values.
Example
The table shows the weights of 15 bald eagles. Use the table to complete the line plot.

Step 1: write the title at the top of the line plot.
_____________
Step 2 : Look at the numbers in the table. Use a scale that shows all of the data values. Draw a number line using the scale. Label the scale.
Step 3 : Mark an X for each data value.

Answer:
A line plot can be defined as a graph that displays the given data as a point above a number line.

Explanation:

The eagles that weigh 3kgs are 2,
The eagles that weigh 4kgs are 6,
The eagles that weigh 5kgs are 4,
The eagles that weigh 6kgs are 2.

Show and Grow

Question 1.
Use the table to complete the line plot.

Answer:
A line plot can be defined as a graph that displays the given data as a point above a number line.

Explanation:

A line plot can be defined as a graph that displays the given data as a point above a number line. As the given values are mostly are multiples of 3, so we will represent the number of inches by multiples of 3. And now we will place the check marks on the given values.

Apply and Grow: Practice

Question 2.
Use the table to complete the line plot.

How many waves were 26 feet tall or taller?
Answer:
There are 11 waves that were 26 feet tall or taller.

Explanation:

A line plot can be defined as a graph that displays the given data as a point above a number line. As the given values are mostly a series of numbers from 25 to 30, so we will represent the number of feet by a series of numbers from 25 to 30. And now we will place the check marks on the given values.

Question 3.
Use the table to complete the line plot.

Which giraffe tongue length is the most common?

Answer:
The most common giraffe tongue lengths are 20 inches.

Explanation:

A line plot can be defined as a graph that displays the given data as a point above a number line. As the given values are mostly a series of numbers from 17 to 21, so we will represent the number of feet by a series of numbers from 17 to 21. And now we will place the check marks on the given values. The most common giraffe tongue lengths are 20 inches.

Precision
How many giraffe tongues are 19 inches long?
Answer:
The 19 inches long giraffe tongues are 3.

Think and Grow: Modeling Real Life

What is the difference in height of the tallest student and the shortest student?
Subtraction equation:

The difference in height is ___ inches.
Answer:
The difference in height is 6 inches.

Explanation:
As we can see in the above number plot, the height of the tallest student is 56 inches, and the height of the shortest student is 50 inches. And the difference between them is 56-50= 6 inches.

Show and Grow

Question 4.
What is the difference between the greatest number of floors and the least number of floors?

How many fewer skyscrapers are there with over 50 floors than skyscrapers with under 50 floors?
Answer:
The fewer skyscrapers are there with over 50 floors than skyscrapers with under 50 floors are 11.

Explanation:
The greatest number of floors is 8 floors and the least number of floors are 2,
so the difference between the greatest number of floors and the least number of floors is 8-2= 6.
The skyscrapers over 50 floors are 32 and the skyscrapers with under 50 floors are 21,
The fewer skyscrapers are there with over 50 floors than skyscrapers with under 50 floors are 32-21= 11.

Make Line Plots Homework & Practice 14.5

Question 1.
Use the table to complete the line plot.
____

How many snakes are longer than 10 feet?
Which snake length is the most common?
Answer:
The snakes which are longer than 10 feet are 1.

Explanation:

The number of snakes which are 5 feet are 1,
The number of snakes which are 6 feet are 3,
The number of snakes which are 7 feet are 3,
The number of snakes which are 8 feet are 0,
The number of snakes which are 9 feet are 2,
The number of snakes which are 10 feet are 4,
The number of snakes which are 11 feet is 1.

Question 2.
Use the table to complete the line plot.

Reasoning
Are most of the students able to complete 30 sit-ups? Explain.

Answer:
No, the students who completed 30 sit-ups are 1 student.

Explanation:

The number of students who completed 27 sit-ups is 6 students.
The number of students who completed 28 sit-ups is 1 student.
The number of students who completed 29 sit-ups is 3 students.
The number of students who completed 30 sit-ups is 1 student.
The number of students who completed 32 sit-ups is 3 students.

DIG DEEPER!
Student H completed more sit-ups than Student F, but fewer sit-ups than Student B. How many sit-ups could Student H have completed?
Answer:

Explanation:

Question 3.
Modeling Real Life
What is the difference of the most number of miles biked and the least number of miles biked?

Answer:
The most number of miles biked and the least number of miles biked 5.

Explanation:
The most number of miles biked are 6 bikes and the least number of miles biked is 1 bike,
so the difference between the most number of miles biked and the least number of miles biked is 6-1= 5.

Review & Refresh

Find the sum or difference. Use the inverse operation to check.

Question 4.