Practice with the help of enVision Math Common Core Grade 5 Answer Key Topic 10 Represent and Interpret Data regularly and improve your accuracy in solving questions.

enVision Math Common Core 5th Grade Answers Key Topic 10 Represent and Interpret Data

enVision STEM Project: Wildfires

Do Research Use the Internet and other sources to learn more about wildfires. Investigate how wildfires affect ecosystems. Explore the costs and benefits of wildfires. List five living things in an ecosystem. Research how long each one takes to recover from a wildfire.

Journal: Write a Report Include what you found. Also in your report:
• Make a pamphlet to show how wildfires affect ecosystems.
• Suggest ways to prevent wildfires.
• Make a line plot to show your data.
• Make up and solve problems using line plots.

Review What You Know

A-Z Vocabulary
Choose the best term from the Word List. Write it on the blank.
• bar graph
• overestimate
• compare
• frequency table
• underestimate

Question 1.
A display that shows how many times an event occurs is a(n) ____
Answer:
A display that shows how many times an event occurs is a frequency table.

Question 2.
A display that uses bars to show data is a(n) ____
Answer:
A display that uses bars to show data is a bar graph.

Question 3.
Rounding each factor in a multiplication to a greater number gives a(n) ____ of the actual product.
Answer:
Rounding each factor in multiplication to a greater number gives a overestimate of the actual product.

Question 4.
You can use the length of the bars in a bar graph to ___ two similar data sets.
Answer:
You can use the length of the bars in a bar graph to compare two similar data sets.

Fraction Computation

Find each answer.

Question 5.
2\(\frac{1}{2}\) + 5\(\frac{1}{3}\)
Answer:
2\(\frac{1}{2}\) + 5\(\frac{1}{3}\) = 7\(\frac{5}{6}\).

Explanation:
The addition of 2\(\frac{1}{2}\) + 5\(\frac{1}{3}\) is \(\frac{5}{2}\) + \(\frac{16}{3}\) which is \(\frac{15+32}{6}\) = \(\frac{47}{6}\) = 7\(\frac{5}{6}\).

Question 6.
13\(\frac{3}{10}\) – 8\(\frac{1}{5}\)
Answer:
13\(\frac{3}{10}\) – 8\(\frac{1}{5}\) = 5\(\frac{1}{10}\).

Explanation:
The subtraction of 13\(\frac{3}{10}\) – 8\(\frac{1}{5}\) is \(\frac{133}{10}\) – \(\frac{41}{5}\) which is \(\frac{133 – 82}{10}\) = \(\frac{51}{10}\) = 5\(\frac{1}{10}\).

Question 7.
8\(\frac{1}{3}\) + 7\(\frac{11}{12}\)
Answer:
8\(\frac{1}{3}\) + 7\(\frac{11}{12}\) = 15\(\frac{11}{12}\).

Explanation:
The addition of 8\(\frac{1}{3}\) + 7\(\frac{11}{12}\) is \(\frac{24}{3}\) + \(\frac{95}{12}\) which is \(\frac{96+95}{12}\)
= \(\frac{191}{12}\)
= 15\(\frac{11}{12}\).

Question 8.
15 – 5\(\frac{2}{9}\)
Answer:
15 – 5\(\frac{2}{9}\) = 9\(\frac{2}{9}\).

Explanation:
The subtraction of 15 – 5\(\frac{2}{9}\) is 15 – \(\frac{47}{9}\)
= \(\frac{135 – 47}{9}\)
= \(\frac{88}{9}\)
= 9\(\frac{2}{9}\).

Question 9.
7\(\frac{5}{8}\) + 13\(\frac{11}{20}\)
Answer:
7\(\frac{5}{8}\) + 13\(\frac{11}{20}\) = 21\(\frac{7}{40}\).

Explanation:
The addition of 7\(\frac{5}{8}\) + 13\(\frac{11}{20}\) is \(\frac{61}{8}\) + \(\frac{271}{20}\) = \(\frac{305+542}{40}\)
= \(\frac{847}{40}\)
= 21\(\frac{7}{40}\).

Question 10.
15\(\frac{4}{5}\) + 1\(\frac{2}{3}\)
Answer:
15\(\frac{4}{5}\) + 1\(\frac{2}{3}\) = 17\(\frac{7}{15}\).

Explanation:
The addition of 15\(\frac{4}{5}\) + 1\(\frac{2}{3}\) is \(\frac{79}{5}\) +  \(\frac{5}{3}\) which is \(\frac{237+25}{15}\)
= \(\frac{262}{15}\)
= 17\(\frac{7}{15}\).

Question 11.
\(\frac{7}{8}\) × 4
Answer:
\(\frac{7}{8}\) × 4 = 7\(\frac{1}{2}\).

Explanation:
The multiplication of \(\frac{7}{8}\) × 4 which is \(\frac{7}{2}\)
= 7\(\frac{1}{2}\).

Question 12.
5 × 1\(\frac{2}{3}\)
Answer:
5 × 1\(\frac{2}{3}\) = 18\(\frac{1}{3}\).

Explanation:
The multiplication of 5 × 1\(\frac{2}{3}\) is 5 × \(\frac{11}{3}\)
= \(\frac{55}{3}\)
= 18\(\frac{1}{3}\).

Question 13.
2\(\frac{1}{8}\) × \(\frac{2}{3}\)
Answer:
2\(\frac{1}{8}\) × \(\frac{2}{3}\) = \(\frac{17}{12}\).

Explanation:
The multiplication of 2\(\frac{1}{8}\) × \(\frac{2}{3}\) is \(\frac{17}{8}\) × \(\frac{2}{3}\) which is \(\frac{17}{12}\).

Bar Graphs
Use the bar graph to answer the questions.

Question 14.
Which animal has about 34 teeth?
Answer:
Hyena.

Explanation:
The animal that has about 34 teeth is Hyena.

Question 15.
About how many more teeth does a dog have than a hyena?
Answer:
The number of many more teeth does a dog has is 8 teeth.

Explanation:
Here, the dog has 42 teeth and a hyena has 34 teeth. So the number of many more teeth does a dog has is 42 – 34 which is 8 teeth.

Question 16.
About how many more teeth does a hyena have than a walrus?
Answer:
The number of many more teeth does a hyena has is 15 teeth.

Explanation:
Here, a hyena has 32 teeth and a walrus has 17 teeth. So the number of many more teeth does a hyena has is 32 – 17 which is 15 teeth.

Pick a Project

PROJECT 10A
How big is Big Data?
Project: Make Line Plots for Data

PROJECT 10B
What was the first U.S. penny?
Project: Design a Coin

PROJECT 10C
Why are sequoia trees so big?
Project: Measure Trees

PROJECT 10D
How are plant leaves different?
Project: Make a Leafy Line Plot

Lesson 10.1 Analyze Line Plots

Solve & Share
Several students were asked how many blocks they walk from home to school each day. The results are shown in the line plot below. Miguel walks the shortest distance. Jila walks the longest distance. How much farther than Miguel does Jila walk?

Answer:
Miguel walks 4.25 blocks more than Jila.

Explanation:
Here, the line plot is shown in the question which represents the distance traveled by the students from their houses to their school. So the distance between 1 and 2 is \(\frac{1}{4}\) which is 0.25 blocks. So the shortest distance traveled by Miguel is 0.25 blocks. And the longest distance traveled by Jila is 4.5 blocks. So the difference between the distances covered by Miguel and Jila is 4.5 – 0.25 = 4.25 blocks. So Miguel walks 4.25 blocks more than Jila.

Make Sense and Persevere How can you find the shortest distance walked?

Look Back! What can you tell about the distance most of the students in the group walk to school each day?

Visual Learning Bridge

Essential Question
How Can You Analyze Data Displayed in a Line Plot?

A.
A line plot, or dot plot, shows data along a number line. Each dot or X represents one value in the data set.
In science class, Abby and her classmates performed an experiment in which they used different amounts of vinegar. The table below shows how much vinegar each person used. What amount was most often used in the experiment?

Answer:
1 cup vinegar was mostly used.

Explanation:
The amount which was most often used in the experiment 1 cup of vinegar.

A line plot shows how often each value occurs.

B.
Read the line plot.
A line plot can be used to organize the amount of vinegar each person used.

C.
Analyze the data. Use the data to answer your original question. What amount was most often used in the experiment?
Most of the data points are at 1, so 1 cup of vinegar was most often used in the experiment.

Convince Me! Reasoning How does the line plot show what the largest amount of vinegar used was?
Answer:
The line plot represents the data using number line. Here, we can see that the 1 cup of vinegar have mostly often used in the experiment.

Guided Practice

Do You Understand?

Question 1.
Why does data need to be ordered from least to greatest before creating a line plot?
Answer:
Here, the data need to be ordered from least to greatest before creating a line plot, so that the data from least to greatest can help us interpret the data to get the better sense of the numbers and scope of numbers we are working with.

Question 2.
In the line plot on the previous page, do any values occur the same number of times? Explain.
Answer:
The values that occur the same number of times is \(\frac{1}{2}\) and 1\(\frac{1}{2}\).

Explanation:
The values that occur the same number of times is \(\frac{1}{2}\) and 1\(\frac{1}{2}\) as they occur two times.

Question 3.
Describe any patterns in the line plot on the previous page.
Answer:
The pattern in the line plot is dot plot.

Do You Know How?

In 4 and 5, use the data set to answer the questions.

Question 4.
Students ran for 10 minutes. They ran the following distances, in miles:
\(\frac{3}{4}\), \(\frac{1}{2}\), 1\(\frac{3}{4}\), \(\frac{3}{4}\), 1
1\(\frac{1}{2}\), 1\(\frac{1}{4}\), 2, 1\(\frac{1}{2}\), 1, 1\(\frac{1}{4}\), \(\frac{3}{4}\), 1
How many distances are recorded?

Answer:
14 distances are recorded.

Explanation:
The number of distances recorded are 14 distances.

Question 5.
Use the line plot that shows the data.

Which distances occurred most often?
Answer:
\(\frac{3}{4}\), 1 and 1\(\frac{1}{4}\).

Explanation:
The distances that occurs most often is \(\frac{3}{4}\), 1 and 1\(\frac{1}{4}\).

Independent Practice

Data in a line plot can be shown with dots or Xs.

In 6-8, use the line plot to answer the questions.

Question 6.
How many orders for cheese does the line plot show?
Answer:
12 orders.

Explanation:
The number of orders for cheese does the line plot shows is 12 orders.

Question 7.
Which amount of cheese was ordered most often?
Answer:
\(\frac{3}{4}\).

Explanation:
\(\frac{3}{4}\) amount of cheese was ordered most often.

Question 8.
How many more orders for \(\frac{3}{4}\) cheese were for a pound or less than for 1 pound or more?

Problem Solving

In 9-11, use the data set and line plot.

Question 9.
Jerome studied the feather lengths of some adult fox sparrows. How long are the longest feathers in the data set?
Answer:
2\(\frac{3}{4}\).

Explanation:
The longest feather length in the given data is 2\(\frac{3}{4}\).

Question 10.
How many feathers are 2\(\frac{1}{4}\) inches or longer? Explain.
Answer:
8 feathers.

Explanation:
The number of feathers that are 2\(\frac{1}{4}\) inches or longer are 8 feathers.

Question 11.
Higher Order Thinking Jerome found another feather that made the difference between the longest and shortest feather 1\(\frac{3}{4}\) inches. What could be the length of the new feather? Explain.

Answer:
The difference is 1.

Explanation:
The difference between the longest and shortest feather is 2\(\frac{3}{4}\) – 1\(\frac{3}{4}\) = \(\frac{11}{4}\) – \(\frac{7}{4}\)
= \(\frac{4}{4}\).
= 1.

Question 12.
Reasoning How can you find the value that occurs most often by looking at a line plot?
Answer:
We can know the value that occurs most often by looking at a line plot by the size of the line plot. If the size is longer then that will be the most occurring on the line plot.

Question 13.
Draw and label a rectangle with a perimeter of 24 inches.
Answer:

Assessment Practice

Question 14.
Use the information shown in the line plot. What is the total weight of the 4 heaviest melons?
A. 6 pounds
B. 11\(\frac{1}{2}\) pounds
C. 22\(\frac{1}{2}\) pounds
D. 24 pounds

Answer:

Lesson 10.2 Make Line Plots

Activity

Solve & Share
A fifth-grade class recorded the height of each student. How could you organize the data? If all the students in the class lay down in a long line, how far would it reach? Make a line plot to solve this problem.

Answer:
Given that a fifth-grade class recorded the height of each student, so the line plot of the fifth-grade class is

Organizing data makes it easier to understand and analyze.

Look Back! Generalize How does organizing the data help you see the height that occurs most often? Explain.
Answer:
Organizing the data helps to communicate the data in a way that helps others to understand the data and interpret it correctly.

Visual Learning Bridge

Essential Question
How Can You Use a Line Plot to Organize and Represent Measurement Data?
Answer:
We can use a line plot to display and organize data and a line plot shows how many times each number appears in the data set.

A.
The dogs in Paulina’s Pet Shop have the following weights. The weights are in pounds.
How can you organize this information in a line plot?

Answer:
The line plot of weights of dogs is

Measurement data that is organized is easier to use.

B.
Organize the data.
Write the weights from least to greatest.
6, 6, 7\(\frac{1}{4}\), 8\(\frac{1}{2}\), 8\(\frac{1}{2}\), 8\(\frac{1}{2}\), 11\(\frac{1}{2}\), 12\(\frac{1}{4}\), 12\(\frac{1}{4}\), 12\(\frac{1}{4}\), 12\(\frac{1}{4}\), 12\(\frac{1}{4}\)
You can also organize the data in a frequency table. The frequency is how many times a given response occurs.

C.
Make a line plot.
First draw the number line using an interval of \(\frac{1}{4}\). Then mark a dot for each value in the data set. Write a title for the line plot.

Convince Me! Reasoning Which weight occurs most often? Which weight occurs least often? How can you tell from the line plot?

Guided Practice

Do You Understand?

Question 1.
In the line plot of dog weights on the previous page, what does each dot represent?
Answer:
Here, each dot represents the weight of the dog.

Question 2.
In a line plot, how do you determine the values to show on the number line?
Answer:

Do You Know How?

Question 3.
Draw a line plot to represent the data.

Answer:
The line plot of weights of pumpkins is

Independent Practice

In 4 and 5, complete the line plot for each data set.

Double check that you have a dot for each value.

Question 4.
11\(\frac{1}{4}\), 12\(\frac{1}{2}\), 11\(\frac{1}{4}\), 14\(\frac{3}{4}\), 10\(\frac{1}{2}\), 11\(\frac{1}{4}\), 12

Answer:
The line plot of the given value

Question 5.
1\(\frac{1}{8}\), 2, 1\(\frac{1}{2}\), 1\(\frac{1}{4}\), 1\(\frac{1}{8}\), 1, 2, 1\(\frac{1}{2}\), 1\(\frac{1}{4}\)

Answer:
The line plot of the given values is

In 6 and 7, construct a line plot for each data set.

Question 6.
\(\frac{1}{2}\), \(\frac{3}{4}\), \(\frac{3}{4}\), 1, 1, 0, \(\frac{1}{2}\), \(\frac{1}{2}\), \(\frac{3}{4}\)
Answer:
The line plot of the given values is

Question 7.
5\(\frac{1}{2}\), 5, 5, 5\(\frac{1}{8}\), 5\(\frac{3}{4}\), 5\(\frac{1}{4}\), 5\(\frac{1}{2}\), 5\(\frac{1}{8}\), 5\(\frac{1}{2}\), 5\(\frac{3}{8}\)
Answer:
The line plot of the given values is

Problem Solving

In 8-10, use the data set.

Question 8.
Make Sense and Persevere Martin’s Tree Service purchased several spruce tree saplings. Draw a line plot of the data showing the heights of the saplings.
Answer:

Question 9.
How many more saplings with a height of 27\(\frac{1}{4}\) inches or less were there than saplings with a height greater than 27\(\frac{1}{4}\) inches?
Answer:
There are 11 saplings.

Explanation:
The number of saplings that are with a height of 27\(\frac{1}{4}\) inches or less was there than saplings with a height greater than 27\(\frac{1}{4}\) inches are 11 saplings.

Question 10.
Higher Order Thinking Suppose Martin’s Tree Service bought two more saplings that were each 27\(\frac{1}{4}\) inches tall. Would the value that occurred most often change? Explain your answer.
Answer:
Yes, the value that occurred most often will change.

Explanation:
Yes, the value that occurred most often will be changed. As there will be an increase of saplings with the height of 27\(\frac{1}{4}\) inches tall, so the value that occurred most often will change.

Question 11.
Why is organizing data helpful?
Answer:
Organizing data is very useful, by this we can find the values easily and can complete our work fastly.

Question 12.
Randall buys 3 tickets for a concert for $14.50 each. He gives the cashier a $50 bill. How much change does he get? Write equations to show your work.
Answer:
The change that will get to Randell is $6.50 and the equation is $50 – $43.50 = $6.50.

Explanation:
Given that Randall buys 3 tickets for a concert for $14.50 each and he gives the cashier a $50 bill, so the total cost for the tickets will be 3 × 14.50 which is $43.50. So the change that will get to Randell is $50 – $43.50 which is $6.5. So the equation is $50 – $43.50 = $6.50.

Assessment Practice

Question 13.
Amy measured how many centimeters the leaves on her houseplants grew in July. Use the leaf growth data below to complete the line plot on the right.
2\(\frac{1}{2}\), 4\(\frac{1}{2}\), 4, 4, 3, 1, 3, 3\(\frac{1}{2}\), 3\(\frac{1}{2}\), 3\(\frac{1}{2}\), 2\(\frac{1}{2}\), 3, 3\(\frac{1}{2}\), 3\(\frac{1}{2}\), 5\(\frac{1}{2}\)

Answer:
The line plot of the given data is

Lesson 10.3 Solve Word Problems Using Measurement Data

Solve & Share
Rainfall for the Amazon was measured and recorded for 30 days. The results were displayed in a line plot. What can you tell about the differences in the amounts of rainfall? Use the line plot to solve this problem.

You can use a representation to analyze data. Show your work!

Look Back! Reasoning What is the difference between the greatest amount of rain in a day and the least amount of rain in a day? How can you tell?

Answer:
The difference is 9 days.

Explanation:
The difference between the greatest amount of rain in a day and the least amount of rain in a day is 11 – 2 which is 9. By using subtraction we can tell that.

Visual Learning Bridge

Essential Question How Can You Use Measurement Data Represented in a Line Plot to Solve Problems?

A.
Bruce measured the daily rainfall while working in Costa Rica. His line plot shows the rainfall for each day in September. What was the total rainfall for the month?

You can use the line plot to make a frequency table.

B.
Multiply each value by the frequency to find the amount of rain for that value. Then add the products to find the number of inches of rainfall for the month.

The table helps you organize the numerical data for your calculations.

Convince Me! Critique Reasoning Rosie says she can find the total rainfall in the example above without multiplying. Do you agree? Explain.
Answer:
Yes, we can find the total rainfall.

Explanation:
Yes, we can find the total rainfall in the example above without multiplying. By adding them we can also find the total rainfall.

Guided Practice

Do You Understand?

In 1-4, use the line plot showing how many grams of salt were left after liquids in various containers evaporated.

Question 1.
How could you find the difference between the greatest amount and the least amount of salt left?
Answer:
The difference is 5.

Explanation:
The difference between the greatest amount and the least amount of salt left is 7 – 2 which is 5.

Do You Know How?

Question 2.
Write a problem that can be answered using the line plot.
Answer:

Question 3.
Write and solve an equation that represents the total number of grams of salt left.
Answer:
The equation is 7 × 1 \(\frac{1}{2}\) + 4 × 2 + 3 × 6 \(\frac{1}{2}\) + 7 × 5 = 73.

Explanation:
The total number of grams of salt is 7 × 1 \(\frac{1}{2}\) + 4 × 2 + 3 × 6 \(\frac{1}{2}\) + 7 × 5 = 7 × \(\frac{3}{2}\) + 8 + 3 × \(\frac{13}{2}\) + 35
= \(\frac{21}{2}\) + 8 + \(\frac{39}{2}\) + 35
= \(\frac{21+16+39+70}{2}\)
= \(\frac{146}{2}\)
= 73.
And the equation is 7 × 1 \(\frac{1}{2}\) + 4 × 2 + 3 × 6 \(\frac{1}{2}\) + 7 × 5 = 73.

Question 4.
How many grams of salt would be left if two of each container were used?
Answer:
The number of grams of salt is 36 \(\frac{1}{2}\).

Explanation:
The number of grams of salt that would be left if two of each container were used is 73 ÷ 2 which is 36 \(\frac{1}{2}\).

Independent Practice

In 5 and 6, use the line plot Allie made to show the lengths of strings she cut for her art project.

Question 5.
Write an equation for the total amount of string.

Answer:
The equation is 2 × 12 \(\frac{5}{8}\) + 7 × 12 \(\frac{3}{4}\) + 5 × 12 \(\frac{7}{8}\) + 5 × 13 + 4 × 13 \(\frac{1}{8}\) = 413 \(\frac{1}{4}\).

Equation:
The equation for the total amount of string is 2 × 12 \(\frac{5}{8}\) + 7 × 12 \(\frac{3}{4}\) + 5 × 12 \(\frac{7}{8}\) + 5 × 13 + 4 × 13 \(\frac{1}{8}\)
= 2 × \(\frac{101}{8}\) + 7 × \(\frac{51}{4}\) + 5 × \(\frac{103}{4}\) + 65 + 4 × \(\frac{105}{4}\)
= \(\frac{202}{8}\) + \(\frac{357}{4}\) + \(\frac{515}{4}\) + 65 + \(\frac{420}{4}\)
= \(\frac{202+714+1030+520+840}{8}\)
= \(\frac{3306}{8}\)
= 413 \(\frac{1}{4}\).

Question 6.
What is the difference in length between the longest and the shortest lengths of string?
Answer:
The difference is 2.

Explanation:
The difference in length between the longest and the shortest lengths of the string is 7 – 2 which is 5.

Problem Solving

In 7 and 8, use the line plot Susannah made to show the amount of rainfall in one week.

Question 7.
Algebra Write and solve an equation for the total amount of rainfall, r, Susannah recorded.

Answer:
The equation will be 2 × \(\frac{1}{16}\) + 1 × \(\frac{1}{8}\) + 1 × \(\frac{1}{4}\) = \(\frac{1}{2}\).

Explanation:
The equation for the total amount of rainfall is 2 × \(\frac{1}{16}\) + 1 × \(\frac{1}{8}\) + 1 × \(\frac{1}{4}\)
= \(\frac{1}{8}\) + \(\frac{1}{8}\) + \(\frac{1}{4}\)
= \(\frac{1+1+2}{8}\)
= \(\frac{4}{8}\)
= \(\frac{1}{2}\).

Question 8.
Higher Order Thinking Suppose the same amount of rain fell the following week, but the same amount of rain fell each day. How much rain fell each day?
Answer:

Question 9.
Make Sense and Persevere The area of a square deck is 81 square feet. How long is each side of the deck?
Answer:
The length of each side is 9.

Explanation:
Given that the area of a square deck is 81 square feet, so the length of each side of the deck is
Area = S × S = 81
S × S = 9 × 9
S = 9.

How does knowing the shape of the deck help you?

Question 10.
Althea recorded the amount she earned from T-shirt sales each day for 14 days. She made a frequency table to organize the data. Write a problem that can be answered by using the frequency table.

Answer:

Assessment Practice

Question 11.
Kurt recorded the amount of snowfall in each month for one year. What was the total snowfall that year?
A. 12 in.
B. 10\(\frac{1}{4}\) in.
C. 7\(\frac{3}{4}\) in.
D. 10\(\frac{1}{2}\) in.

Answer:
The total snowfall of that year is 10\(\frac{1}{4}\).

Equation:
Given that Kurt recorded the amount of snowfall in each month for one year, so the total snowfall of that year is 1 × \(\frac{1}{4}\) + 2 × \(\frac{1}{2}\) + 1 × \(\frac{3}{4}\) + 3 × 1 + 1 × 2\(\frac{1}{2}\) + 1 × 2\(\frac{3}{4}\)
= \(\frac{1}{4}\) + \(\frac{2}{2}\) + \(\frac{3}{4}\) + 3 + \(\frac{9}{4}\) + \(\frac{11}{4}\)
= \(\frac{1+4+3+12+10+11}{4}\)
= \(\frac{41}{4}\)
= 10\(\frac{1}{4}\).

Lesson 10.4 Critique Reasoning

Activity

Problem Solving

Solve & Share
A cross country coach recorded the team’s practice runs and made the line plot below. The coach had each runner analyze the line plot and write an observation. Read the statements and explain whether you think each runner’s reasoning makes sense.

Thinking Habits
Be a good thinker! These questions can help you.
• What questions can I ask to understand other people’s thinking?
• Are there mistakes in other people’s thinking?
• Can I improve other people’s thinking?

Look Back! Critique Reasoning Quinn said that to find the total distance the team ran in September, you add each number that has an X above it: \(\frac{1}{2}\) + 1 + 1\(\frac{1}{2}\) + 1\(\frac{3}{4}\) + 2 + 2\(\frac{1}{2}\) + 2\(\frac{3}{4}\) + 3. Do you agree? Explain why or why not.

Visual Learning Bridge

Essential Question How Can You Critique the Reasoning of Others?

A.
Ms. Kelly’s class made a line plot showing how many hours each student spent watching television the previous evening. Amanda said, “No one watched TV for 3 hours.” Drake said, “No, 3 of us watched no TV.” Who is correct? Explain your reasoning.

What information do Amanda and Drake use for reasoning?
Amanda and Drake base their reasoning on their analysis of the data displayed on the line plot.

B.
How can I critique the reasoning of others?
I can
• decide if the statements make sense.
• look for mistakes in calculations.
• clarify or correct flaws in reasoning.

Here’s my thinking.

C.
Amanda’s statement is incorrect. Her reasoning has flaws. She sees the 3 Xs above the zero and thinks that means that zero people watched 3 hours of TV. The labels on the number line tell how many hours, and the Xs tell how many students. So, Amanda should have said, “There are 3 people who watched 0 hours of TV.”
Drake’s statement is correct. Since there are 3 Xs above the o, he is correct when he says that 3 people watched no TV.

Convince Me! Critique Reasoning Andre said, “More than half of us watched TV for less than 2 hours.” Explain how you can critique Andre’s reasoning to see if his thinking makes sense.

Guided Practice
Renee works for a sand and gravel company. She made a line plot to show the weight of the gravel in last week’s orders. She concluded that one third of the orders were for more than 6 tons.

Question 1.
What is Renee’s conclusion? How did she support it?
Answer:

Question 2.
Describe at least one thing you would do to critique Renee’s reasoning.
Answer:

Question 3.
Does Renee’s conclusion make sense? Explain.
Answer:

Independent Practice

Critique Reasoning
Aaron made a line plot showing the weights of the heads of cabbage he picked from his garden. He said that since 1\(\frac{1}{2}\) + 2 + 2\(\frac{1}{4}\) + 2\(\frac{3}{4}\) = 8\(\frac{1}{2}\), the total weight of the cabbages is 8\(\frac{1}{2}\) pounds.

Question 4.
Describe at least one thing you would do to critique Aaron’s reasoning
Answer:

Question 5.
Is Aaron’s addition accurate? Show how you know.
Answer:

When you critique reasoning, you need to explain if someone’s method makes sense.

Question 6.
Can you identify any flaws in Aaron’s thinking? Explain.
Answer:

Question 7.
Does Aaron’s conclusion make sense? Explain.
Answer:

Problem Solving

Performance Task
Television Commercials
Ms. Fazio is the manager of a television station. She prepared a line plot to show the lengths of the commercials aired during a recent broadcast. She concluded that the longest commercials were 3 times as long as the shortest ones because 3 × \(\frac{1}{2}\) = 1\(\frac{1}{2}\).

Question 8.
Make Sense and Persevere Which information in the line plot did Ms. Fazio need to use in order to draw her conclusion?
Answer:

Question 9.
Reasoning Did the number of Xs above the number line affect Ms. Fazio’s conclusion? Explain.
Answer:

To use math precisely, you need to check that the words, numbers, symbols, and units you use are correct and that your calculations are accurate.

Question 10.
Model With Math Did Ms. Fazio use the correct operation to support her conclusion? Explain.
Answer:

Question 11.
Be Precise Are Ms. Fazio’s calculations accurate? Show how you know.
Answer:

Question 12.
Critique Reasoning Is Ms. Fazio’s conclusion logical? How did you decide? If not, what can you do to improve her reasoning?
Answer:

Topic 10 Fluency Practice

Activity

Follow the Path
Solve each problem. Follow problems with an answer of 29,160 to shade a path from START to FINISH. You can only move up, down, right, or left.

Topic 10 Vocabulary Review

Glossary

Understand Vocabulary

Choose the best term from the Word List. Write it on the blank.

Word List
• bar graph
• data
• frequency table
• line plot

Question 1.
Another name for collected information is _____
Answer:
Another name for collected information is data.

Question 2.
A ___ uses tally marks to show how many times a data value occurs in a data set.
Answer:
A frequency table uses tally marks to show how many times a data value occurs in a data set.

Question 3.
A(n) ___ is a display of responses on a number line with a dot or X used to show each time a response occurs.
Answer:
A line plot is a display of responses on a number line with a dot or X used to show each time a response occurs.

Round the data to the nearest whole number and order from least to greatest.

Question 4.
2.3, 8.6, 5.5, 4.9
Answer:
2.3, 4.9, 5.5, 8.6.

Explanation:
The nearest whole number and order from least to greatest are 2.3, 4.9, 5.5, 8.6.

Question 5.
42.1, 50, 37.2, 76.5, 43.9
Answer:
37.2, 42.1, 43.9, 50, 76.5.

Explanation:
The nearest whole number and order from least to greatest are 37.2, 42.1, 43.9, 50, 76.5.

Order each data set from least to greatest.

Question 6.
1\(\frac{1}{2}\), 0, 1\(\frac{3}{4}\), 13\(\frac{1}{2}\), 1\(\frac{2}{3}\)
Answer:
0, 1\(\frac{1}{2}\), 1\(\frac{2}{3}\), 13\(\frac{1}{2}\).

Explanation:
The nearest whole number and order from least to greatest are 0, 1\(\frac{1}{2}\), 1\(\frac{2}{3}\), 13\(\frac{1}{2}\).

Question 7.
\(\frac{6}{7}\), 2\(\frac{1}{2}\), \(\frac{3}{4}\), 1, 1\(\frac{3}{4}\)
Answer:
\(\frac{3}{4}\), \(\frac{6}{7}\), 1, 1\(\frac{3}{4}\), 2\(\frac{1}{2}\).

Explanation:
The nearest whole number and order from least to greatest are \(\frac{3}{4}\), \(\frac{6}{7}\), 1, 1\(\frac{3}{4}\), 2\(\frac{1}{2}\).

Write always, sometimes, or never.

Question 8.
A line plot is ____ the best way to display data.
Answer:
A line plot is always the best way to display data.

Question 9.
A data display ___ uses fractions or decimals.
Answer:
A data display sometimes uses fractions or decimals.

Question 10.
Cross out the words that are NOT examples of items that contain data.
Encyclopedia Alphabet Email address MP3 Player Phone number Grocery list

Use Vocabulary in Writing

Question 11.
Twenty students measured their heights for an experiment in their science class. How can a line plot help the students analyze their results?
Answer:
Given that twenty students measured their heights for an experiment in their science class, here line plot helps the students analyze their results by comparing sets of data and track changes over time.

Topic 10 Reteaching

Set A
pages 429-432

The data set below shows the number of cartons of milk drank by 20 students in a week.
2, 4, 3\(\frac{1}{2}\), \(\frac{1}{2}\), 1\(\frac{1}{2}\), 1\(\frac{1}{2}\), 3\(\frac{1}{2}\), 2, 3, \(\frac{1}{2}\), 1, 3\(\frac{1}{2}\), 3, 2, 1, 3\(\frac{1}{2}\), 1, 3, 3\(\frac{1}{2}\), 2

The line plot shows how often each data value occurs.

Use the Cartons Drank line plot.

Question 1.
How many students drank 1\(\frac{1}{2}\) cartons?
Answer:
2 students.

Explanation:
2 students drank 1\(\frac{1}{2}\) cartons.

Question 2.
How many students drank more than 2\(\frac{1}{2}\) cartons?
Answer:
9 students.

Explanation:
9 students drank more than 2\(\frac{1}{2}\) cartons.

Question 3.
What was the greatest number of cartons drank by a student?
Answer:
4 cartons.

Explanation:
The greatest number of cartons drank by a student is 4 cartons.

Question 4.
How many students drank only 1 carton?
Answer:
3 students.

Explanation:
3 students drank only 1 carton.

Question 5.
What is the difference between the greatest and least number of cartons drank?
Answer:
The difference is 5 – 1 = 4.

Explanation:
The difference between the greatest and least number of cartons drank is 5 – 1 which is 4.

Set B
pages 433-436

Twelve people were surveyed about the number of hours they spend reading books on a Saturday. The results are:

Draw a number line from 0 to 3. Mark the number line in fourths because the survey results are given in \(\frac{1}{4}\) hours. Then for each response, place a dot above the value on the number line.

Remember that you can make a line plot to show and compare data.

Use the information below to make a line plot about Patrick’s plants.

Patrick listed how many inches his plants grew in one week: 1 \(\frac{1}{2}\) \(\frac{3}{4}\) 1\(\frac{1}{2}\) \(\frac{1}{2}\) 1\(\frac{1}{4}\) \(\frac{1}{2}\) 1

Question 1.
Complete the line plot.
Answer:
The line plot is

Question 2.
How many dots are on the line plot?
Answer:
7 dots.

Explanation:
There are 7 dots on the line plot.

Question 3.
How many inches of plant growth was the most common?
Answer:
\(\frac{1}{2}\) and 1\(\frac{1}{2}\).

Explanation:
The most common plant growth in inches is \(\frac{1}{2}\) and 1\(\frac{1}{2}\).

Set C
pages 437-440
This line plot shows the amount of flour Cheyenne needs for several different recipes. She organizes the data in a frequency table to calculate the total amount of flour she needs.

Remember that you can multiply each data value by its frequency to find the total amount.
Use the line plot and frequency table at the left.

Question 1.
What values are multiplied in the third column of the table?
Answer:
\(\frac{1}{2}\) × 7 = 3\(\frac{1}{2}\).

Explanation:
The value multiplied in the third column of the table is \(\frac{1}{2}\) × 7 which is 3\(\frac{1}{2}\).

Question 2.
Write and solve an equation to find the total amount of flour Cheyenne needs.
Answer:

Set D
pages 441-444
Think about these questions to help you critique the reasoning of others.

Thinking Habits
• What questions can I ask to understand other people’s thinking?
• Are there mistakes in other people’s thinking?
• Can I improve other people’s thinking?

Remember you need to carefully consider all parts of an argument.

Question 1.
Justin says the line plot shows that the daily rainfall for the past two weeks is about an inch. Do you agree with his reasoning? Why or why not?
Answer:

Topic 10 Assessment Practice

Question 1.
Which line plot shows the data?


Answer:
Option A shows correct line plot.

Question 2.
The line plot shows the results from a survey asking parents how many children they have in school. How many parents have two children in school?

Answer:
6 parents.

Explanation:
The number of parents have two children in the school are 6 parents.

Question 3.
Georgiana made a line plot of the amount of time she spent practicing her violin each day in the past two weeks.

A. What is the difference between the greatest and least times spent practicing?
B. What is the most common amount of time she spent practicing?
C. What is the total amount of time Georgiana practiced? Write and solve an equation to show your work.
Answer:
The difference is 5.
The common amount of time is \(\frac{3}{4}\).
The equation is 3 \(\frac{1}{2}\) + 6 × \(\frac{3}{4}\) + 3 × 1 + 1 × 1\(\frac{1}{4}\) + 1 × 1\(\frac{1}{2}\) = 11\(\frac{3}{4}\).

Explanation:
The difference between the greatest and least times spent practicing is 6 – 1 which is 5.
The most common amount of time she spent practicing is \(\frac{3}{4}\).
The total amount of time Georgiana practiced is 3 \(\frac{1}{2}\) + 6 × \(\frac{3}{4}\) + 3 × 1 + 1 × 1\(\frac{1}{4}\) + 1 × 1\(\frac{1}{2}\) on solving we will get 11\(\frac{3}{4}\).

Ashraf and Melanie cut rope into different lengths for an art project. They made a line plot to display their data. Use the line plot to answer 4-6.

Question 4.
Which is the most common length of rope?
A. 1\(\frac{5}{8}\) ft
B. 1\(\frac{3}{4}\) ft
C. 1\(\frac{7}{8}\) ft
D. 2\(\frac{3}{8}\) ft
Answer:
The most common length of the rope is 1\(\frac{5}{8}\) ft

Question 5.
Which is the total length of rope represented by the data?
A. 25\(\frac{1}{8}\) ft
B. 27\(\frac{5}{8}\) ft
C. 27\(\frac{7}{8}\) ft
D. 28 ft
Answer:

Question 6.
Suppose 2 more pieces of rope are cut to be 2 feet. Would the value that occurred most often change? Explain your reasoning.
Answer:

Question 7.
Terry works at a bakery. This morning he recorded how many ounces each loaf of bread weighed.

A. Make a line plot for the data set.

Answer:
The line plot is

B. Morgan said that the difference between the heaviest loaf of bread and the lightest loaf of bread is 1\(\frac{1}{2}\) ounces. Do you agree with Morgan? Explain.
C. What is the combined weight of all the loaves of bread? Show your work.
Answer:

Topic 10 Performance Task

Measuring Bugs
Ms. Wolk’s class measured the length and weight of Madagascar Hissing Cockroaches.

Question 1.
The Cockroach Lengths line plot shows the lengths the class found.
Part A
Which length did the most students get? How can you tell from the line plot?

Answer:
The length did the most students get is 2\(\frac{1}{4}\).

Explanation:
The length did the most students get is 2\(\frac{1}{4}\), we can say by seeing the number of length more.
Part B
Jordan said that all of the Madagascar Hissing Cockroaches were between 2 and 3 inches long. Is he correct? Explain your reasoning.
Part C
Ginny said that twice as many students found a length of 2\(\frac{1}{2}\) inches as found a length of 1\(\frac{3}{4}\) inches. Is she correct? Explain your reasoning.
Part D
How many cockroaches were measured for the data set? How do you know?
Answer:
28 cockroaches.

Explanation:
The number of cockroaches were measured for the data set is 28 cockroaches.

Question 2.
The Cockroach Weights table shows the weights the class found.
Part A
Complete the line plot to represent the cockroach weights.

Part B
What is the total weight of all the cockroaches the students measured? Complete the table to help you. Show your work.

Answer:

Practice with the help of enVision Math Common Core Grade 5 Answer Key Topic 7 Use Equivalent Fractions to Add and Subtract Fractions regularly and improve your accuracy in solving questions.

enVision Math Common Core 5th Grade Answers Key Topic 7 Use Equivalent Fractions to Add and Subtract Fractions

Use Equivalent Fractions to Add and Subtract Fractions

Essential Questions: How can sums and differences of fractions and mixed numbers be estimated? What are common procedures for adding and subtracting fractions and mixed numbers?

enVision STEM Project: Fossils Tell Story

Do Research Use the Internet or other sources to find out more about fossils. What are fossils? How and where do we find them? What do they tell us about the past? What can they tell us about the future? Pay particular attention to fossils from the Eocene epoch.

Journal: Write a Report Include what you found. Also in your report:
• Describe a fossil that you have seen or would like to find.
• Tell if there are any fossils where you live.
• Make up and solve addition and subtraction problems about fossils. Use fractions and mixed numbers in your problems.

Review What You Know

Choose the best term from the box. Write it on the blank.

Vocabulary

• denominator
• numerator
• unit fraction
• fraction
• mixed number

Question 1.
A ___ has a whole number part and a fraction part.
Answer:
A Mixed Number has a whole number part and a fraction part.

Question 2.
A ___ represents the number of equal parts in one whole.
Answer:
A Denominator represents the number of equal parts in one whole

Question 3.
A ___ has a numerator of 1.
Answer:
A Unit Fraction has a numerator of 1.

Question 4.
A symbol used to name one or more parts of a whole or a set, or a location on the number line, is a ____.
Answer:
A symbol used to name one or more parts of a whole or a set, or a location on the number line, is a Fraction.

Compare Fractions
Compare. Write >, <, or = for each .

Question 5.
\(\frac{1}{5}\) \(\frac{1}{15}\)
Answer:
\(\frac{1}{5}\) \(\frac{1}{15}\)

Question 6.
\(\frac{17}{10}\) \(\frac{17}{5}\)
Answer:
\(\frac{17}{10}\) \(\frac{17}{5}\)

Question 7.
\(\frac{5}{25}\) \(\frac{2}{5}\)
Answer:
\(\frac{5}{25}\) \(\frac{2}{5}\)

Question 8.
\(\frac{12}{27}\) \(\frac{6}{9}\)
Answer:
\(\frac{12}{27}\) \(\frac{6}{9}\)

Question 9.
\(\frac{11}{16}\) \(\frac{2}{8}\)
Answer:
\(\frac{11}{16}\) \(\frac{2}{8}\)

Question 10.
\(\frac{2}{7}\) \(\frac{1}{5}\)
Answer:
\(\frac{2}{7}\) \(\frac{1}{5}\)

Question 11.
Liam bought \(\frac{5}{8}\) pound of cherries. Harrison bought more cherries than Liam. Which could be the amount of cherries that Harrison bought?
A. \(\frac{1}{2}\) pound
B. \(\frac{2}{5}\) pound
C. \(\frac{2}{3}\) pound
D. \(\frac{3}{5}\) pound
Answer:
D. \(\frac{3}{5}\) pound

Question 12.
Jamie has read \(\frac{1}{4}\) of a book. Raul has read of the same book. Who is closer to reading the whole book? Explain.
Answer:  Jamie and Raul are in the same position to complete the book because Jamie has read \(\frac{1}{4}\) of a book and also Raul has read \(\frac{1}{4}\) of a book. They both completed the same book with same amount. Both are equal.

Equivalent Fractions
Write a fraction equivalent to each fraction.

Question 13.
\(\frac{6}{18}\)
Answer:
\(\frac{6}{18}\) x 2 = \(\frac{12}{32}\)

Question 14.
\(\frac{12}{22}\)
Answer:
\(\frac{12}{22}\) x 2 = \(\frac{24}{44}\)

Question 15.
\(\frac{15}{25}\)
Answer:
\(\frac{15}{25}\) x 2 = \(\frac{30}{50}\)

Question 16.
\(\frac{8}{26}\)
Answer:
\(\frac{8}{26}\) x 2 = \(\frac{16}{52}\)

Question 17.
\(\frac{14}{35}\)
Answer:
\(\frac{14}{35}\) x 2 = \(\frac{28}{70}\)

Question 18.
\(\frac{4}{18}\)
Answer:
\(\frac{4}{18}\) x 2 = \(\frac{8}{36}\)

Question 19.
\(\frac{1}{7}\)
Answer:
\(\frac{1}{7}\) x 2 = \(\frac{1}{14}\)

Question 20.
\(\frac{4}{11}\)
Answer:
\(\frac{4}{11}\) x 2 = \(\frac{8}{22}\)

Pick a Project

PROJECT 7A
What’s in your gumbo?
Project: Record a Cooking Show

PROJECT 7B
Does this story sound fishy?
Project: Write a Tall Tale about Fishing Friends

PROJECT 7C
How many cups of juice can you get from 5 oranges?
Project: Get the Juice from Oranges

3-ACT MATH PREVIEW

Math Modeling

The Gif Recipe

Before watching the video, think: Some recipes are easier to follow than others. I’ve never made this one before. Maybe I should read the entire recipe before starting to cook.

Lesson 7.1 Estimate Sums and Differences of Fractions

Activity

Solve&Share
Jack needs about 1\(\frac{1}{2}\) yards of string. He has three pieces of string that are different lengths. Without finding the exact amount, which two pieces should he choose to get closest to 1\(\frac{1}{2}\) yards of string? Solve this problem any way you choose.

Reasoning You can use number sense to estimate the answer. Show your work!

Answer:

Look Back! How can a number line help you estimate?
Answer: It helps me to estimate the appropriate answer. It makes easy to add the yards and easy to find the yards which are closet to the given yard. It is somewhat in a elaborate way.

Visual Learning Bridge

Essential Question How Can You Estimate the Sum of Two Fractions?

A.
Mr. Fish is welding together two copper pipes to repair a leak. He will use the pipes shown. Is the new pipe closer to \(\frac{1}{2}\) foot or 1 foot long? Explain.

B.
Step 1
Replace each fraction with the nearest half or whole. A number line can make it easy to decide if each fraction is closest to 0, \(\frac{1}{2}\), or 1.

\(\frac{1}{6}\) is between 0 and 2, but is closer to 0.
\(\frac{5}{12}\) is also between 0 and 2, but is closer to the benchmark fraction \(\frac{1}{2}\).

C.
Step 2
Add to find the estimate.
A good estimate of \(\frac{1}{6}\) + \(\frac{5}{12}\) is 0 + \(\frac{1}{2}\), or \(\frac{1}{2}\). So, the welded pipes will be closer to 2 foot than 1 foot long

Since each addend is less than 5, it is reasonable that their sum is less than 1.

Convince Me! Critique Reasoning Nolini says that if the denominator is more than twice the numerator, the fraction can always be replaced with 0. Is she correct? Give an example in your explanation.
Answer: Yes,
It depends on how accurate you want your answer to be, but basically yes, imagine we set our numerator to be x, and our denominator to be 2x, if we built our fraction we will get that:  If the denominator of a fraction is exactly twice the numerator, then the fraction will be simplified to 1/2 or 0.5.Now, the greater the denominator is, the closer the fraction will get to zero.
Take for example I had an fraction like this:

which approximates to 0. If we made the denominator bigger, let’s say 7, we would get: 
notice the answer is closer to 0 this time, so it’s valid to round it to zero. If we made the denominator greater, let’s say 25, we would get: and so on. So it is true that if the denominator of a fraction is greater than twice the numerator, we can always replace the fraction with a 0
(depending on how accurate you want your answer to be).

Guided Practice

Do You Understand?

Question 1.
In the problem at the top of page 270, would you get the same estimate if Mr. Fish’s pipes measured foot and \(\frac{7}{12}\) foot?
Answer:
Yes, The estimation would be the same because Mr. Fish’s pipes measured foot is \(\frac{1}{6}\) + \(\frac{5}{12}\) = 0.5 and the \(\frac{7}{12}\) = 0.5

Question 2.
Number Sense If a fraction has a 1 in the numerator and a number greater than 2 in the denominator, will the fraction be closer to 0, \(\frac{1}{2}\), or 1? Explain.
Answer:
If a fraction has a 1 in the numerator and a number greater than 2 in the denominator, will the fraction be closer to \(\frac{1}{2}\) because the numerator is 1 and the denominator is 2 that means \(\frac{1}{2}\) = 0.5

Do You Know How?

In 3 and 4, use a number line to tell if each fraction is closest to 0, \(\frac{1}{2}\), or 1. Then estimate the sum or difference.

Question 3.

a. \(\frac{11}{12}\) Closest to: ___
b. \(\frac{1}{6}\) Closest to: _____
Estimate the sum \(\frac{11}{12}\) + \(\frac{1}{6}\)
c. 1 + __ = __
Answer:
a. \(\frac{11}{12}\) Closest to: 1
b. \(\frac{1}{6}\) Closest to:  \(\frac{1}{2}\)
Estimate the sum \(\frac{11}{12}\) + \(\frac{1}{6}\)
c. 1 +\(\frac{1}{2}\) = 1.5

Question 4.

a. \(\frac{14}{16}\) Closest to: ___
b. \(\frac{5}{8}\) Closest to: ___
Estimate the difference \(\frac{14}{16}\) – \(\frac{5}{8}\).
c. ___ – ___ = ___
Answer:
a. \(\frac{14}{16}\) Closest to: 1
b. \(\frac{5}{8}\) Closest to:  \(\frac{1}{2}\)
Estimate the difference \(\frac{14}{16}\) – \(\frac{5}{8}\).
c. 1 – \(\frac{1}{2}\) = 0.5

Independent Practice

Leveled Practice In 5, use a number line to tell if each fraction is closest to 0, \(\frac{1}{2}\), or 1
In 6-11, estimate the sum or difference by replacing each fraction with 0, \(\frac{1}{2}\), or 1.

Question 5.

a. \(\frac{7}{8}\) Closest to: ____
b. \(\frac{5}{12}\) Closest to: ___
Estimate the difference \(\frac{7}{8}\) – \(\frac{5}{12}\)
c. ___ – ___ = ___
Answer:
a. \(\frac{7}{8}\) Closest to: 1
b. \(\frac{5}{12}\) Closest to: \(\frac{1}{2}\)
Estimate the difference \(\frac{7}{8}\) – \(\frac{5}{12}\)
c. 1 – 0.5 = 0.5

Question 6.
\(\frac{9}{10}\) + \(\frac{5}{6}\)
Answer:
\(\frac{9}{10}\) + \(\frac{5}{6}\) =  1.7

Question 7.
\(\frac{11}{18}\) – \(\frac{2}{9}\)
Answer:
\(\frac{11}{18}\) – \(\frac{2}{9}\) = 0.2

Question 8.
\(\frac{1}{16}\) + \(\frac{2}{15}\)
Answer:
\(\frac{1}{16}\) + \(\frac{2}{15}\) = 0.16

Question 9.
\(\frac{24}{25}\) – \(\frac{1}{9}\)
Answer:
\(\frac{24}{25}\) – \(\frac{1}{9}\) = 0.86

Question 10.
\(\frac{3}{36}\) + \(\frac{1}{10}\)
Answer:
\(\frac{3}{36}\) + \(\frac{1}{10}\) = 0.18

Question 11.
\(\frac{37}{40}\) – \(\frac{26}{50}\)
Answer:
\(\frac{37}{40}\) – \(\frac{26}{50}\) = 0.4

Problem Solving

Question 12.
Number Sense Name two fractions that are closer to 1 than to \(\frac{1}{2}\). Then, name two fractions that are closer to \(\frac{1}{2}\) than to 0 or 1 and two other fractions that are closer to 0 than to \(\frac{1}{2}\). Find two of your fractions that have a sum of about 1\(\frac{1}{2}\).

Answer:
a(1). \(\frac{25}{30}\) = 0.8 which is closet to 1.
(2). \(\frac{9}{10}\) =  0.9 which is closet to 1.
b(1). \(\frac{5}{12}\) = 0.4 which is closet to \(\frac{1}{2}\).
(2). \(\frac{3}{6}\) = 0.5 which is closet to \(\frac{1}{2}\).
c(1).\(\frac{1}{16}\) = 0.06 which is closet to 0
(2).\(\frac{2}{30}\) = 0.06 which is closet to 0
The two fractions that have a sum of about 1\(\frac{1}{2}\) = \(\frac{25}{30}\) + \(\frac{3}{6}\) = 1.4

Question 13.
Higher Order Thinking How would you estimate whether \(\frac{27}{50}\) is closer to c or 1 without using a number line? Explain.
Answer: \(\frac{27}{50}\) is closer to \(\frac{1}{2}\).
Because I find it by dividing them \(\frac{27}{50}\) = 0.5 and \(\frac{1}{2}\) = 0.5

Question 14.
Katie made a bag of trail mix with \(\frac{1}{2}\) cup of raisins, \(\frac{3}{5}\) cup of banana chips, and \(\frac{3}{8}\) cup of peanuts. About how much trail mix did Katie make?
Answer: \(\frac{1}{2}\) + \(\frac{3}{5}\) + \(\frac{3}{8}\) = 1\(\frac{1}{2}\)

Question 15.
Reasoning The Annual Mug Race is the longest river sailboat race in the world. The event is run along the St. Johns River, which is 310 miles long. About how many times as long as the race is the river?

Answer: The Annual Mug Race is 42 miles and the St. Johns River, which is 310 miles long. So to find out the answer for this we need to divide the River by race so then we can find the times as answer.
310/42 = 7.4 times the river is long as the race

Assessment Practice

Question 16.
Part A
Steve is making breakfast. The recipes call for a cup of milk for grits and cup for biscuits. He only has 2 cups of milk. Does he have enough to make his breakfast? Explain.
Answer: Yes! He has enough cups of milk to make breakfast that means he has 2 cups of milk and one cup is for grits and another cup is for biscuits. So, Total 2 cups of milk.
Part B
If he has enough milk, about how much milk will he have left? If he doesn’t have enough milk, about how much will he need?
Answer: He has enough cups of milk to make breakfast that means he has 2 cups of milk and one cup is for grits and another cup is for biscuits. So, Total 2 cups of milk. He has enough cups of milk he will be left nothing at the end. If he has only one cup of milk means he need other cup of milk for biscuits.

Lesson 7.2 Find Common Denominators

Solve&Share

Sue wants \(\frac{1}{2}\) of a rectangular pan of cornbread. Dena wants of the same pan of cornbread. How should you cut the cornbread so that each girl gets the size portion she wants? Solve this problem any way you choose.

You can draw a picture to represent the pan as 1 whole. Then solve. Show your work!

Answer:-

Look Back! Construct Arguments Is there more than one way to divide the pan of cornbread into equal-sized parts? Explain how you know.
Answer: Sue wants half of the cornbread and Dena wants same amount. Here, we have 1 whole pan from this we gonna share into two for that we divided the whole pan into two. i.e., \(\frac{1}{2}\) = 0.5 and we have shared it to Sue and Dena.

Visual Learning Bridge

Essential Question
How Can You Find Common Denominators?

A.
Tyrone partitioned a rectangle into thirds. Sally partitioned a rectangle of the same size into fourths. How could you partition a rectangle of the same size so that you see both thirds and fourths?
You can partition a rectangle to show thirds or fourths.

B.
This rectangle is partitioned into thirds and fourths.

The rectangle is partitioned into 12 equal parts. Each part is \(\frac{1}{12}\)

C.
The fractions \(\frac{1}{3}\) and \(\frac{1}{4}\) can be renamed with equivalent fractions.

Fractions that have the same denominators, such as \(\frac{4}{12}\) and \(\frac{3}{12}\), are said to have common denominators.

Convince Me! Use Appropriate Tools Draw rectangles such as the ones above to find fractions equivalent to \(\frac{2}{5}\) and \(\frac{1}{3}\) that have the same denominator.
Answer:

Another Example
Find a common denominator for \(\frac{2}{3}\) and \(\frac{5}{6}\). Then rename each fraction with an equivalent fraction.

One Way
Multiply the denominators to find a common denominator: 3 × 6 = 18.
Write equivalent fractions with denominators of 18.

Another Way
Use the fact that one denominator is a multiple of the other.
You know that 6 is a multiple of 3.

So, \(\frac{4}{6}\) and \(\frac{5}{6}\) is one way to rename \(\frac{2}{3}\) and \(\frac{5}{6}\) with a common denominator.

Guided Practice

Do You Understand?

Question 1.
In the example on the previous page, how many twelfths are in \(\frac{1}{3}\) each section of Tyrone’s rectangle? How many twelfths are in each \(\frac{1}{4}\) section of Sally’s rectangle?
Answer:
In Tyrone’s rectangle there are \(\frac{4}{12}\) twelfths in \(\frac{1}{3}\) and there are \(\frac{3}{12}\)  are there in \(\frac{1}{4}\)

Do You Know How?

In 2 and 3, find a common denominator for each pair of fractions.

Question 2.
\(\frac{3}{8}\) and \(\frac{2}{3}\)
Answer:

Question 3.
\(\frac{1}{6}\) and \(\frac{4}{3}\)
Answer:

Independent Practice

In 4-11, find a common denominator for each pair of fractions. Then write equivalent fractions with the common denominator.

Question 4.
\(\frac{2}{5}\) and \(\frac{1}{6}\)
Answer:
\(\frac{2}{5}\) = \(\frac{2}{5}\) x \(\frac{6}{6}\) = \(\frac{12}{30}\)
\(\frac{1}{6}\) = \(\frac{1}{6}\) x \(\frac{5}{5}\) = \(\frac{5}{30}\)

Question 5.
\(\frac{1}{3}\) and \(\frac{4}{5}\)
Answer:
\(\frac{1}{3}\) = \(\frac{1}{3}\) x \(\frac{5}{5}\) = \(\frac{5}{5}\)
\(\frac{4}{5}\) = \(\frac{4}{5}\) x \(\frac{3}{3}\) = \(\frac{12}{15}\)

Question 6.
\(\frac{5}{8}\) and \(\frac{3}{4}\)
Answer:
\(\frac{5}{8}\)v = \(\frac{5}{8}\) x \(\frac{4}{4}\) = \(\frac{20}{32}\)
\(\frac{3}{4}\) = \(\frac{3}{4}\) x \(\frac{8}{8}\) = \(\frac{24}{32}\)

Question 7.
\(\frac{3}{10}\) and \(\frac{9}{8}\)
Answer:
\(\frac{3}{10}\) = \(\frac{3}{10}\) x \(\frac{8}{8}\) = \(\frac{24}{80}\)
\(\frac{9}{8}\) = \(\frac{9}{8}\) x \(\frac{10}{10}\) = \(\frac{90}{80}\)

Question 8.
\(\frac{3}{7}\) and \(\frac{1}{2}\)
Answer:
\(\frac{3}{7}\) = \(\frac{3}{7}\) x \(\frac{2}{2}\) = \(\frac{6}{14}\)
\(\frac{1}{2}\) = \(\frac{1}{2}\) x \(\frac{7}{7}\) = \(\frac{7}{14}\)

Question 9.
\(\frac{5}{12}\) and \(\frac{3}{5}\)
Answer:
\(\frac{5}{12}\) = \(\frac{5}{12}\) x \(\frac{5}{5}\) = \(\frac{25}{60}\)
\(\frac{3}{5}\) = \(\frac{3}{5}\) x \(\frac{12}{12}\) = \(\frac{36}{60}\)

Question 10.
\(\frac{7}{9}\) and \(\frac{2}{3}\)
Answer:
\(\frac{7}{9}\) = \(\frac{7}{9}\)  x \(\frac{3}{3}\)  = \(\frac{21}{27}\)
\(\frac{2}{3}\) = \(\frac{2}{3}\) x \(\frac{9}{9}\) = \(\frac{18}{27}\)

Question 11.
\(\frac{3}{8}\) and \(\frac{9}{20}\)
Answer:
\(\frac{3}{8}\) = \(\frac{3}{8}\) x \(\frac{20}{20}\) = \(\frac{60}{160}\)
\(\frac{9}{20}\) = \(\frac{9}{20}\) x \(\frac{8}{8}\) = \(\frac{72}{160}\)

Problem Solving

Question 12.
Critique Reasoning Explain any mistakes in the renaming of the fractions below. Show the correct renaming.
\(\frac{3}{4}\) = \(\frac{9}{12}\) \(\frac{2}{3}\) = \(\frac{6}{12}\)
Answer:
\(\frac{3}{4}\) = \(\frac{9}{12}\)
\(\frac{2}{3}\) = \(\frac{8}{12}\)

Question 13.
Higher Order Thinking For keeping business records, every three months of a year is called a quarter. How many months are equal to three-quarters of a year? Explain how you found your answer.
Answer: One quarter = \(\frac{3}{12}\) So, 3 quarters = \(\frac{9}{12}\)
Here, 12 = 12 months and 3 months = 1 quarter. Therefore, 3 + 3 + 3 = 9 months

Question 14.
Nelda baked two kinds of pasta in pans. Each pan was the same size. She sliced one pan of pasta into 6 equal pieces. She sliced the other pan into 8 equal pieces. How can the pans of pasta now be sliced so that both pans have the same-sized pieces? Draw on the pictures to show your work. If Nelda has served 6 pieces from one pan so far, what fraction of one pan has she served?

Answer:

To reach her goal that both pans should have same sized pieces. So,
\(\frac{1}{6}\) x 4 pieces = \(\frac{1}{24}\). Therefore to set goal for the other pan also she needs to multiply with 3 pieces \(\frac{1}{8}\) x 3 = \(\frac{1}{24}\). So, Finally she sets her goal.
If Nelda has served 6 pieces from one pan so far, then there will be \(\frac{1}{24}\) – \(\frac{1}{6}\) = \(\frac{1}{18}\) will be left.

Question 15.
Number Sense What is the price of premium gasoline rounded to the nearest dollar? rounded to the nearest dime? rounded to the nearest penny?

Answer:

$4.409 to the nearest dime: $4.40

to the nearest penny: $4.41

to the nearest dollar: $4.00

Assessment Practice

Question 16.
Choose all the numbers that could be common denominators for and
8
12
16
36
48
Answer:

Question 17.
Choose all the numbers that could be common denominators for \(\frac{11}{12}\) and \(\frac{4}{5}\)
12
17
30
60
125
Answer:

Lesson 7.3 Add Fractions with Unlike Denominators

Activity

Solve & Share

Over the weekend, Eleni ate \(\frac{1}{4}\) box of cereal, and Freddie ate \(\frac{3}{8}\) of the same box. What portion of the box of cereal did they eat in all? Solve this problem any way you choose.

You can use fraction strips to represent adding fractions. Show your work!

Answer:

Look Back! Make Sense and Persevere What steps did you take to solve this problem?
Answer: Step 1: Finding their common fractions for common denominator
Step2: Multiply with the factor to get common denominator.
Step3: Adding the fractions after getting the common denominator.

Visual Learning Bridge

Essential Question
How Can You Add Fractions with Unlike Denominators?

A.
Alex rode his scooter from his house to the park. Later, he rode from the park to baseball practice. How far did Alex ride?

You can add to find the total distance that Alex rode his scooter.

B.
Step 1
Change the fractions to equivalent fractions with a common, or like, denominator

The number 6 is a common multiple of 2 and 3, so \(\frac{1}{2}\) and \(\frac{1}{3}\) can both be rewritten with a common denominator of 6.

Step 2
Write equivalent fractions with a common denominator.

Step 3
Add the fractions to find the total number of sixths.

Convince Me! Construct Arguments in the example above, would you get the same sum if you used 12 as the common denominator? Explain.
Answer:
Step 1:
Place 112 fractions strips under the 1 whole strip on your Matboard. Then place a 14 fraction strip beside the 112 strips.
Step 2:
Find fraction strips, all with the same denominator, that are equivalent to 112 and 14. Place the fraction strips under the sum. At the right, draw a picture of the model and write the equivalent fractions.
14 x 33 = 312
112
Step 3:
Add the fractions with like denominators. Use the 1 whole strip to rename the sum in the simplest form.
312 + 112 = 412

Another Example
Find \(\frac{5}{12}\) + \(\frac{1}{4}\).
\(\frac{5}{12}\) + \(\frac{1}{4}\) = \(\frac{5}{12}\) + \(\frac{3}{12}\) Write equivalent fractions with common denominators.
= \(\frac{5+3}{12}\) = \(\frac{8}{12}\) or \(\frac{2}{3}\) Find the total number of twelfths by adding the numerators.

Guided Practice

Do You Understand?

Question 1.
In the example at the top of page 278, if the park was \(\frac{1}{8}\) mile from baseball practice instead of \(\frac{1}{3}\) mile, how far would Alex ride his scooter in all?
Answer:
\(\frac{1}{8}\) mile from baseball and \(\frac{1}{2}\) park to home then:
\(\frac{1}{8}\) = \(\frac{1}{8}\) x \(\frac{2}{2}\) = \(\frac{2}{16}\)
\(\frac{1}{2}\) = \(\frac{1}{2}\) x \(\frac{8}{8}\) = \(\frac{8}{16}\)
So, \(\frac{2}{16}\) + \(\frac{8}{16}\) = \(\frac{10}{16}\) or \(\frac{5}{8}\)

Question 2.
Vocabulary Rico and Nita solved the same problem. Rico got \(\frac{6}{8}\) for an answer, and Nita got \(\frac{3}{4}\). Which answer is correct? Use the term equivalent fraction in your explanation.
Answer:
No, They both were wrong. Rico and Nita got wrong answer. The answer is \(\frac{5}{8}\).
\(\frac{1}{8}\) mile from baseball and \(\frac{1}{2}\) park to home then:
\(\frac{1}{8}\) = \(\frac{1}{8}\) x \(\frac{2}{2}\) = \(\frac{2}{16}\)
\(\frac{1}{2}\) = \(\frac{1}{2}\) x \(\frac{8}{8}\) = \(\frac{8}{16}\)
So, \(\frac{2}{16}\) + \(\frac{8}{16}\) = \(\frac{10}{16}\) or \(\frac{5}{8}\)

 

Do You Know How?

Find the sum. Use fraction strips to help.

Question 3.

Answer:

Independent Practice

In 4 and 5, find each sum. Use fraction strips to help.

Remember that you can use multiples to find a common denominator.

Question 4.

Answer:

Question 5.

Answer:

Problem Solving

Question 6.
Explain why the denominator 6 in \(\frac{3}{6}\) is not changed when adding the fractions.

Answer:
A denominator indicates how many equal pieces make one unit. If you add the denominators when adding fractions, the new denominator will not show how many equal pieces are in one unit.

Question 7.
Model with Math About \(\frac{1}{10}\) of the bones in your body are in your skull. Your hands have about \(\frac{1}{4}\) of the bones in your body. Write and solve an equation to find the fraction of the bones in your body that are in your hands or skull.
Answer:
\(\frac{1}{10}\) = \(\frac{1}{10}\) x \(\frac{4}{4}\) = \(\frac{4}{40}\)
\(\frac{1}{4}\) = \(\frac{1}{4}\) x \(\frac{10}{10}\) = \(\frac{10}{40}\)
the fraction of the bones in your body that are in your hands or skull are: \(\frac{4}{40}\) + \(\frac{10}{40}\) = \(\frac{14}{40}\)

Question 8.
enVision® STEM Of 36 chemical elements, 2 are named for women scientists and 25 are named for places. What fraction of these 36 elements are named for women or places? Show your work.
Answer:

Question 9.
Higher Order Thinking Roger made a table showing how he spends his time in one day. How many days will go by before Roger has slept the equivalent of one day? Explain how you found your answer.

Answer:

Assessment Practice

Question 10.
Which equations are true when is \(\frac{1}{2}\) is placed in the box?

Answer:

Question 11.
Which equations are true when \(\frac{4}{7}\) placed in the box?

Answer:

Lesson 7.4 Subtract Fractions with Unlike Denominators

Activity

Solve & Share

Rose bought the length of copper pipe shown below. She used 5 yard to repair a water line in her house. How much pipe does she have left? Solve this problem any way you choose.

You can use mental math to find equivalent fractions so that \(\frac{1}{2}\) and \(\frac{4}{6}\) will have like denominators. Show your work!

Answer:-

Look Back! Generalize How is subtracting fractions with unlike denominators similar to adding fractions with unlike denominators?
Answer:- Subtracting fractions with unlike denominators similar to adding fractions with unlike denominators because in addition firstly we do to set denominators alike and then we add them. Same here also of unlike denominators are there we do them alike and then we subtract. Adding and subtracting is the only difference remaining all the steps are same.

Visual Learning Bridge

Essential Question How Can You Subtract Fractions with Unlike Denominators?

A.
Linda used \(\frac{1}{4}\) yard of the fabric she bought for a sewing project. How much fabric did she have left?

You can use subtraction to find how much fabric was left.

B.
Step 1
Find a common multiple of the denominators.
Multiples of 3: 3,6,9,12,…
Multiples of 4: 4, 8, 12,…
The number 12 is a multiple of 3 and 4. Write equivalent fractions with a denominator of 12 for \(\frac{2}{3}\) and \(\frac{1}{4}\).

C.
Step 2
Use the Identity Property to rename the fractions with a common denominator.

C.
Step 3
Subtract the numerators.

Linda has \(\frac{5}{12}\) yard of fabric left.

Convince Me! Critique Reasoning Suppose Linda had \(\frac{2}{3}\) of a yard of fabric and told Sandra that she used of a yard. Šandra says this is not possible. Do you agree? Explain your answer.
Answer:-  No, That is not possible because Linda had \(\frac{2}{3}\) of a yard that is 0.6 yard then how can she use 1 yard of fabric that is why Sandra says this is not possible. So, here Sandra is correct in her way and Linda cannot use the 1 yard of fabric.

Guided Practice

Do You Understand?

Question 1.
In the example on page 282, is it possible to use a common denominator greater than 12 and get the correct answer? Why or why not?
Answer: We use the common denominator by figuring common multiply factors in the fraction if in the above example if we got common factor which is greater than 12 then it is always possible to get denominator which is greater than 12.
In the above example if we got common factor as 20 then it is easy to get denominator which is actually greater than 12 and if not we have to go with the common factor either 12 or below 12.

Question 2.
In the example on page 282, if Linda had started with one yard of fabric and used \(\frac{5}{8}\) of a yard, how much fabric would be left?
Answer: Linda had one yard of fabric and she used \(\frac{5}{8}\) of a yard, The she left with 1 – \(\frac{5}{8}\) = 0.4

Do You Know How?

For 3-6, find each difference.

Question 3.

Answer:

Question 4.

Answer:

Question 5.

Answer:

Question 6.

Answer:

Independent Practice

Leveled Practice In 7-16, find each difference.

Question 7.

Answer:

Question 8.

Answer:

Question 9.

Answer:

Question 10.

Answer:

Question 11.

Answer:

Question 12.

Answer:

Question 13.
\(\frac{7}{10}\) – \(\frac{2}{5}\)
Answer:
\(\frac{7}{10}\) = \(\frac{7}{10}\) x \(\frac{5}{5}\) = \(\frac{21}{50}\)
\(\frac{2}{5}\) = \(\frac{2}{5}\) x \(\frac{10}{10}\) = \(\frac{20}{50}\)
\(\frac{21}{50}\) – \(\frac{20}{50}\) = \(\frac{1}{50}\)

Question 14.
\(\frac{13}{16}\) – \(\frac{1}{4}\)
Answer:
\(\frac{13}{16}\) = \(\frac{13}{16}\) x \(\frac{4}{4}\) = \(\frac{52}{64}\)
\(\frac{1}{4}\) = \(\frac{1}{4}\) x \(\frac{16}{16}\) = \(\frac{16}{64}\)
\(\frac{52}{64}\) – \(\frac{16}{64}\) = \(\frac{36}{64}\)

Question 15.
\(\frac{2}{9}\) – \(\frac{1}{6}\)
Answer:
\(\frac{2}{9}\) = \(\frac{2}{9}\) x \(\frac{6}{6}\) = \(\frac{12}{54}\)
\(\frac{1}{6}\) = \(\frac{1}{6}\) x \(\frac{9}{9}\) = \(\frac{9}{54}\)
\(\frac{12}{54}\) – \(\frac{9}{54}\) = \(\frac{3}{54}\)

Question 1
\(\frac{6}{5}\) – \(\frac{3}{8}\)
Answer:
\(\frac{6}{5}\) = \(\frac{6}{5}\) x \(\frac{8}{8}\) = \(\frac{48}{40}\)
\(\frac{3}{8}\) = \(\frac{3}{8}\) x \(\frac{5}{5}\) = \(\frac{15}{40}\)
\(\frac{48}{40}\)  – \(\frac{15}{40}\) = \(\frac{33}{40}\)

Problem Solving

Question 17.
Model with Math Write and solve an equation to find the difference between the location of Point A and Point B on the ruler.

Answer:
Point A: \(\frac{1}{4}\)
Point B: \(\frac{2}{5}\)
Point A: \(\frac{1}{4}\) x \(\frac{5}{5}\) = \(\frac{5}{20}\)
Point B: \(\frac{2}{5}\) x \(\frac{4}{4}\) = \(\frac{8}{20}\)
Point A – Point B = \(\frac{5}{20}\) – \(\frac{8}{20}\) = \(\frac{3}{20}\)

Question 18.
Algebra Write an addition and a subtraction equation for the diagram. Then, find the missing value.

Answer:

Question 19.
Why do fractions need to have a common denominator before you add or subtract them?
Answer:
In order to add fractions, the fractions must have a common denominator. We need the pieces of each fraction to be the same size to combine them together. Since the pieces are all the same size, we can add these two fractions together.

Question 20.
Number Sense Without using paper and pencil, how would you find the sum of 9.8 and 2.6?
Answer: The sum of 9.8 and 2.6 is 9.8 + 2.6 = 12.4

Question 21.
Higher Order Thinking Find two fractions with a difference of \(\frac{1}{5}\) but with neither denominator equal to 5.
Answer:
There an infinite family of solutions. Any two fractions of the form:
(a+ b)/5a and b/5a
will work. Thus,
(a+b)/5a – b/5a = (a+b-b)/5a = a/5a = 1/5
of course, you can multiply each of the above fraction by 1, written as c/c, so you have c(a+b)/5ac if you wish to further obfuscate things. Similarly, you can also multiply the second fraction by 1 to get bd/5bd.
I’m sure there are other general answers as well, if an infinity of solutions is insufficient. You may wish to be sure that the product ac is even, so 5 won’t appear at all in the fraction as the last digit.

Assessment Practice

Question 22.
Choose the correct numbers from the box below to complete the subtraction sentence that follows.

Answer:

Question 23.
Choose the correct numbers from the box below to complete the subtraction sentence that follows.

Answer:

Lesson 7.5 Add and Subtract Fractions

Activity

Solver & Share
Tyler and Dean ordered pizza. Tyler ate \(\frac{1}{2}\) of the pizza and Dean ate \(\frac{1}{3}\) of the pizza. How much of the pizza was eaten, and how much is left? Solve this problem any way you choose.

Reasoning You can use number sense to help you solve this problem. Show your work!

Answer:

Look Back! How can you check that your answer makes sense?
Answer: It is easy to identify the answer because if we want how much pizza did they eat we can directly add the amount which they have ate and if we want to know how much pizza left we easily get by subtracting both.

Visual Learning Bridge

Essential Question
How Can Adding and Subtracting Fractions Help You Solve Problems?

A.
Kayla had \(\frac{9}{10}\) gallon of paint. She painted the ceilings in her bedroom and bathroom. How much paint does she have left after painting the two ceilings?

You can use both addition and subtraction to find how much paint she has left.

B.
Step 1
Add to find out how much paint Kayla used for the two ceilings.

Kayla used \(\frac{13}{15}\) gallon of paint

C.
Step 2
Subtract the amount of paint Kayla used from the amount she started with.

Kayla used \(\frac{1}{30}\) gallon of paint left.

Convince Me! Make Sense and Persevere For the problem above, how would you use estimation to check that the answer is reasonable?

Add to find out how much paint Kayla used for the two ceilings.

Kayla used \(\frac{13}{15}\) gallon of paint.
Subtract the amount of paint Kayla used from the amount she started with.

Kayla used \(\frac{1}{30}\) gallon of paint left.

Guided Practice

Do You Understand?

Question 1.
In the example on page 286, how much more paint did Kayla use to paint the bedroom ceiling than the bathroom ceiling?
Answer:
Add to find out how much paint Kayla used for the two ceilings.

Question 2.
Number Sense Kevin estimated the difference of \(\frac{9}{10}\) – \(\frac{4}{8}\) to be 0. Is his estimate reasonable? Explain.
Answer:
\(\frac{9}{10}\) = \(\frac{9}{10}\) x \(\frac{8}{8}\) = \(\frac{72}{80}\)
\(\frac{4}{8}\) = \(\frac{4}{8}\)  x \(\frac{10}{10}\)  = \(\frac{40}{80}\)
\(\frac{72}{80}\) – \(\frac{40}{80}\) = \(\frac{32}{80}\) = 0.4
Kevin estimated the difference of \(\frac{9}{10}\) – \(\frac{4}{8}\) to be 0 and his estimation was wrong because 0.4 is closest to \(\frac{1}{2}\) not closest to 0

Do You Know How?

For 3-6, find the sum or difference.

Question 3.

Answer:

Question 4.

Answer:

Question 5.
\(\frac{7}{8}\) – \(\frac{3}{6}\)
Answer:
\(\frac{7}{8}\) = \(\frac{7}{8}\) x \(\frac{6}{6}\) = \(\frac{42}{48}\)
\(\frac{3}{6}\) = \(\frac{3}{6}\) x \(\frac{8}{8}\) = \(\frac{24}{48}\)
\(\frac{42}{48}\) – \(\frac{24}{48}\) = \(\frac{18}{48}\)

Question 6.
\(\frac{7}{8}\) + (\(\frac{4}{8}\) – \(\frac{2}{4}\))
Answer:
\(\frac{4}{8}\) = \(\frac{4}{8}\) x \(\frac{4}{4}\) = \(\frac{16}{32}\)
\(\frac{2}{4}\) = \(\frac{2}{4}\) x \(\frac{8}{8}\) = \(\frac{16}{32}\)
\(\frac{16}{32}\) – \(\frac{16}{32}\) = \(\frac{16}{32}\) = 0
\(\frac{7}{8}\) + (0) = \(\frac{7}{8}\)

Independent Practice

In 7-22, find the sum or difference.

Question 7.

Answer:

Question 8.

Answer:

Question 9.

Answer:

Question 10.

Answer:

Question 11.
\(\frac{17}{15}\) – \(\frac{1}{3}\)
Answer:
\(\frac{17}{15}\) = \(\frac{17}{15}\)  x \(\frac{3}{3}\) = \(\frac{51}{45}\)
\(\frac{1}{3}\) = \(\frac{1}{3}\) x \(\frac{15}{15}\) = \(\frac{15}{45}\)
\(\frac{51}{45}\) –  \(\frac{15}{45}\) = \(\frac{36}{45}\)

Question 12.
\(\frac{7}{16}\) + \(\frac{3}{8}\)
Answer:
\(\frac{7}{16}\) = \(\frac{7}{16}\)  x \(\frac{8}{8}\)  = \(\frac{56}{128}\)
\(\frac{3}{8}\) = \(\frac{3}{8}\) x \(\frac{16}{16}\) = \(\frac{48}{128}\)
\(\frac{56}{128}\)  + \(\frac{48}{128}\)  = \(\frac{8}{128}\)

Question 13.
\(\frac{2}{5}\) + \(\frac{1}{4}\)
Answer:
\(\frac{8}{20}\) + \(\frac{5}{20}\) = \(\frac{13}{20}\)

Question 14.
\(\frac{1}{7}\) + \(\frac{1}{2}\)
Answer:
\(\frac{2}{14}\) + \(\frac{7}{14}\) = \(\frac{9}{14}\)

Question 15.
\(\frac{1}{2}\) – \(\frac{3}{16}\)
Answer:
\(\frac{16}{32}\) – \(\frac{6}{32}\) = \(\frac{10}{32}\)

Question 16.
\(\frac{7}{8}\) – \(\frac{2}{3}\)
Answer:
\(\frac{14}{24}\) – \(\frac{16}{24}\) = – \(\frac{2}{24}\)

Question 17.
\(\frac{11}{12}\) – \(\frac{4}{6}\)
Answer:
\(\frac{66}{72}\) – \(\frac{48}{72}\) = \(\frac{18}{72}\)

Question 18.
\(\frac{7}{18}\) + \(\frac{5}{9}\)
Answer:
\(\frac{63}{162}\) + \(\frac{90}{162}\) = \(\frac{153}{162}\)

Question 19.
(\(\frac{7}{8}\) + \(\frac{1}{12}\)) – \(\frac{1}{2}\)
Answer:
\(\frac{84}{108}\) + \(\frac{8}{108}\) = \(\frac{92}{108}\)

\(\frac{184}{216}\) – \(\frac{92}{216}\) = \(\frac{92}{216}\)

Question 20.
(\(\frac{11}{18}\) – \(\frac{4}{9}\)) + \(\frac{1}{6}\)
Answer:
\(\frac{99}{162}\) – \(\frac{72}{162}\) = \(\frac{29}{162}\)
\(\frac{174}{648}\) + \(\frac{162}{648}\) = \(\frac{12}{648}\)

Question 21.
\(\frac{13}{14}\) – (\(\frac{1}{2}\) + \(\frac{2}{7}\))
Answer:
\(\frac{7}{14}\) + \(\frac{4}{14}\) = \(\frac{11}{14}\)
\(\frac{13}{14}\) – \(\frac{11}{14}\) = \(\frac{2}{14}\)

Question 22.
\(\frac{1}{6}\) + (\(\frac{15}{15}\) – \(\frac{7}{10}\))
Answer:
\(\frac{150}{150}\) – \(\frac{105}{150}\) = \(\frac{45}{150}\)
\(\frac{150}{900}\) + \(\frac{270}{900}\) = \(\frac{420}{900}\)

Problem Solving

Question 23.
The table shows the amounts of ingredients needed to make a pizza. How much more cheese do you need than pepperoni and mushrooms combined? Show how you solved the problem.

Answer:

Question 24.
Charlie’s goal is to use less than 50 gallons of water per day. His water bill for the month showed that he used 1,524 gallons of water in 30 days. Did Charlie meet his goal this month? Explain how you decided.
Answer:

Question 25.
Construct Arguments Jereen spent \(\frac{1}{4}\) hour on homework before school, another \(\frac{1}{2}\) hour after she got home, and a final \(\frac{1}{3}\) hour after dinner. Did she spend more or less than 1 hour on homework in all? Explain.
Answer:

Question 26.
Carl has three lengths of cable, \(\frac{5}{6}\) yard long, \(\frac{1}{4}\) yard long, and \(\frac{2}{3}\) yard long. If he uses 1 yard of cable, how much cable is left? Explain your work.

Answer:

Question 27.
Higher Order Thinking Find two fractions with a sum of \(\frac{2}{3}\) but with neither denominator equal to 3.
Answer:

Assessment Practice

Question 28.
What fraction is missing from the following equation?

A. \(\frac{4}{12}\)
B. \(\frac{3}{8}\)
C. \(\frac{5}{12}\)
D. \(\frac{8}{8}\)
Answer:

Question 29.
What is the value of the expression?
\(\frac{1}{4}\) + \(\frac{1}{4}\) + \(\frac{3}{8}\)
A. \(\frac{5}{8}\)
B. \(\frac{7}{8}\)
C. \(\frac{7}{16}\)
D. \(\frac{7}{32}\)
Answer:

Lesson 7.6 Estimate Sums and Differences of Mixed Numbers

Activity

Solve&Share

Alex has five cups of strawberries. He wants to use 1\(\frac{3}{4}\) cups of strawberries for a fruit salad and 3\(\frac{1}{2}\) cups for jam. Does Alex have enough strawberries to make both recipes? Solve this problem any way you choose.

Generalize You can estimate because you just need to know if Alex has enough. Show your work!

Look Back! Does it make sense to use 1 cup and 3 cups to estimate if Alex has enough strawberries? Explain.

Visual Learning Bridge

Essential Question What Are Some Ways to Estimate?

A.
Jamila’s mom wants to make a size 10 dress and jacket. About how many yards of fabric does she need?

B.
One Way
Use a number line to round fractions and mixed numbers to the nearest whole number

Jamila’s mom needs about 4 yards of fabric.

C.
Another Way
Use \(\frac{1}{2}\) as a benchmark fraction.
Replace each fraction with the nearest \(\frac{1}{2}\) unit.
1\(\frac{5}{8}\) is close to 1\(\frac{1}{2}\).
2\(\frac{1}{4}\) is halfway between 2 and 2\(\frac{1}{2}\)

You can replace 2\(\frac{1}{4}\) with 2\(\frac{1}{2}\).

So, 2\(\frac{1}{4}\) + 1\(\frac{5}{8}\) is about 2\(\frac{1}{2}\) + 1\(\frac{1}{2}\) = 4.

Convince Me! Critique Reasoning In Box C above, why does it make sense to replace 2\(\frac{1}{4}\) with 2\(\frac{1}{2}\) rather than 2?

Guided Practice

Do You Understand?

Question 1.
To estimate with mixed numbers, when should you round to the next greater whole number?
Answer:

Question 2.
When should you estimate a sum or difference?
Answer:

Do You Know How?

In 3-5, round to the nearest whole number.

Question 3.
2\(\frac{3}{4}\)
Answer:

Question 4.
1\(\frac{5}{7}\)
Answer:

Question 5.
2\(\frac{3}{10}\)
Answer:

In 6 and 7, estimate each sum or difference using benchmark fractions.

Question 6.
2\(\frac{5}{9}\) – 1\(\frac{1}{3}\)
Answer:

Question 7.
2\(\frac{4}{10}\) + 3\(\frac{5}{8}\)
Answer:

Independent Practice

Leveled Practice In 8-11, use the number line to round the mixed numbers to the nearest whole numbers.

Question 8.
11\(\frac{4}{6}\)
Answer:

Question 9.
11\(\frac{2}{8}\)
Answer:

Question 10.
11\(\frac{8}{12}\)
Answer:

Question 11.
11\(\frac{4}{10}\)
Answer:

In 12-20, estimate each sum or difference.

Question 12.
2\(\frac{1}{8}\) – \(\frac{5}{7}\)
Answer:

Question 13.
12\(\frac{1}{3}\) + 2\(\frac{1}{4}\)
Answer:

Question 14.
2\(\frac{2}{3}\) + \(\frac{7}{8}\) + 6\(\frac{7}{12}\)
Answer:

Question 15.
1\(\frac{10}{15}\) – \(\frac{8}{9}\)
Answer:

Question 16.
10\(\frac{5}{6}\) – 2\(\frac{3}{8}\)
Answer:

Question 17.
12\(\frac{8}{25}\) + 13\(\frac{5}{9}\)
Answer:

Question 18.
48\(\frac{1}{10}\) – 2\(\frac{7}{9}\)
Answer:

Question 19.
33\(\frac{14}{15}\) + 23\(\frac{9}{25}\)
Answer:

Question 20.
14\(\frac{4}{9}\) + 25\(\frac{1}{6}\) + 7\(\frac{11}{18}\)
Answer:

Problem Solving

Question 21.
Use the recipes to answer the questions.
a. Estimate how many cups of Fruit Trail Mix the recipe can make.

b Estimate how many cups of Traditional Trail Mix the recipe can make.
c Estimate how much trail mix you would have if you made both recipes.
Answer:

Question 22.
Kim is 3\(\frac{5}{8}\) inches taller than Colleen. If Kim is 60 inches tall, what is the best estimate of Colleen’s height?
Answer:

Question 23.
Higher Order Thinking Last week Jason walked 3\(\frac{1}{4}\) miles each day for 3 days and 4\(\frac{5}{8}\). miles each day for 4 days. About how many miles did Jason walk last week?
Answer:

Question 24.
Make Sense and Persevere Cal has $12.50 to spend. He wants to ride the roller coaster twice and the Ferris wheel once. Does Cal have enough money? Explain. What are 3 possible combinations of rides Cal can take using the money he has?

Answer:

Assessment Practice

Question 25.
Liam used 2\(\frac{2}{9}\) cups of milk for a pancake recipe and drank another 93 cups of milk. About how much milk did he use in all?
A. 8 cups
B. 10 cups
C. 12 cups
D. 13 cups
Answer:

Question 26.
Annie has 13\(\frac{1}{2}\) yards of string. She uses 1\(\frac{9}{10}\) yards to fix her backpack. About how much string does she have left?
A. 11 yards
B. 12 yards
C. 14 yards
D. 15 yards
Answer:

Lesson 7.7 Use Models to Add Mixed Numbers

Solve&Share
Martina is baking bread. She mixes 1\(\frac{3}{4}\) cups of flour with other ingredients. Then she adds 4\(\frac{1}{2}\) cups of flour to the mixture. How many cups of flour does she need? Solve this problem any way you choose.

Use Appropriate Tools You can use fraction strips to help add mixed numbers. Show your work!

Look Back! Explain how you can estimate the sum above.

Visual Learning Bridge

Essential Question How Can You Model Addition master of Mixed Numbers?

A.
Bill has 2 boards he will use to make picture frames. What is the total length of the boards Bill has to make picture frames?

You can find a common denominator to add the fractions.

B.
Step 1
Rename the fractional parts as equivalent fractions with a like denominator. Add the fractions.

C.
Step 2
Add the whole number parts.

Then add the sum of the fractional parts.

The total length of the boards is 4\(\frac{1}{4}\) feet.

Convince Me! Critique Reasoning Tom has 2 boards that are the same length as Bill’s. He says that he found the total length of the boards by adding 28 twelfths and 23 twelfths. Does his method work? Explain.

Guided Practice

Do You Understand?

Question 1.
When adding two mixed numbers, does it ever make sense to rename the fractional sum? Explain.
Answer:

Do You Know How?

In 2-5, use fraction strips to find each sum.

Question 2.
1\(\frac{1}{10}\) + 2\(\frac{4}{5}\)
Answer:

Question 3.
1\(\frac{1}{2}\) + 2\(\frac{3}{4}\)
Answer:

Question 4.
3\(\frac{2}{3}\) + 1\(\frac{4}{6}\)
Answer:

Question 5.
3\(\frac{1}{6}\) + 2\(\frac{2}{3}\)
Answer:

Independent Practice

Leveled Practice In 6 and 7, use each model to find the sum.

Question 6.
Charles used 1\(\frac{2}{3}\) cups of walnuts and 2\(\frac{1}{6}\) cups of cranberries to make breakfast bread. How many cups of walnuts and cranberries did he use in all?

Answer:

Question 7.
Mary worked 2\(\frac{3}{4}\) hours on Monday and 1\(\frac{1}{2}\) hours on Tuesday. How many hours did she work in all on Monday and Tuesday?

Answer:

In 8-16, use fraction strips to find each sum.

Question 8.
2\(\frac{6}{10}\) + 1\(\frac{3}{5}\)
Answer:

Question 9.
4\(\frac{5}{6}\) + 1\(\frac{7}{12}\)
Answer:

Question 10.
4\(\frac{2}{5}\) + 3\(\frac{7}{10}\)
Answer:

Question 11.
3\(\frac{1}{2}\) +1\(\frac{3}{4}\)
Answer:

Question 12.
1\(\frac{7}{8}\) + 5\(\frac{1}{4}\)
Answer:

Question 13.
2\(\frac{6}{12}\) + 1\(\frac{1}{2}\)
Answer:

Question 14.
3\(\frac{2}{5}\) + 1\(\frac{9}{10}\)
Answer:

Question 15.
2\(\frac{7}{12}\) + 1\(\frac{3}{4}\)
Answer:

Question 16.
2\(\frac{7}{8}\) + 5\(\frac{1}{2}\)
Answer:

Problem Solving

Question 17.
Lindsey used 1\(\frac{1}{4}\) gallons of tan paint for the ceiling and 4\(\frac{3}{8}\). gallons of green paint for the walls of her kitchen. How much paint did Lindsey use in all? Use fraction strips to help
Answer:

Question 18.
Paul said, “I walked 2\(\frac{1}{2}\) miles on Saturday and 2\(\frac{3}{4}\) miles on Sunday.” How many miles is that in all?
Answer:

Question 19.
Higher Order Thinking Tori is making muffins. The recipe calls for 2\(\frac{5}{6}\) cups of brown sugar for the muffins and 1\(\frac{1}{3}\) cups of brown sugar for the topping. Tori has 4 cups of brown sugar. Does she have enough brown sugar to make the muffins and the topping? Explain.

You can use fraction strips or a number line to compare amounts.

In 20 and 21, use the map. Each unit represents one block.

Question 20.
Ben left the museum and walked 4 blocks to his next destination. What was Ben’s destination?

Answer:

Question 21.
Make Sense and Persevere Ben walked from the restaurant to the bus stop. Then, he took the bus to the stadium. If he took the shortest route, how many blocks did Ben travel? Note that Ben can only travel along the grid lines.
Answer:

Assessment Practice

Question 22.
Marta used 2\(\frac{3}{4}\) cups of milk and 15 cups of cheese in a recipe. How many cups of cheese and milk did Marta use?
A. 3 cups
B. 3\(\frac{4}{6}\) cups
C. 4\(\frac{1}{4}\) cups
D. 4\(\frac{3}{4}\) cups
Answer:

Question 23.
Garrett ran 21\(\frac{1}{2}\) miles last week. He ran 17\(\frac{7}{8}\) miles this week. How many miles did he run in all?
A. 38 miles
B. 38\(\frac{1}{2}\) miles
C. 39\(\frac{3}{8}\) miles
D. 39\(\frac{7}{8}\) miles
Answer:

Lesson 7.8 Add Mixed Numbers

Solve & Share
Joaquin used two types of flour in a muffin recipe. How much flour did he use in all? Solve any way you choose.

Use Structure Use what you know about adding fractions. Show your work!

Look Back! How is adding mixed numbers with unlike denominators the same as adding fractions with unlike denominators? How is it different?

Visual Learning Bridge

Essential Question How Can You Add Mixed Numbers?

You can use addition to find the total amount of soil.

A.
Rhoda mixes 1\(\frac{1}{2}\) cups of sand with 2\(\frac{2}{3}\) cups of potting mixture to prepare soil for her cactus plants. After mixing them together, how many cups of soil does Rhoda have?

B.
Step 1
Find 2\(\frac{2}{3}\) + 1\(\frac{1}{2}\).
Write equivalent fractions with a common denominator.

C.
Step 2
Add the fractions.

C.
Step 3
Add the whole numbers.

Convince Me! Critique Reasoning Kyle used 9 as an estimate for 3\(\frac{1}{6}\) + 5\(\frac{7}{8}\). He got 9\(\frac{1}{24}\) for the exact sum. Is his calculated answer reasonable? Explain.

Guided Practice

Do You Understand?

Question 1.
How is adding mixed numbers like adding fractions and whole numbers?
Answer:

Question 2.
Look at the example on page 298. Why is the denominator 6 used in the equivalent fractions?
Answer:

Do You Know How?

In 3-6, estimate and then find each sum.

Question 3.

Answer:

Question 4.

Answer:

Question 5.
4\(\frac{1}{9}\) + 1\(\frac{1}{3}\)
Answer:

Question 6.
6\(\frac{5}{12}\) + 4\(\frac{5}{8}\)
Answer:

Independent Practice

Leveled Practice In 7-18, estimate and then find each sum.

Remember, fractions must have a common, or like, denominator before they can be added.

Question 7.

Answer:

Question 8.

Answer:

Question 9.

Answer:

Question 10.

Answer:

Question 11.

Answer:

Question 12.

Answer:

Question 13.

Answer:

Question 14.

Answer:

Question 15.
2\(\frac{3}{4}\) + 7\(\frac{3}{5}\)
Answer:

Question 16.
3\(\frac{8}{9}\) + 8\(\frac{1}{2}\)
Answer:

Question 17.
1\(\frac{7}{12}\) + 2\(\frac{3}{8}\)
Answer:

Question 18.
3\(\frac{11}{12}\) + 9\(\frac{1}{16}\)
Answer:

Problem Solving

Question 19.
Use the map to find the answer.

a What is the distance from the start to the end of the trail?
b Louise walked from the start of the trail to the bird lookout and back. Did she walk a longer or shorter distance than if she had walked from the start of the trail to the end? Explain.
c Another day, Louise walked from the start of the trail to the end. At the end, she realized she forgot her binoculars at the bird lookout. She walked from the end of the trail to the bird lookout and back. What is the total distance she walked?
Answer:

Question 20.
Higher Order Thinking Twice a day Cameron’s cat eats 4 ounces of dry cat food and 2 ounces of wet cat food. Dry food comes in 5-pound bags. Wet food comes in 6-ounce cans.
a How many cans of wet food should he buy to feed his cat for a week?
b How many ounces of wet cat food will be left over at the end of the week?
c How many days can he feed his cat from a 5-pound bag of dry food?

Remember: There are 16 ounces in a pound.

Question 21.
Julia bought 12 bags of cucumber seeds. Each bag contains 42 seeds. If she plants one half of the seeds, how many seeds does she have left?
Answer:

Question 22.
Critique Reasoning John added 2\(\frac{7}{12}\) and 5\(\frac{2}{3}\) and got 7\(\frac{1}{4}\) as the sum. Is John’s answer reasonable? Explain.
Answer:

Assessment Practice

Question 23.
What is the missing number in the following equation?

Answer:

Question 24.
Arnie skated 1\(\frac{3}{4}\) miles from home to the lake. He skated 1\(\frac{1}{3}\) miles around the lake, and then skated back home. Write an addition sentence to show how many miles Arnie skated in all.
Answer:

Lesson 7.9 Use Models to Subtract Mixed Numbers

Solve & Share
Clara and Erin volunteered at an animal shelter a total of 9\(\frac{5}{6}\) hours. Clara worked for 4\(\frac{1}{3}\) hours. How many hours did Erin work? You can use fraction strips to solve this problem.

Generalize How can you use what you know about adding mixed numbers to help you subtract mixed numbers? Show your work!

Look Back! How can you estimate the difference for the problem above? Explain your thinking.

Visual Learning Bridge

Essential Question How Can You Model Subtraction question of Mixed Numbers?

A.
James needs 1\(\frac{11}{12}\) inches of pipe to repair a small part of a bicycle frame. He has a pipe that is 2\(\frac{1}{2}\) inches long. Does he have enough pipe left over to fix a \(\frac{3}{4}\)-inch piece of frame on another bike?

Rename 2\(\frac{1}{2}\) as 2\(\frac{6}{12}\) that the fractions have a common denominator.

B.
Step 1
Model the number you are subtracting from, 2\(\frac{6}{12}\)

If the fraction you will be subtracting is greater than the fraction part of the number you model, rename 1 whole.
Since \(\frac{11}{12}\) > \(\frac{6}{12}\), rename 1 whole as \(\frac{12}{12}\).

Step 2
Use your renamed model to cross out the number that you are subtracting, 1\(\frac{11}{12}\)
There are \(\frac{7}{12}\) left.

So, 2\(\frac{1}{2}\) – 1\(\frac{11}{12}\) = \(\frac{7}{12}\)
James will have \(\frac{7}{12}\) inch of pipe left. He does not have enough for the other bike.

Convince Me! Use Appropriate Tools Use fraction strips to find 5\(\frac{1}{2}\) – 2\(\frac{3}{4}\).

Guided Practice

Do You Understand?

Question 1.
When subtracting two mixed numbers, is it always necessary to rename one of the wholes? Explain.

Do You Know How?

In 2-5, use fraction strips to find each difference.

Question 2.
4\(\frac{5}{6}\) – 2\(\frac{1}{3}\)
Answer:

Question 3.
4\(\frac{1}{8}\) – 3\(\frac{3}{4}\)
Answer:

Question 4.
5\(\frac{1}{2}\) – 2\(\frac{5}{6}\)
Answer:

Question 5.
5\(\frac{4}{10}\) – 3\(\frac{4}{5}\)
Answer:

Independent Practice

In 6 and 7, use each model to find the difference.

Question 6.
Terrell lives 2\(\frac{5}{6}\) blocks away from his best friend. His school is 4\(\frac{1}{3}\) blocks away in the same direction. If he stops at his best friend’s house first, how much farther do they have to walk to school?

Answer:

Question 7.
Tina bought 3\(\frac{1}{2}\) pounds of turkey and 2\(\frac{1}{4}\) pounds of cheese. She used 1\(\frac{1}{2}\) pounds of cheese to make macaroni and cheese. How much cheese does she have left?

Answer:

In 8-15, use fraction strips to find each difference.

Question 8.
12\(\frac{3}{4}\) – 9\(\frac{5}{8}\)
Answer:

Question 9.
8\(\frac{1}{6}\) – 7\(\frac{2}{3}\)
Answer:

Question 10.
13\(\frac{7}{9}\) – 10\(\frac{2}{3}\)
Answer:

Question 11.
3\(\frac{1}{12}\) – 2\(\frac{3}{4}\)
Answer:

Question 12.
6\(\frac{3}{4}\) – 3\(\frac{11}{12}\)
Answer:

Question 13.
4\(\frac{3}{5}\) – 1\(\frac{1}{10}\)
Answer:

Question 14.
6\(\frac{1}{2}\) – 3\(\frac{7}{10}\)
Answer:

Question 15.
6\(\frac{2}{3}\) – 4\(\frac{2}{9}\)
Answer:

Problem Solving

For 16 and 17, use the table at the right.

Question 16.
How much longer is a Red Oak leaf than a Sugar Maple leaf? Write an equation to model your work.
Answer:

Question 17.
How much longer is a Red Oak leaf than a Paper Birch leaf? Write an equation to model your work.

Question 18.
Higher Order Thinking Lemmy walked 3\(\frac{1}{2}\) miles on Saturday and 4\(\frac{3}{4}\) miles on Sunday. Ronnie walked 5\(\frac{3}{8}\) miles on Saturday. Who walked farther? How much farther?
Answer:

Question 19.
Model with Math Jamal is buying lunch for his family. He buys 4 drinks that each cost $1.75 and 4 sandwiches that each cost $7.50. If the prices include tax and he also leaves a $7 tip, how much does he spend in all? Write equations to show your work.
Answer:

Assessment Practice

Question 20.
What is the missing number in the following equation?

Answer:

Question 21.
What is the missing number in the following equation?

Answer:

Lesson 7.10 Subtract Mixed Numbers

Solve&Share
Evan walks 2-miles to his aunt’s house. He has already walked mile. How much farther does he have to go? Solve this problem any way you choose.

Use Structure Use what you know about subtracting fractions. Show your work!

Look Back! Jon said, “Changing \(\frac{3}{4}\) to \(\frac{6}{8}\) makes this problem easier.” What do you think Jon meant?

Visual Learning Bridge

Essential Question How Can You Subtract Mixed Numbers?

A.
A golf ball measures about 1\(\frac{2}{3}\) inches across the center. What is the difference between the distance across the center of the hole and the golf ball?

You can use subtraction to find the difference.

B.
Step 1
Write equivalent fractions with a common denominator.

C.
Step 2
Rename 4\(\frac{3}{12}\) to show more twelfths.

D.
Step 3
Subtract the fractions. Then subtract the whole numbers.

The hole is 2\(\frac{7}{12}\) inches wider.

Convince Me! Critique Reasoning Estimate 8\(\frac{1}{3}\) – 3\(\frac{3}{4}\). Tell how you got your estimate. Susi subtracted and found the actual difference to be 5\(\frac{7}{12}\). Is her answer reasonable? Explain.

Another Example
Sometimes you may have to rename a whole number to subtract. Find the difference of 6 – 2\(\frac{3}{8}\).

Guided Practice

Do You Understand?

Question 1.
In the example above, why do you need to rename the 6?
Answer:

Question 2.
In the example on page 306, could two golf balls fall into the hole at the same time? Explain your reasoning.
Answer:

In 3-6, estimate and then find each difference.

Question 3.

Answer:

Question 4.

Answer:

Question 5.
6\(\frac{3}{10}\) – 1\(\frac{4}{5}\)
Answer:

Question 6.
9\(\frac{1}{3}\) – 4\(\frac{3}{4}\)
Answer:

Independent Practice

In 7-18, estimate and then find each difference.

Remember to check that your answer makes sense by comparing it to the estimate.

Question 7.

Answer:

Question 8.

Answer:

Question 9.

Answer:

Question 10.

Answer:

Question 11.
6\(\frac{1}{3}\) – 5\(\frac{2}{3}\)
Answer:

Question 12.
9\(\frac{1}{2}\) – 6\(\frac{3}{4}\)
Answer:

Question 13.
8\(\frac{3}{16}\) – 3\(\frac{5}{8}\)
Answer:

Question 14.
7\(\frac{1}{2}\) – \(\frac{7}{10}\)
Answer:

Question 15.
15\(\frac{1}{6}\) – 4\(\frac{3}{8}\)
Answer:

Question 16.
13\(\frac{1}{12}\) – 8\(\frac{1}{4}\)
Answer:

Question 17.
6\(\frac{1}{3}\) – 2\(\frac{3}{5}\)
Answer:

Question 18.
10\(\frac{5}{12}\) – 4\(\frac{7}{8}\)
Answer:

Problem Solving

Question 19.
The average weight of a basketball is 21\(\frac{1}{10}\) ounces. The average weight of a baseball is 5\(\frac{1}{4}\) ounces. How many more ounces does the basketball weigh? Write the missing numbers in the diagram.

Answer:

Question 20.
enVision® STEM The smallest mammals on Earth are the bumblebee bat and the Etruscan pygmy shrew. The length of a certain bumblebee bat is 1\(\frac{9}{50}\) inches. The length of a certain Etruscan pygmy shrew is 1\(\frac{21}{50}\) inches. How much smaller is the bat than the shrew?
Answer:

Question 21.
Be Precise How are the purple quadrilateral and the green quadrilateral alike? How are they different?

Answer:

Question 22.
Higher Order Thinking Sam used the model to find 2\(\frac{5}{12}\) – 1\(\frac{7}{12}\). Did Sam model the problem correctly? Explain. If not, show how the problem should have been modeled and find the difference.

Answer:

Assessment Practice

Question 23.
Choose the correct number from the box below to complete the subtraction sentence that follows.

Answer:

Question 24.
Choose the correct number from the box below to complete the subtraction sentence that follows.

Answer:

Lesson 7.11 Add and Subtract Mixed Numbers

Activity

Solve&Share
Tim has 15 feet of wrapping paper. He uses 4\(\frac{1}{3}\) feet for his daughter’s present and 5\(\frac{3}{8}\) feet for his niece’s present. How much wrapping paper does Tim have left? Solve this problem any way you choose.

Reasoning What steps are needed to solve the problem? Show your work!

Look Back! In the problem above, how could you have estimated the amount of wrapping paper that is left?

Visual Learning Bridge

Essential Question
How Can Adding and Subtracting Mixed Numbers Help You Solve Problems?

A.
Clarisse has two lengths of fabric to make covers for a sofa and chair. The covers require 9\(\frac{2}{3}\) yards of fabric. How much fabric will Clarisse have left?

Find a common denominator when adding and subtracting fractions.

B.
Step 1
Add to find out how much fabric Clarisse has in all.

Clarisse has 13\(\frac{7}{12}\) yards of fabric in all.

C.
Step 2
Subtract the amount she will use from the total length of fabric.

Clarisse will have 3\(\frac{11}{12}\) yards of fabric left.

Convince Me! Make Sense and Persevere Clarisse has 14\(\frac{3}{4}\) yards of fabric to cover another sofa and chair. The new sofa needs 9\(\frac{1}{6}\) yards of fabric and the new chair needs 4\(\frac{1}{3}\) yards of fabric. Estimate to decide if Clarisse has enough fabric. If so, how much fabric will she have left?

Guided Practices

Do You Understand?

Question 1.
In the example on page 310, why do you add before you subtract?
Answer:

Question 2.
In the example on page 310, does Clarisse have enough fabric left over to make two cushions that each use 2\(\frac{7}{12}\) yards of fabric? Explain.
Answer:

Do You Know How?

In 3-5, find the sum or difference.

Question 3.

Answer:

Question 4.

Answer:

Question 5.

Answer:

In 6-9, solve. Do the addition in the parentheses first.

Question 6.
4\(\frac{3}{5}\) + 11\(\frac{2}{15}\)
Answer:

Question 7.
8\(\frac{2}{3}\) – 3\(\frac{3}{4}\)
Answer:

Question 8.

Answer:

Question 9.

Answer:

Independent Practice

In 10-14, find each sum or difference.

Question 10.

Answer:

Question 11.

Answer:

Question 12.

Answer:

Question 13.

Answer:

Question 14.

Answer:

In 15-20, solve. Do the operation in the parentheses first.

Question 15.

Answer:

Question 16.

Answer:

Question 17.

Answer:

Question 18.

Answer:

Question 19.

Answer:

Question 20.

Answer:

Problem Solving

In 21-23, use the table below.

Question 21.
Be Precise How much longer is the maximum jump of a South African sharp-nosed frog than the maximum jump of a leopard frog?
Answer:

Question 22.
How many centimeters long is a bullfrog? Round to the nearest whole number.
Answer:

Question 23.
Higher Order Thinking Which frog jumps about 10 times its body length? Explain how you found your answer.
Answer:

Question 24.
A-Z Vocabulary Write three numbers that are common denominators of \(\frac{7}{15}\) and \(\frac{3}{5}\).
Answer:

Question 25.
Marie plants 12 packages of vegetable seeds in a community garden. Each package costs $1.97 with tax. What is the total cost of the seeds?
Answer:

Assessment Practice

Question 26.
Which equations are true when 5\(\frac{3}{8}\) is placed in the box?

Answer:

Question 27.
Which equations are true when 3\(\frac{1}{3}\) is placed in the box?

Answer:

Lesson 7.12 Model with Math

Problem Solving

Solve & Share
Annie found three seashells at the beach. How much shorter is the Scotch Bonnet seashell than the combined lengths of the two Alphabet Cone seashells? Solve this problem any way you choose. Use a diagram to help.

Thinking Habits
Be a good thinker! These questions can help you.
• How can I use math I know to help solve this problem?
• How can I use pictures, objects, or an equation to represent the problem?
• Can I write an equation to show the problem?

Look Back! Model with Mathematics What is another way to represent this problem?

Visual Learning Bridge

Essential Question How Can You Represent a Problem Question with a Bar Diagram?

A.
The first step of a recipe is to mix the flour, white sugar, and brown sugar. Will a bowl that holds 4 cups be large enough?

Use a model to represent the problem.

What do I need to do to solve the problem?
I need to find the total amount of the first three ingredients and compare that amount to 4 cups.

Here’s my thinking.

B.
How can I model with math?

I can
• use math I know to help solve this problem.
• use a diagram to represent and solve this problem.
• write an equation involving fractions or mixed numbers.
• decide if my results make sense.

C.
I will use a bar diagram and an equation to represent the situation.

I can write this answer as a mixed number. 2\(\frac{6}{4}\) = 3\(\frac{2}{4}\) or 3\(\frac{1}{2}\)
There are 3\(\frac{1}{2}\) cups of ingredients, and 3. is less than 4. So, the 4-cup bowl is large enough.

Convince Me! Model with Mathematics How many more cups of ingredients could still fit in the bowl? Use a bar diagram and an equation to represent the problem.

Guided Practice

Phillip wants to run a total of 3 miles each day. Monday morning, he ran 1\(\frac{7}{8}\) miles. How many more miles does he still need to run?

Question 1.
Draw a diagram to represent the problem.
Answer:

Bar diagrams show how the quantities in a problem are related.

Question 2.
Write and solve an equation for this problem. How did you find the solution?
Answer:

Question 3.
How many more miles does Phillip still need to run?
Answer:

Independent Practice

Model with Math

A landscaper used 2\(\frac{1}{2}\) tons of sunburst pebbles, 3\(\frac{1}{4}\) tons of black polished pebbles, and a ton of river pebbles. What was the total weight of the pebbles?

Question 4.
Draw a diagram and write an equation to represent the problem.
Answer:

Question 5.
Solve the equation. What fraction computations did you do?
Answer:

Question 6.
How many tons of pebbles did the landscaper use?
Answer:

Problem Solving

Performance Task

Camp Activities
During the 6-hour session at day camp, Roland participated in boating, hiking, and lunch. The rest of the session was free time. How much time did Roland spend on the three activities? How much free time did he have?

Question 7.
Make Sense and Persevere What do you know and what do you need to find?
Answer:

When you model with math, you use the math you know to solve new problems.

Question 8.
Reasoning Describe the quantities and operations you will use to find how much time Roland spent on the planned activities. Which quantities and operations will you use to find how much free time Roland had?
Answer:

Question 9.
Model with Math Draw a diagram and use an equation to help you find how much time Roland spent on the activities. Then, draw a diagram and use an equation to help you find how much free time Roland had.
Answer:

Topic 7 Fluency Practice

Activity

Find a Match

Work with a partner. Point to a clue.
Read the clue.
Look below the clues to find a match. Write the clue letter in the box next to the match.
Find a match for every clue.

Clues

Topic 7 Vocabulary Review

Word List

Glossary

Understand Vocabulary

Write always, sometimes, or never.

• benchmark fractions
• common denominator
• equivalent fractions
• mixed number

Question 1.
A fraction can ____ be renamed as a mixed number.
Answer:

Question 2.
The sum of a mixed number and a whole number is ____ a mixed number
Answer:

Question 3.
\(\frac{1}{5}\) is__ used as a benchmark fraction.
Answer:

Question 4.
Equivalent fractions ___ have the same value.
Answer:

For each of these terms, give an example and a non-example.

Draw a line from each number in Column A to the same value in Column B.

Use Vocabulary in Writing

Question 12.
How can you write a fraction equivalent to \(\frac{60}{80}\) with a denominator that is less than 80?
Answer:

Topic 7 Reteaching

Set A
pages 269-272
Estimate the sum or difference by replacing each fraction with 0, \(\frac{1}{2}\), or 1.
Estimate \(\frac{4}{5}\) + \(\frac{5}{8}\)
Step 1
\(\frac{4}{5}\) is close to 1.
Step 2
\(\frac{5}{8}\) is close to \(\frac{4}{8}\) or \(\frac{1}{2}\)
Step 3
1 + \(\frac{1}{2}\) = 1\(\frac{1}{2}\)
Step 1
\(\frac{7}{12}\) is closes to \(\frac{6}{12}\) or \(\frac{1}{2}\).
Step 2
\(\frac{1}{8}\) is close to 0.
Step 3
\(\frac{1}{2}\) – 0 = \(\frac{1}{2}\)
So, \(\frac{7}{12}\) – \(\frac{1}{8}\) is about \(\frac{1}{2}\).

Remember that you can use as number line to decide if a fraction is closest to 0, \(\frac{1}{2}\), or 1.

Estimate each sum or difference.

Question 1.
\(\frac{2}{3}\) + \(\frac{5}{6}\)
Answer:

Question 2.
\(\frac{7}{8}\) – \(\frac{5}{12}\)
Answer:

Question 3.
\(\frac{1}{8}\) + \(\frac{1}{16}\)
Answer:

Question 4.
\(\frac{5}{8}\) – \(\frac{1}{6}\)
Answer:

Question 5.
\(\frac{1}{5}\) + \(\frac{1}{3}\)
Answer:

Question 6.
\(\frac{11}{12}\) – \(\frac{1}{10}\)
Answer:

Question 7.
\(\frac{9}{10}\) + \(\frac{1}{5}\)
Answer:

Question 8.
\(\frac{3}{5}\) – \(\frac{1}{12}\)
Answer:

Set B
pages 273-276

Find a common denominator for \(\frac{4}{9}\) and \(\frac{1}{3}\). Then rename each fraction as an equivalent fraction with the common denominator.

Step 1
Multiply the denominators:
9 × 3 = 27, so 27 is a common denominator.
Step 2
Rename the fractions:
\(\frac{4}{9}\) = \(\frac{4}{9}\) × \(\frac{3}{3}\) = \(\frac{4}{9}\)
\(\frac{1}{3}\) = \(\frac{1}{3}\) × \(\frac{9}{9}\) = \(\frac{9}{27}\)
So, \(\frac{4}{9}\) = \(\frac{12}{27}\) and \(\frac{1}{3}\) = \(\frac{9}{27}\).

Remember you can check to see if one denominator is a multiple of the other. Since 9 is a multiple of 3, another common denominator for the fractions and is 9. Find a common denominator. Then rename each fraction as an equivalent fraction with the common denominator.

Question 1.
\(\frac{3}{5}\) and \(\frac{7}{10}\)
Answer:

Question 2.
\(\frac{5}{6}\) and \(\frac{7}{18}\)
Answer:

Question 3.
\(\frac{3}{7}\) and \(\frac{1}{4}\)
Answer:

Set C
pages 277-280, 281-284, 285-288

Find \(\frac{5}{6}\) – \(\frac{3}{4}\)

Step 1
Find a common denominator by listing multiples of 6 and 4.
6:6, 12, 18, 24, 30, 36, 42
4:4, 8, 12, 16, 20, 24, 28, 32
12 is a common multiple of 6 and 4, so use 12 as the common denominator.

Step 2
Use the Identity Property to write equivalent fractions.
\(\frac{5}{6}\) = \(\frac{5 \times 2}{6 \times 2}\) = \(\frac{10}{12}\)
\(\frac{3}{4}\) = \(\frac{3 \times 3}{4 \times 3}\) = \(\frac{9}{12}\)

Step 3
Subtract.
\(\frac{10}{12}\) – \(\frac{9}{12}\) = \(\frac{1}{12}\)

Remember to multiply the numerator and denominator by the same number when writing an equivalent fraction.

Question 1.
\(\frac{2}{5}\) + \(\frac{3}{10}\)
Answer:

Question 2.
\(\frac{1}{9}\) + \(\frac{5}{6}\)
Answer:

Question 3.
\(\frac{3}{4}\) – \(\frac{5}{12}\)
Answer:

Question 4.
\(\frac{7}{8}\) – \(\frac{2}{3}\)
Answer:

Question 5.
\(\frac{1}{12}\) + \(\frac{3}{8}\)
Answer:

Question 6.
\(\frac{4}{5}\) – \(\frac{2}{15}\)
Answer:

Question 7.
Teresa spends \(\frac{1}{3}\) of her day at school. She spends \(\frac{1}{12}\) of her day eating meals. What fraction of the day does Teresa spend at school or eating meals?
Answer:

Set D
pages 289-292

Estimate 5\(\frac{1}{3}\) + 9\(\frac{9}{11}\)
To round a mixed number to the nearest whole number, compare the fraction part of the mixed number to \(\frac{1}{2}\).
If the fraction part is less than \(\frac{1}{2}\), round to the nearest lesser whole number.
5\(\frac{1}{3}\) rounds to 5.
If the fraction part is greater than or equal to \(\frac{1}{2}\), round to the nearest greater whole number.
9\(\frac{9}{11}\) rounds to 10.
So, 5\(\frac{1}{3}\) + 9\(\frac{9}{11}\) ≈ 5 + 10 = 15.

Remember that means ≈ is approximately equal to.”

Remember that you can also use benchmark fractions such as \(\frac{1}{4}\), \(\frac{1}{3}\), \(\frac{1}{2}\), and \(\frac{3}{4}\) to help you estimate.

Estimate each sum or difference.

Question 1.
3\(\frac{1}{4}\) – 1\(\frac{1}{2}\)
Answer:

Question 2.
5\(\frac{2}{9}\) + 4\(\frac{11}{13}\)
Answer:

Question 3.
2\(\frac{3}{8}\) + 5\(\frac{3}{5}\).
Answer:

Question 4.
9\(\frac{3}{7}\) – 6\(\frac{2}{5}\)
Answer:

Question 5.
8\(\frac{5}{6}\) – 2\(\frac{1}{2}\)
Answer:

Question 6.
7\(\frac{3}{4}\) + 5\(\frac{1}{8}\)
Answer:

Question 7.
11\(\frac{5}{12}\) + \(\frac{7}{8}\)
Answer:

Question 8.
13\(\frac{4}{5}\) – 8\(\frac{1}{6}\)
Answer:

Question 9.
A mark on the side of a pier shows the water is 4\(\frac{7}{8}\) feet deep. At high tide, the water level rises 2\(\frac{1}{4}\) feet. About how deep is the water at high tide?
Answer:

Set E
pages 293-296

Find 1\(\frac{1}{4}\) + 1\(\frac{7}{8}\)
Step 1
Rename the fractions with a common denominator. Model the addends and add the fractional parts.

Step 2
Add the whole numbers to the regrouped fractions.

So, 1\(\frac{1}{4}\) + 1\(\frac{7}{8}\) = 3\(\frac{1}{8}\)

Remember that you may need to rename a fraction as a mixed number.

Use a model to find each sum.

Question 1.
2\(\frac{5}{6}\) + 1\(\frac{5}{6}\)
Answer:

Question 2.
1\(\frac{1}{2}\) + 3\(\frac{3}{4}\)
Answer:

Question 3.
2\(\frac{3}{10}\) + 2\(\frac{4}{5}\)
Answer:

Question 4.
2\(\frac{1}{4}\) + 5\(\frac{11}{12}\)
Answer:

Question 5.
6\(\frac{2}{3}\) + 5\(\frac{5}{6}\)
Answer:

Question 6.
7\(\frac{1}{3}\) + 8\(\frac{7}{9}\)
Answer:

Question 7.
8\(\frac{4}{10}\) + 2\(\frac{3}{5}\)
Answer:

Question 8.
3\(\frac{1}{3}\) + 9\(\frac{11}{12}\)
Answer:

Set F
pages 301-304
Find 2\(\frac{1}{3}\) – 1\(\frac{5}{6}\). Rename 2\(\frac{1}{3}\) as 2\(\frac{2}{6}\)
Step 1
Model the number you are subtracting from, 2\(\frac{1}{3}\) or 2\(\frac{2}{6}\)
Since \(\frac{5}{6}\) > \(\frac{2}{6}\), rename 1 whole as \(\frac{6}{6}\)

Step 2
Cross out the number you are subtracting, 1\(\frac{5}{6}\).

The answer is the amount that is left.
So, 2\(\frac{1}{3}\) – 1\(\frac{5}{6}\) = \(\frac{3}{6}\) or \(\frac{1}{2}\)

Remember that the difference is the part of the model that is not crossed out. Use a model to find each difference.

Question 1.
15\(\frac{6}{10}\) – 3\(\frac{4}{5}\)
Answer:

Question 2.
6\(\frac{3}{4}\) – 5\(\frac{1}{2}\)
Answer:

Question 3.
4\(\frac{1}{6}\) – 1\(\frac{2}{3}\)
Answer:

Question 4.
12\(\frac{1}{4}\) – 7\(\frac{1}{2}\)
Answer:

Question 5.
9\(\frac{7}{10}\) – 3\(\frac{4}{5}\)
Answer:

Question 6.
5\(\frac{5}{8}\) – 3\(\frac{1}{4}\)
Answer:

Set G
pages 297-300, 305-308, 309-312

Gil had two lengths of wallpaper, 2\(\frac{3}{4}\) yards and 1\(\frac{7}{8}\) yards long. He used some and now has 1\(\frac{5}{6}\) yards left. How many yards of wallpaper did Gil use?
Step 1
Add to find the total amount of wallpaper Gil had.

Step 2
Subtract to find the amount of wallpaper Gil used.

Gil used 2\(\frac{19}{24}\) yards of wallpaper.

Remember when you add or subtract mixed numbers, rename the fractional parts to have a common denominator.

Solve. Do the operation in the parentheses first.

Question 1.
5\(\frac{1}{2}\) + 2\(\frac{1}{8}\)
Answer:

Question 2.
7\(\frac{5}{6}\) – 3\(\frac{2}{3}\)
Answer:

Question 3.
3\(\frac{1}{4}\) + 1\(\frac{5}{6}\)
Answer:

Question 4.
9 – 3\(\frac{3}{8}\)
Answer:

Question 5.

Answer:

Question 6.

Answer:

Set H
pages 313-316
Think about these questions to help you model with math.
Thinking Habits
• How can I use math I know to help solve this problem?
• How can I use pictures, objects, or an equation to represent the problem?
• How can I use numbers, words, and symbols to solve the problem?

Remember that a bar diagram can help you write an addition or a subtraction equation. Draw a bar diagram and write an equation to solve.

Question 1.
Justin jogs 3\(\frac{2}{5}\) miles every morning. He jogs 4\(\frac{6}{10}\) miles every evening. How many miles does he jog every day?
Answer:

Question 2.
Last year Mia planted a tree that was 5\(\frac{11}{12}\) feet tall. This year the tree is 7\(\frac{2}{3}\) feet tall. How many feet did the tree grow?
Answer:

Topic 7 Assessment Practice

Question 1.
Estimate the sum of \(\frac{3}{4}\) and \(\frac{1}{5}\). Write an equation.
Answer:

Question 2.
Select all the expressions that are equal to \(\frac{2}{3}\). Explain.
\(\frac{1}{6}\) + \(\frac{1}{2}\); I found a common denominator and then added the
numerators to get \(\frac{2}{3}\)
\(\frac{2}{9}\) + \(\frac{7}{18}\); I added the numerators and denominators to get \(\frac{2}{3}\)
\(\frac{5}{12}\) + \(\frac{1}{4}\); I added the numerators and denominators to get \(\frac{2}{3}\)
1\(\frac{1}{6}\) – \(\frac{1}{3}\); I found a common denominator and then subtracted the numerators to get \(\frac{2}{3}\)
2 – 1\(\frac{1}{3}\) found a common denominator and then subtracted the numerators to get \(\frac{2}{3}\)
Answer:

Question 3.
Tim has \(\frac{5}{12}\) of a jar of blackberry jam and \(\frac{3}{8}\) of a jar of strawberry jam. Write \(\frac{5}{12}\) and \(\frac{3}{8}\) using a common denominator. What fraction represents the total amount of jam Tim has?
Answer:

Question 4.
Sandra drove for \(\frac{1}{3}\) hour to get to the store. Then she drove \(\frac{1}{5}\) hour to get to the library. What fraction of an hour did Sandra drive in all? Explain.
Answer:

Question 5.
The bar diagram below shows the fractional parts of a pizza eaten by Pablo and Jamie.

A. Rename each fraction using a common denominator.
B. Use the renamed fractions to write and solve an equation to find the total amount of pizza eaten.
Answer:

Question 6.
Choose the correct sum for each expression.

Answer:

Question 7.
Benjamin and his sister shared a large sandwich. Benjamin ate of the sandwich and his sister ate of the sandwich.
A. Estimate how much more Benjamin ate than his sister. Explain how you found your estimate.
B. How much more did Benjamin eat than his sister? Find the exact amount.
Answer:

Question 8.
Alicia had 3\(\frac{1}{8}\) feet of wood. She used 1\(\frac{3}{4}\) feet of wood. Estimate the amount of wood Alicia has left.
A. 2 feet
B. 1 foot
C. 0 feet
D. 3 feet
Answer:

Question 9.
Explain why you must rename 2\(\frac{7}{12}\) in order to find 2\(\frac{7}{12}\) – \(\frac{5}{6}\)
Answer:

Question 10.
Mona bought 3\(\frac{3}{8}\) pounds of cheddar cheese. She used 2\(\frac{3}{4}\) pounds to make sandwiches. Write and solve an expression to find how much cheese is left.
Answer:

Question 11.
Marie needs 2\(\frac{1}{4}\) yards of fabric. She already has 1\(\frac{3}{8}\) yards. Which equation shows how many more yards of fabric Marie has to buy?
A. 2\(\frac{1}{4}\) + 1\(\frac{3}{8}\) = 1\(\frac{1}{8}\)
B. \(\frac{1}{4}\) + \(\frac{3}{8}\) = \(\frac{3}{4}\)
C. 2\(\frac{1}{4}\) – 1\(\frac{3}{8}\) = \(\frac{7}{8}\)
D. \(\frac{3}{8}\) – \(\frac{1}{4}\) = \(\frac{1}{8}\)
Answer:

Question 12.
During a trip, Martha drove \(\frac{1}{6}\) of the time, Chris drove \(\frac{1}{4}\) a of the time, and Juan drove the rest of the time. What fraction of the time did Juan drive?
Answer:

Question 13.
Gilberto worked 3\(\frac{1}{4}\) hours on Thursday, 4\(\frac{2}{5}\) hours on Friday, and 6\(\frac{1}{2}\) hours on Saturday. How many hours did he work in all during the three days?
A. 13\(\frac{1}{10}\) hours
B. 13\(\frac{3}{20}\) hours
C. 14\(\frac{1}{10}\) hours
D. 14\(\frac{3}{20}\) hours
Answer:

Question 14.
The model below can be used to find the sum of two mixed numbers. What is the sum? What is the difference? Show your work.

Answer:

Question 15.
Estimate the sum of 1\(\frac{1}{3}\) and 2\(\frac{3}{4}\). Explain how you found your estimate.
Answer:

Question 16.
Find 4\(\frac{1}{5}\) – \(\frac{7}{10}\)
A. Explain why 4\(\frac{1}{5}\) must be renamed in order to do the subtraction.
B. Explain how to rename 4\(\frac{1}{5}\) in order to do the subtraction.
Answer:

Question 17.
Mark is making a small frame in the shape of an equilateral triangle with the dimensions shown below. What is the perimeter of the frame?

A. 6\(\frac{1}{2}\) cm
B. 9\(\frac{1}{2}\) cm
C. 9\(\frac{1}{6}\) cm
D. 10\(\frac{1}{2}\) cm
Answer:

Question 18.
A baker uses food coloring to color cake batter. He needs 4\(\frac{1}{8}\) ounces of green food coloring. The baker only has 2\(\frac{1}{3}\) ounces. How much more green food coloring does he need? If the baker only finds 1 ounce of food coloring at the store, how many more ounces does the baker need?
Answer:

Question 19.
Models for two mixed numbers are shown below. What is the sum of the numbers? What is the difference? Show your work.

Answer:

Question 20.
Dawson says that the expression is equal to a whole number. Do you agree? Explain.
Answer:

Topic 7 Performance Task

Tying Knots
Liam and Pam each have a length of thick rope. Liam has tied an overhand knot in his rope. The overhand knot is a basic knot often used as a basis for other types of knots.

Question 1.
Liam untied the overhand knot. The full length of the rope is shown below. How much rope did the knot use?

Answer:

Question 2.
Liam laid his untied rope end-to-end with Pam’s rope.
Part A
About how long would the two ropes be? Explain how you got your estimate.
Part B
Explain whether the actual length would be greater or less than your estimate.
Answer:

Question 3.
Liam and Pam tied their two ropes together with a square knot. The knot used 15 feet of rope. How long is their rope? Explain.

Answer:

Question 4.
Marco has a rope that is 16 feet long. He ties his rope to Liam and Pam’s rope with a square knot that uses 1\(\frac{1}{8}\) feet of rope.
Part A
How long are the three ropes tied together? Write an equation to model the problem. Then solve the equation.
Part B
Liam, Pam, and Marco decide to shorten the tied ropes by cutting off \(\frac{2}{5}\) foot from one end and \(\frac{1}{6}\) foot from the other end. About how much rope is cut off in all? Explain.
Part C
How long are the tied ropes now? Show your work.
Answer:

Practice with the help of enVision Math Common Core Grade 5 Answer Key Topic 4 Use Models and Strategies to Multiply Decimals regularly and improve your accuracy in solving questions.

enVision Math Common Core 5th Grade Answers Key Topic 4 Use Models and Strategies to Multiply Decimals

Essential Question: What are some common procedures for estimating and finding products involving decimals?

enVision STEM Project: Solar Energy
Do Research Use the Internet or other sources to learn about solar energy. Find at least five ways that we use the Sun’s energy today.

Answer:
To dry your clothes.
To grow your food.
To heat your water.
To power your car.
To generate your electricity.

Explanation:
In the above-given question,
given that,
The sun has gone to a lot of trouble to send us its energy.
we can use solar energy in many ways.
they are:
to dry your clothes.
to grow your food.
to heat your water.
to power your car.
to generate your electricity.

Journal: Write a Report Include what you found. Also in your report:
• Describe at least one way that you could use solar energy. Could it save you money?
• Estimate how much your family pays for energy costs such as lights, gasoline, heating, and cooling.
• Make up and solve problems by multiplying whole numbers and decimals.

Review What You Know

Vocabulary

Choose the best term from the box. Write it on the blank.
• exponent
• hundredths
• overestimate
• partial products
• power
• round
• tenths
• thousandths
• underestimate

Question 1.
One way to estimate a number is to ____ the number.

Answer:
One way to estimate a number is to round the number.

Explanation:
In the above-given question,
given that,
one way to estimate a number is to round the number.
for example:
2.456 round to tenths.
2.556.

Question 2.
Using 50 for the number of weeks in a year is a(n) _____.

Answer:
Using 50 for the number of weeks in a year is a(n) is exponents.

Explanation:
In the above-given question,
given that,
using 50 for the number of weeks in a year is a(n) is exponents.
for example:
a(n) = 50.
a = 5.
50/5 = 10.

Question 3.
In the number 3.072, the digit 7 is in the ___ place and the digit 2 is in the ____ place.

Answer:
In the number 3.072, the digit 7 is in the hundredths place and the digit 2 is in the thousandths place.

Explanation:
In the above-given question,
given that,
In the number 3.072, the digit 7 is in the hundredths place and the digit 2 is in the thousandths place.
for example:
3.072.
7 is in the hundredths place.
2 is in the thousands place.

Question 4.
10,000 is a(n) ____ of 10 because 10 × 10 × 10 × 10 = 10,000.

Answer:
10,000 is a(n) power of 10 because 10 x 10 x 10 x 10 = 10,000.

Explanation:
In the above-given question,
given that,
for example:
10 x 10 x 10 x 10.
100 x 100 = 10,000.

Whole Number Multiplication Find each product.

Question 5.
64 × 100

Answer:
The product is 6400.

Explanation:
In the above-given question,
given that,
the two numbers are 64 and 100.
multiply the two numbers.
64 x 100 = 6400.
so the product is 6400.

Question 6.
7,823 × 10³

Answer:
The product is 7823000.

Explanation:
In the above-given question,
given that,
the two numbers are 7823 and 1000.
multiply the two numbers.
7823 x 1000 = 7823000.
so the product is 7823000.

Question 7.
10 × 1,405

Answer:
The product is 14050.

Explanation:
In the above-given question,
given that,
the two numbers are 10 and 1405.
multiply the two numbers.
10 x 1405 = 14050.
so the product is 14050.

Question 8.
53 × 413

Answer:
The product is 21889.

Explanation:
In the above-given question,
given that,
the two numbers are 53 and 413.
multiply the two numbers.
53 x 413 = 21889.
so the product is 21889.

Question 9.
906 × 57

Answer:
The product is 51,642.

Explanation:
In the above-given question,
given that,
the two numbers are 906 and 57.
multiply the two numbers.
906 x 57 = 51,642.
so the product is 51,642.

Question 10.
1,037 × 80

Answer:
The product is 82,960.

Explanation:
In the above-given question,
given that,
the two numbers are 1037 and 80.
multiply the two numbers.
1037 x 80 = 82,960.
so the product is 82,960.

Round Decimals

Round each number to the nearest tenth.

Question 11.
842.121

Answer:
The number to the nearest tenth = 842.10.

Explanation:
In the above-given question,
given that,
the number is 842.121.
to round a number to the nearest tenth look at the number of ones.
if this is 5 or more round up.
if it is 4 or less round down.
842.10.
so the number to the nearest tenth = 842.10.

Question 12.
10,386.145

Answer:
The number to the nearest tenth = 10386.10.

Explanation:
In the above-given question,
given that,
the number is 10,386.145.
to round a number to the nearest tenth look at the number of ones.
if this is 5 or more round up.
if it is 4 or less round down.
10386.10.
so the number to the nearest tenth = 10386.10.

Question 13.
585.055

Answer:
The number to the nearest tenth = 585.155.

Explanation:
In the above-given question,
given that,
the number is 585.055.
to round a number to the nearest tenth look at the number of ones.
if this is 5 or more round up.
if it is 4 or less round down.
585.055.
so the number to the nearest tenth = 585.155.

Properties of Multiplication
Use the Commutative and Associative Properties of Multiplication to complete each multiplication.

Question 14.
96 × 42 = 4,032 so 42 × 96 = ___

Answer:
42 x 96 = 4032.

Explanation:
In the above-given question,
given that,
the two numbers are 96 and 42.
multiply the two numbers.
96 x 42 = 4032.
42 x 96 = 4032.

Question 15.
4 (58 × 25) = 4 × (25 × ___) = (___ × ___) × 58 = ___

Answer:
4(58 x 25) = 4 x (25 x 58) = (25 x 4) x 58 = 5800.

Explanation:
In the above-given question,
given that,
the two numbers are 58, 4, and 25.
multiply the two numbers.
4 x 25 x 58.
25 x 4 x 8 = 5800.

Question 16.
(293 × 50) × 20 = 293 × (50 × ___) = ___

Answer:
293 x 50 x 20 = 293 x 50 x 20 = 2,93,000.

Explanation:
In the above-given question,
given that,
the two numbers are 293, 50, and 20.
multiply the two numbers.
293 x 50 x 20.
50 x 293 x 20 = 2,93,000.

pick a Project

PROJECT 4A

How can you set up an exercise plan?
Project: Plan an exercise Program

PROJECT 4B
How much does it cost to dress a team?
Project: Budget a Team

PROJECT 4C
How far can a rocket go in 100 seconds?
Project: Make a Poster

Answer:
The rocket can go in 100 sec = 790 km.

Explanation:
The above-given question,
given that,
the rocket can go in 100 sec is:
1 minute = 60 sec.
1 sec = 7.9 km.
100sec = 7.9 x 100.
790 km.

PROJECT 4D
How much extra do you have to pay?
Project: Make a Data Display

Lesson 4.1 Multiply Decimals by Powers of 10

Activity

Solve & Share

Javier is helping his parents put up posters in their movie theater. Each poster has a thickness of 0.012 inch. How thick is a stack of 10 posters? 100 posters? 1,000 posters? Solve this problem any way you choose.

Answer:
The thick is a stack of 10 posters = 0.12.
the thick is a stack of 100 posters = 1.2.
the thick is a stack of 1000 posters = 12.

Explanation:
In the above-given question,
given that,
Javier is helping his parents put up posters in their movie theater.
Each poster has a thickness of 0.012.
0.012 x 10 = 0.12.
0.012 x 100 = 1.2.
0.012 x 1000 = 12.

You can use the structure of our number system and mental math to help you.

Look Back! Use Structure How is your answer for 1,000 posters similar to 0.012? How is it different?

Visual Learning Bridge

Essential Question
What Patterns Can Help You Multiply Decimals by Powers of 10?

A.
You can use place value and what you know about whole numbers to multiply decimals by powers of 10. What patterns can you find?

We already know what happens when a whole number is multiplied by powers of 10.

B.
In a place-value chart, the same pattern appears when a decimal is multiplied by powers of 10.

A digit in one place is worth 10 times more when moved to the place on its left. Every time a number is multiplied by 10, the digits of the number shift to the left.

C.
Holding the numbers still, another pattern appears.
3.63 × 1 = 3.63
3.63 × 10¹ = 36.3
3.63 × 10² = 363.0
3.63 × 10³ = 3630.0
It looks like the decimal point moves to the right each time.

Convince Me! Use Structure Complete the chart. What patterns can you use to place the decimal point?

Answer:
1.275 x 10 = 12.75, 1.275 x 100 = 127.5, 1.275 x 1000 = 1275.
26.014 x 10 = 260.14, 26.014 x 100 = 2601.4, 26.014x 1000 = 26014.
0.4 x 10 = 4, 0.4 x 100 = 40, 0.4 x 1000 = 400.

Explanation:
In the above-given question,
given that,
the numbers are 1.275, 26.014, and 0.4.
multiply the numbers by 10, 100, and 1000.
1.275 x 10 = 12.75, 1.275 x 100 = 127.5, 1.275 x 1000 = 1275.
26.014 x 10 = 260.14, 26.014 x 100 = 2601.4, 26.014x 1000 = 26014.
0.4 x 10 = 4, 0.4 x 100 = 40, 0.4 x 1000 = 400.

Guided Practice

Do You Understand?

Question 1.
When multiplying by a power of 10, like 4.58 × 10³, how do you know you are moving the decimal in the correct direction?

Answer:
The number is 4580.

Explanation:
In the above-given question,
given that,
4.58 × 10^3.
4.58 x 10 x 10 x 10.
4.58 x 1000 = 4580.
so the number is 4580.

Do You Know How?

In 2-5, find each product.

Question 2.
0.009 × 10

Answer:
The product is 0.09.

Explanation:
In the above-given question,
given that,
the numbers are 0.009  and 10.
multiply the numbers.
0.009 x 10 = 0.09.
so the product is 0.09.

Question 3.
3.1 × 10³

Answer:
The product is 3100.

Explanation:
In the above-given question,
given that,
the numbers are 3.1 and 1000.
multiply the numbers.
3.1 x 1000 = 3100.
so the product is 3100.

Question 4.
0.062 × 10²

Answer:
The product is 6.2.

Explanation:
In the above-given question,
given that,
the numbers are 0.062  and 100.
multiply the numbers.
0.062 x 100 = 6.2.
so the product is 6.2.

Question 5.
1.24 × 10⁴

Answer:
The product is 1240.

Explanation:
In the above-given question,
given that,
the numbers are 1.24  and 10000.
multiply the numbers.
1.24 x 10000 = 1240.
so the product is 1240.

Independent Practice

Leveled Practice in 6 and 7, find each product.

Place-value patterns can help you solve these problems.

Question 6.
42.3 ×1 = ___
42.3 × 10 = ___
42.3 × 10² = ___

Answer:
42.3 x 1 = 42.3.
42.3 x 10 = 423.
42.3 x 100 = 4230.

Explanation:
In the above-given question,
given that,
the numbers are 42.3, 42.2 x 10, and 42.3 x 100.
42.3 x 1 = 42.3.
42.3 x 10 = 423.
42.3 x 100 = 4230.

Question 7.
____ = 0.086 × 10¹
___ = 0.086 × 100
____ = 0.086 × 1,000

Answer:
0.086 x 10 = 0.86.
0.086 x 100 = 8.6.
0.086 x 1000 = 86.

Explanation:
In the above-given question,
given that,
the numbers are 0.086 x 10, 0.086 x 100, and 0.086 x 1000.
0.086 x 10 = 0.86.
0.086 x 100 = 8.6.
0.086 x 1000 = 86.

In 8-15, find each product.

Question 8.
63.7 × 10

Answer:
The product is 637.

Explanation:
In the above-given question,
given that,
the numbers are 63.7  and 10.
multiply the numbers.
63.7 x 10 = 637.
so the product is 637.

Question 9.
563.7 × 10²

Answer:
The product is 56370.

Explanation:
In the above-given question,
given that,
the numbers are 563.7  and 100.
multiply the numbers.
563.7 x 100 = 56370.
so the product is 56370.

Question 10.
0.365 × 10⁴

Answer:
The product is 3650.

Explanation:
In the above-given question,
given that,
the numbers are 0.365  and 10000.
multiply the numbers.
0.365 x 10000 = 3650.
so the product is 3650.

Question 11.
5.02 × 100

Answer:
The product is 502.

Explanation:
In the above-given question,
given that,
the numbers are 5.02  and 100.
multiply the numbers.
5.02 x 100 = 502.
so the product is 502.

Question 12.
94.6 × 10³

Answer:
The product is 94600.

Explanation:
In the above-given question,
given that,
the numbers are 94.6  and 1000.
multiply the numbers.
94.6 x 1000 = 94600.
so the product is 94600.

Question 13.
0.9463 × 10²

Answer:
The product is 94.63.

Explanation:
In the above-given question,
given that,
the numbers are 0.9463  and 100.
multiply the numbers.
0.9463 x 100 = 94.63.
so the product is 94.63.

Question 14.
0.678 × 100

Answer:
The product is 67.8.

Explanation:
In the above-given question,
given that,
the numbers are 0.678  and 100.
multiply the numbers.
0.678 x 100 = 67.8.
so the product is 67.8.

Question 15.
681.7 × 10⁴

Answer:
The product is 6817000.

Explanation:
In the above-given question,
given that,
the numbers are 681.7 and 10000.
multiply the numbers.
681.7 x 10000 = 6817000.
so the product is 6817000.

In 16-18, find the missing exponent.

Question 16.
0.629 × = 62.9

Answer:
The missing exponent is 2.

Explanation:
In the above-given question,
given that,
0.629 x 10 x 10.
0.629 x 100 = 62.9.
0.629

Question 17.
× 0.056 = 560

Answer:
The missing exponent is 4.

Explanation:
In the above-given question,
given that,
10 x 10 x 10 x 10 x 0.056.
100 x 100 x 0.056.
10000 x 0.056.
560.

Question 18.
1.23 = × 0.123

Answer:
The missing exponent is 0.

Explanation:
In the above-given question,
given that,
1.23 x 10.
1.23.

Problem Solving

In 19-21, use the table to find the answers.

Question 19.
Monroe uses a microscope to observe specimens in science class. The microscope enlarges objects to 100 times their actual size. Find the size of each specimen as seen in the microscope.

Answer:
The size of each specimen as seen in the microscope = 0.8, 11, 0.25, and 0.4.

Explanation:
In the above-given question,
given that,
Monroe uses a microscope to observe specimens in science class.
The microscope enlarges objects to 100 times their actual size.
0.008 x 100 = 0.8.
0.011 x 100 = 11.
0.0025 x 100 = 0.25.
0.004 x 100 = 0.4.

Question 20.
Monroe’s teacher wants each student to draw a sketch of the longest specimen. Which specimen is the longest?

Answer:
Specimen B is the longest.

Explanation:
In the above-given question,
given that,
Monroe’s teacher wants each student to draw a sketch of the longest specimen.
0.008 x 100 = 0.8.
0.011 x 100 = 11.
0.0025 x 100 = 0.25.
0.004 x 100 = 0.4.
so specimen B is the longest.

Question 21.
Seen through the microscope, a specimen is 0.75 cm long. What is its actual length?

Answer:
The actual length = 75 cm.

Explanation:
In the above-given question,
given that,
a specimen is 0.75 cm long.
0.75 x 100 = 75.
so the actual length of the specimen is 75 cm.

Question 22.
Jon’s binoculars enlarge objects to 10 times their actual size. If the length of an ant is 0.43 inch, what is the length as seen up close through his binoculars?

Answer:
The length as seen up close through his binoculars = 4.3 inches.

Explanation:
In the above-given question,
given that,
Jon’s binoculars enlarge objects to 10 times their actual size.
If the length of an ant is 0.43 inches.
0.43 x 10 = 4.3 inches.
so the length as seen up close through his binoculars = 4.3 inches.

Question 23.
Higher Order Thinking Jefferson drew a line 9.5 inches long. Brittany drew a line 10 times as long. What is the difference in length between the two lines?

Answer:
The difference in length between the two lines = 85.5 inches.

Explanation:
In the above-given question,
given that,
Jefferson drew a line 9.5 inches long.
Brittany drew a line 10 times as long.
9.5 x 10 = 95.
95 – 9.5 = 85.5.
so the difference in length between the two lines = 85.5 inches.

Question 24.
Construct Arguments José ran 2.6 miles. Pavel ran 2.60 miles. Who ran farther? Explain your reasoning.

Answer:
Jose and Pavel ran the same.

Explanation:
In the above-given question,
given that,
José ran 2.6 miles. Pavel ran 2.60 miles.
2.6 is equal to 2.60.
so Jose and Pavel ran the same.

Assessment Practice

Question 25.
Choose all equations that are true.
4.82 × 1,000 = 482,000
4.82 × 10² = 482
482 × 10¹ = 48.2
482 × 10³ = 482
48.2 × 10⁴ = 4,820

Answer:
4.82 x 100 = 482.

Explanation:
In the above-given question,
given that,
The equations are:
4.82 x 1000 = 482000.
4.82 x 10 = 482.
482 x 1000 = 482.
48.2 x 10000 = 4820.
so the equation that is true is 4.82 x 100 = 482.

Question 26.
Choose all equations that are true when 102 is placed in the box.
o 37 = × 0.37 0
0.37 = × 0.037
0370 = × 3.7
0.37 = × 3.7
3.7 = × 0.037

Answer:
None of the equations are true.

Explanation:
In the above-given question,
given that,
37 x 102 = 0.370.
0.37 x 102 = 0.037.
0370 x 102 = 3.7.
3.7 x 102 = 0.037.
so none of the equations are true.

Lesson 4.2 Estimate the Product of a Decimal and a Whole Number

Activity

Solve & Share

Renee needs 32 strands of twine for an art project. Each strand must be 1.25 centimeters long. About how many centimeters of twine does she need? Solve this problem any way you choose!

Answer:
The centimeters of twine does she need = 40 cm.

Explanation:
In the above-given question,
given that,
Renee needs 32 strands of twine for an art project.
Each strand must be 1.25 centimeters long.
32 x 1.25 = 40.
so the centimeters of twine does she need = 40 cm.

Generalize How can you relate what you know about estimating with whole numbers to estimating with decimals? Show
your work!

Look Back! Is your estimate an overestimate or an underestimate? How can you tell?

Visual Learning Bridge

Essential Question
What Are Some Ways to Estimate Products of Decimals and Whole Numbers?

A.
A wedding planner needs to buy 16 pounds of sliced cheddar cheese. About how much will the cheese cost?

B.
One Way
Round each number to the nearest dollar and nearest ten.

$2 × 20 = $40
The cheese will cost about $40.

C.
Another Way
Use compatible numbers that you can multiply mentally.

$2 × 15 = $30
The cheese will cost about $30.

Convince Me! Reasoning About how much money would 18 pounds of cheese cost if the price is $3.95 per pound? Use two different ways to estimate the product. Are your estimates overestimates or underestimates? Explain.

Another Example
Manuel walks a total of 0.75 mile to and from school each day. If there have been 105 school days so far this year, about how many miles has he walked in all?
Round to the nearest whole number.

Use compatible numbers.

Be sure to place the decimal point correctly.

Both methods provide reasonable estimates of how far Manuel has walked.

Guided Practice

Do You Understand?

Question 1.
Number Sense There are about 20 school days in a month. In the problem above, about how many miles does Manuel walk each month? Write an equation to show your work.

Answer:
The number of miles does Manuel walks each month = 16.

Explanation:
In the above-given question,
given that,
There are about 20 school days in a month.
0.75 x 20.
0.8 x 20 = 16.
so the number of miles does Manuel walks each month = 16.

Question 2.
Without multiplying, which estimate in the Another example do you think is closer to the exact answer? Explain your reasoning

Answer:
The number of miles does Manuel walks each month = 16.

Explanation:
In the above-given question,
given that,
There are about 20 school days in a month.
0.75 x 20.
0.8 x 20 = 16.
so the number of miles does Manuel walks each month = 16.

Do You Know How?

In 3-8, estimate each product using rounding or compatible numbers.

Question 3.
0.87 × 112

Answer:
The product is 112.

Explanation:
In the above-given question,
given that,
the numbers are 0.87 and 112.
0.87 x 112.
0.8 is equal to 1.
112 x 1 = 112.
so the product is 112.

Question 4.
104 × 0.33

Answer:
The product is 52.

Explanation:
In the above-given question,
given that,
the numbers are 104 and 0.33.
104 x 0.33.
0.33 is equal to 0.5.
104 x 0.5 = 52.
so the product is 52.

Question 5.
9.02 × 80

Answer:
The product is 720.

Explanation:
In the above-given question,
given that,
the numbers are 9.02 and 80.
9.02 x 80.
9.02 is equal to 9.
9 x 80 = 720.
so the product is 720.

Question 6.
0.54 × 24

Answer:
The product is 12.

Explanation:
In the above-given question,
given that,
the numbers are 0.54 and 24.
0.54 x 24.
0.54 is equal to 0.5.
0.5 x 24 = 12.
so the product is 12.

Question 7.
33.05 × 200

Answer:
The product is 6600.

Explanation:
In the above-given question,
given that,
the numbers are 33.05 and 200.
33.05 x 200.
33.05 is equal to 33.
33 x 200 = 6600.
so the product is 6600.

Question 8.
0.79 × 51

Answer:
The product is 51.

Explanation:
In the above-given question,
given that,
the numbers are 0.70 and 51.
0.7 x 51.
0.7 is equal to 1.
51 x 1 = 51.
so the product is 51.

Independent Practice

In 9-16, estimate each product.

Question 9.
0.12 × 105

Answer:
The product is 10.5.

Explanation:
In the above-given question,
given that,
the numbers are 0.12 and 105.
0.12 x 105.
0.12 is equal to 0.1.
105 x 0.1 = 10.5.
so the product is 10.5.

Question 10.
45.3 × 4

Answer:
The product is 180.

Explanation:
In the above-given question,
given that,
the numbers are 45.3 and 4.
45.3 x 4.
45.3 is equal to 45.
45 x 4 = 180.
so the product is 180.

Question 11.
99.2 × 82

Answer:
The product is 8118.

Explanation:
In the above-given question,
given that,
the numbers are 99.2 and 82.
99.2 x 82.
99.2 is equal to 99.
99 x 82 = 8118.
so the product is 8118.

Question 12.
37 × 0.93

Answer:
The product is 37.

Explanation:
In the above-given question,
given that,
the numbers are 37 and 0.93.
37 x 0.93.
0.93 is equal to 1.
37 x 1 = 37.
so the product is 37.

Question 13.
1.67 × 4

Answer:
The product is 8.

Explanation:
In the above-given question,
given that,
the numbers are 1.67 and 4.
1.67 x 4.
1.67 is equal to 2.
4 x 2 = 8.
so the product is 8.

Question 14.
3.2 × 184

Answer:
The product is 552.

Explanation:
In the above-given question,
given that,
the numbers are 3.2 and 184.
3.2 x 184.
3.2 is equal to 3.
184 x 3 = 552.
so the product is 552.

Question 15.
12 × 0.37

Answer:
The product is 3.6.

Explanation:
In the above-given question,
given that,
the numbers are 12 and 0.3.
0.3 x 12.
0.3 is equal to 0.3.
12 x 0.3 = 3.6.
so the product is 3.6.

Question 16.
0.904 × 75

Answer:
The product is 75.

Explanation:
In the above-given question,
given that,
the numbers are 0.904 and 75.
0.904 x 75.
0.9 is equal to 1.
75 x 1 = 75.
so the product is 75.

Problem Solving

Question 17.
About how much money does Stan need to buy 5 T-shirts and 10 buttons?

Answer:
The cost to buy 5 T-shirts and 10 buttons = $82.

Explanation:
In the above-given question,
given that,
the cost of a button is $1.95.
the cost of the T-shirt is $12.50.
1.95 x 10 = 19.5.
12.50 x 5 = 62.5.
19.5 + 62.5 = 82.
so the cost to buy 5 T-shirts and 10 buttons = $82.

Question 18.
Joseph buys a pair of shorts for $17.95 and 4 T-shirts. About how much money does he spend?

Answer:
The amount of money he spends = $67.95.

Explanation:
In the above-given question,
given that,
Joseph buys a pair of shorts for $17.95 and 4 T-shirts.
$12.50 x 4 = 50.
$17.95 + 50 = 67.95.
so the money he spend = $67.95.

Question 19.
Marcy picked 18.8 pounds of peaches at the pick-your-own orchard. Each pound costs $1.28. About how much did Marcy pay for the peaches? Write an equation to model your work.

Answer:
The amount did Marcy pay for the peaches = $24.064.

Explanation:
In the above-given question,
given that,
Marcy picked 18.8 pounds of peaches at the pick-your-own orchard.
Each pound costs $1.28.
18.8 x 1.28 = 24.064.
so the amount did Marcy pay for the peaches = $24.064.

Question 20.
Be Precise Joshua had $20. He spent $4.58 on Friday, $7.43 on Saturday, and $3.50 on Sunday. How much money does he have left? Show how you found the answer.

Answer:
The money does he have left = $15.51.

Explanation:
In the above-given question,
given that,
Be Precise Joshua had $20.
He spent $4.58 on Friday.
$7.43 on Saturday, and $3.50 on Sunday.
4.58 + 7.43 + 3.50 = 15.51.
so the money does he have left = $15.51.

Question 21.
Higher Order Thinking Ms. Webster works 4 days a week at her office and 1 day a week at home. The route to Ms. Webster’s office is 23.7 miles. The route home is 21.8 miles. About how many miles does she drive for work each week? Explain how you found your answer.

Answer:
The number of miles does she drive for work each week = $116.6.

Explanation:
In the above-given question,
given that,
Ms. Webster works 4 days a week at her office and 1 day a week at home.
The route to Ms. Webster’s office is 23.7 miles.
The route home is 21.8 miles.
23.7 x 4 = 94.8.
21.8 x 1 = 21.8.
94.8 + 21.8 = 116.6
so the number of miles does she drive for work each week = $116.6.

Assessment Practice

Question 22.
Rounding to the nearest tenth, which of the following give an underestimate?
39.45 × 1.7
27.54 × 0.74
9.91 × 8.74
78.95 × 1.26
18.19 × 2.28

Answer:
The equations are 39.45 x 1.7 and 27.54 x 0.74.

Explanation:
In the above-given question,
given that,
the equations are:
39.45 x 1.7 = 67.065.
27.54 x 0.74 = 20.3796.
9.91 x 8.74 = 86.6134.
78.95 x 1.26 = 99.477.
18.19 x 2.28 = 41.4732.

Question 23.
Rounding to the nearest whole number, which of the following give an overestimate?
11.6 × 9.5
4.49 × 8.3
12.9 × 0.9
0.62 × 1.5
8.46 × 7.38

Answer:
The equation is 8.46 x 7.38.

Explanation:
In the above-given question,
given that,
the equations are:
11.6 x 9.5 = 110.2.
4.49 x 8.3 = 37.267.
12.9 x 0.9 = 11.61.
0.62 x 1.5 = 0.93.
8.46 x 7.38 = 62.4348.

Lesson 4.3 Use Models to Multiply a Decimal and a Whole Number

Activity

Solve & Share
Mara has 4 garden plots. Each is 0.7 acre in area. What is the total area of the garden plots? Use objects or the grids below to show your work.

How can you represent multiplying a decimal and a whole number?

Look Back! Critique Reasoning Ed says a decimal grid shows 10 tenths. Monica says a decimal grid shows 100 hundredths. Who is correct? Explain.

Visual Learning Bridge

Essential Question How Can You Model Multiplying a Decimal by a Whole Number?

A.
How can you use models to find 4 × 0.36?
When showing decimals, it is important to establish which type of block represents 1.

You can use place-value blocks to show multiplication of a decimal by a whole number.

B.
Multiplying 4 × 0.36 is like combining 4 groups each containing 0.36.

Regrouping after combining the blocks gives:

Convince Me! Make Sense and Persevere Bari made a train with 5 cars that are each 1.27 meter long. What is the total length of the train? Use place-value blocks to model the problem. Then find the product using an equation and compare the answers.

Guided Practice

Do You Understand?

Question 1.
Without multiplying, is 4 × 0.36 less than or greater than 4? Explain.

Answer:
4 x 0.36 is greater than 4.

Explanation:
In the above-given question,
given that,
the numbers are 0.36 and 4.
4 x 0.36 = 1.44.
1.44 is greater than 4.
so 4 x 0.36 is greater than 4.

Do You Know How?

In 2-5, find the product. You may use place-value blocks to help.

Question 2.
0.8 × 4

Answer:
The product is 3.2.

Explanation:
In the above-given question,
given that,
the numbers are 0.8 and 4.
multiply the two numbers.
0.8 x 4 = 3.2.
so the product is 3.2.

Question 3.
0.7 × 3

Answer:
The product is 2.1.

Explanation:
In the above-given question,
given that,
the numbers are 0.7 and 3.
multiply the two numbers.
0.7 x 3 = 2.1.
so the product is 2.1.

Question 4.
0.5 × 6

Answer:
The product is 3.0.

Explanation:
In the above-given question,
given that,
the numbers are 0.5 and 6.
multiply the two numbers.
0.5 x 6 = 3.0.
so the product is 3.0.

Question 5.
0.6 × 5

Answer:
The product is 3.0.

Explanation:
In the above-given question,
given that,
the numbers are 0.6 and 5.
multiply the two numbers.
0.6 x 5 = 3.0.
so the product is 3.0.

Independent Practice

Use or draw place-value blocks to help you model the problem.

In 6 and 7, find the product. Use place-value blocks to help.

Question 6.
0.55 × 3 = ___

Answer:
The product is 1.65.

Explanation:
In the above-given question,
given that,
the numbers are 0.55 and 3.
multiply the two numbers.
0.55 x 3 = 1.65.
so the product is 1.65.

Question 7.
___ = 0.45 × 2

Answer:
The product is 0.9.

Explanation:
In the above-given question,
given that,
the numbers are 0.45 and 2.
multiply the two numbers.
0.45 x 2 = 0.9.
so the product is 0.9.

In 8-19, find the product. Use place-value blocks to help.

Question 8.
5 × 0.5

Answer:
The product is 2.5.

Explanation:
In the above-given question,
given that,
the numbers are 0.5 and 5.
multiply the two numbers.
0.5 x 5 = 2.5.
so the product is 2.5.

Question 9.
4 × 0.27

Answer:
The product is 1.08.

Explanation:
In the above-given question,
given that,
the numbers are 0.27 and 4.
multiply the two numbers.
0.27 x 4 = 1.08.
so the product is 1.08.

Question 10.
6 × 0.13

Answer:
The product is 0.78.

Explanation:
In the above-given question,
given that,
the numbers are 0.13 and 6.
multiply the two numbers.
0.13 x 6 = 0.78.
so the product is 0.78.

Question 11.
0.78 × 5

Answer:
The product is 3.9.

Explanation:
In the above-given question,
given that,
the numbers are 0.78 and 5.
multiply the two numbers.
0.78 x 5 = 3.9.
so the product is 3.9.

Question 12.
10 × 0.32

Answer:
The product is 3.2.

Explanation:
In the above-given question,
given that,
the numbers are 0.32 and 10.
multiply the two numbers.
0.32 x 10 = 3.2.
so the product is 3.2.

Question 13.
6 × 2.03

Answer:
The product is 12.18.

Explanation:
In the above-given question,
given that,
the numbers are 6 and 2.03.
multiply the two numbers.
6 x 2.03 = 12.18.
so the product is 12.18.

Question 14.
1.35 × 5

Answer:
The product is 6.75.

Explanation:
In the above-given question,
given that,
the numbers are 1.35 and 5.
multiply the two numbers.
1.35 x 5 = 6.75.
so the product is 6.75.

Question 15.
100 × 0.12

Answer:
The product is 12.

Explanation:
In the above-given question,
given that,
the numbers are 100 and 0.12.
multiply the two numbers.
100 x 0.12 = 12.
so the product is 12.

Question 16.
4 × 0.15

Answer:
The product is 0.6.

Explanation:
In the above-given question,
given that,
the numbers are 0.15 and 4.
multiply the two numbers.
0.15 x 4 = 0.6.
so the product is 0.6.

Question 17.
3 × 2.5

Answer:
The product is 7.5.

Explanation:
In the above-given question,
given that,
the numbers are 3 and 2.5.
multiply the two numbers.
3 x 2.5 = 7.5.
so the product is 7.5.

Question 18.
0.9 × 7

Answer:
The product is 6.3.

Explanation:
In the above-given question,
given that,
the numbers are 0.9 and 7.
multiply the two numbers.
0.9 x 7 = 6.3.
so the product is 6.3.

Question 19.
0.35 × 3

Answer:
The product is 1.05.

Explanation:
In the above-given question,
given that,
the numbers are 0.35 and 3.
multiply the two numbers.
0.35 x 3 = 1.05.
so the product is 1.05.

Problem Solving

Question 20.
A city is building 3 parks in a new subdivision. Each park will be 1.25 acres. How many total acres will the 3 parks be? Use place-value blocks to model the problem if you need help.

Answer:
The total number of acres will the 3 parks be 3.75 acres.

Explanation:
In the above-given question,
given that,
A city is building 3 parks in a new subdivision.
Each park will be 1.25 acres.
1.25 x 3 = 3.75.
so the total number of acres will the 3 parks be 3.75 acres.

How is multiplying with decimals like multiplying whole numbers?

Question 21.
Higher Order Thinking The city acquired more land next to the subdivision. If it decides to make each park 12.5 acres, how many additional acres would the parks occupy?

Answer:
The number of additional acres would the parks occupy = 8.75 acres.

Explanation:
In the above-given question,
given that,
The city acquired more land next to the subdivision.
If it decides to make each park 12.5 acres.
3.75 – 12.5 = 8.75.
so the number of additional acres would the parks occupy = 8.75 acres.

Question 22.
Write a multiplication equation that matches the shading on the grid.

Answer:
2 + 0.4 = 2.4.

Explanation:
In the above-given question,
given that,
there are 2 hundred blocks.
1 + 1 = 2.
there are 4 tens blocks.
0.4.
2 + 0.4 = 2.4

Question 23.
Critique Reasoning Jen multiplied 9 by 0.989 and got an answer of 89.01. How can you use estimation to show that Jen’s answer is wrong? What mistake do you think she made?

Answer:
Yes, the estimation was wrong.

Explanation:
In the above-given question,
given that,
Jen multiplied 9 by 0.989 and got an answer of 89.01.
0.989 x 9 = 8.901.
so the estimation shows that Jen’s answer is wrong.

Assessment Practice

Question 24.
Anita needs 5 pounds of bananas to make banana bread for a bake sale. Each pound of bananas costs $0.50.
Part A
How can Anita use place-value blocks to find the total cost of the bananas? What is the total cost?

Part B
How can Anita use what she knows about whole-number multiplication to check her answer?

Answer:
The total cost of the bananas = $2.5.

Explanation:
In the above-given question,
given that,
Anita needs 5 pounds of bananas to make banana bread for a bake sale.
Each pound of bananas costs $0.50.
$0.50 x 5 = $2.5.
so the total cost of the bananas = $2.5.

Lesson 4.4 Multiply a Decimal and a Whole Number

Solve & Share

A car travels 1.15 kilometers in 1 minute. If it travels at a constant speed, how far will it travel in 3 minutes? in 5 minutes? Solve this problem any way you choose!

Answer:
The far will it travel in 3 minutes = 3.45.
the far will it travel in 5 minutes = 5.75.

Explanation:
In the above-given question,
given that,
A car travels 1.15 kilometers in 1 minute.
If it travels at a constant speed,
1.15 x 3 = 3.45.
1.15 x 5 = 5.75.

Generalize You can connect what you know about whole-number multiplication to multiplying a decimal by a whole number.

Look Back! How can addition be used to answer the questions above?

Visual Learning Bridge

Essential Question How Do You Multiply a Decimal by a Whole Number?

A.
To raise money for a charity, $0.15 was collected from every ticket sold to a Lions baseball game. If you bought 12 tickets, how much money would go to the charity?

You can multiply 0.15 × 12 by thinking about 15 × 12 and using place-value patterns.

B.
One Way
Add 0.15 twelve times.

C.
Another Way
First, multiply as you would with whole numbers.

Use number sense to place the decimal point.

Notice the digits are the same.

Convince Me! Generalize Here are two similar problems:

Place the decimal point correctly in each answer. Explain your thinking.

Guided Practice

Do You Understand?

Question 1.
What is the difference between multiplying a whole number by a decimal and multiplying two whole numbers?

Answer:
The answer is the same.

Explanation:
In the above-given question,
given that,
33 x 19 = 627.
0.33 x 19 = 627.
so the answer is the same.

Question 2.
If a group bought 24 tickets, how much money would go to charity? Explain how you found your answer.

Answer:
The money would go to charity = 3.36.

Explanation:
In the above-given question,
given that,
If a group bought 24 tickets.
24 x 0.14 = 3.36.
so the money would go to charity = 3.36.

Do You Know How?

For 3-8, find each product.

Question 3.

Answer:
The product is 19.6.

Explanation:
In the above-given question,
given that,
the two numbers are 9.8 and 2.
multiply the two numbers.
9.8 x 2 = 19.6.
so the product is 19.6.

Question 4.

Answer:

The product is 5.36.

Explanation:
In the above-given question,
given that,
the two numbers are 0.67 and 8.
multiply the two numbers.
0.67 x 8 = 5.36.
so the product is 5.36.

Question 5.
34 × 5.3

Answer:
The product is 180.2.

Explanation:
In the above-given question,
given that,
the two numbers are 34 and 5.3.
multiply the two numbers.
34 x 5.3 = 180.2.
so the product is 19.6.

Question 6.
4.6 × 21

Answer:
The product is 96.6.

Explanation:
In the above-given question,
given that,
the two numbers are 4.6 and 21.
multiply the two numbers.
4.6 x 21 = 96.6.
so the product is 96.6.

Question 7.
0.6 × 15
Answer:
The product is 9.

Explanation:
In the above-given question,
given that,
the two numbers are 0.6 and 15.
multiply the two numbers.
0.6 x 15 = 9.
so the product is 9.

Question 8.
55 × 1.1

Answer:
The product is 60.5.

Explanation:
In the above-given question,
given that,
the two numbers are 55 and 1.1.
multiply the two numbers.
55 x 1.1 = 60.5.
so the product is 60.5.

Independent Practice

For 9-20, find each product.

Use what you know about whole-number multiplication and place value to help you!

Question 9.

Answer:
The product is 311.4.

Explanation:
In the above-given question,
given that,
the two numbers are 34.6 and 9.
multiply the two numbers.
34.6 x 9 = 311.4.
so the product is 311.4.

Question 10.

Answer:
The product is 1284.

Explanation:
In the above-given question,
given that,
the two numbers are 64.2 and 20.
multiply the two numbers.
64.2 x 20 = 1284.
so the product is 1284.

Question 11.

Answer:
The product is 8.8.

Explanation:
In the above-given question,
given that,
the two numbers are 40 and 0.22.
multiply the two numbers.
40 x 0.22 = 8.8.
so the product is 8.8.

Question 12.

Answer:
The product is 131.1.

Explanation:
In the above-given question,
given that,
the two numbers are 57 and 2.3.
multiply the two numbers.
57 x 2.3 = 131.1.
so the product is 131.1.

Question 13.
5.8 × 11

Answer:
The product is 63.8.

Explanation:
In the above-given question,
given that,
the two numbers are 5.8 and 11.
multiply the two numbers.
5.8 x 11 = 63.8.
so the product is 63.8.

Question 14.
56 × 0.4

Answer:
The product is 22.4.

Explanation:
In the above-given question,
given that,
the two numbers are 56 and 0.4.
multiply the two numbers.
56 x 0.4 = 22.4.
so the product is 22.4.

Question 15.
170 × 0.003

Answer:
The product is 0.51.

Explanation:
In the above-given question,
given that,
the two numbers are 170 and 0.003.
multiply the two numbers.
170 x 0.003 = 0.51.
so the product is 0.51.

Question 16.
0.3 × 99

Answer:
The product is 29.7.

Explanation:
In the above-given question,
given that,
the two numbers are 0.3 and 99.
multiply the two numbers.
0.3 x 99 = 29.7.
so the product is 29.7.

Question 17.
26 × 1.61

Answer:
The product is 41.86.

Explanation:
In the above-given question,
given that,
the two numbers are 26 and 1.61.
multiply the two numbers.
26 x 1.61 = 41.86.
so the product is 41.86.

Question 18.
50 × 0.914

Answer:
The product is 45.7.

Explanation:
In the above-given question,
given that,
the two numbers are 50 and 0.914.
multiply the two numbers.
50 x 0.914 = 45.7.
so the product is 45.7.

Question 19.
10.76 × 100

Answer:
The product is 1076.

Explanation:
In the above-given question,
given that,
the two numbers are 10.76 and 100.
multiply the two numbers.
10.76 x 100 = 1076.
so the product is 1076.

Question 20.
2.54 × 12

Answer:
The product is 30.48.

Explanation:
In the above-given question,
given that,
the two numbers are 2.54 and 12.
multiply the two numbers.
2.54 x 12 = 30.48.
so the product is 30.48.

Problem Solving

Question 21.
enVision® STEM To meet peak energy demand, an electric power cooperative buys back electricity generated locally. They pay $0.07 per solar-powered kWh (kilowatt-hour). How much money does a school make when it sells back 956 kWh to the cooperative?

Round and estimate to check for reasonableness.

Answer:
The money does a school makes when it sells back 956 kWh to the cooperative = $66.92.

Explanation:
In the above-given question,
given that,
an electric power cooperative buys back electricity generated locally.
They pay $0.07 per solar-powered kWh (kilowatt-hour).
956 x 0.07 = 66.92.
so the money doe a school makes when it sells back 956 kWh to the cooperative = $66.92.

Question 22.
The airline that Vince is using has a baggage weight limit of 41 pounds. He has two green bags, each weighing 18.4 pounds, and one blue bag weighing 3.7 pounds. Are his bags within the weight limit? Explain.

Answer:
Yes, the bags within the weight limit.

Explanation:
In the above-given question,
given that,
The airline that Vince is using has a baggage weight limit of 41 pounds.
He has two green bags, each weighing 18.4 pounds.
18.4 + 3.7 = 22.1.
so the bags are within the limit.

Question 23.
Michael keeps track of how much time he uses his family’s computer each week for 10 weeks. He created the frequency table with the data he collected. How many hours did Michael spend on the computer?

Answer:
The number of hours did Michael spend on the computer = 17.

Explanation:
In the above-given question,
given that,
Michael keeps track of how much time he uses his family’s computer each week for 10 weeks.
3.5 + 4 + 4.5 + 5 = 17.
so the number of hours did Micheal spend on the computer = 17.

Question 24.
Critique Reasoning Sara is multiplying two factors, one is a whole number and one has two decimal places. She says the product could have one decimal place. Is she correct? Give an example and explain your reasoning.

Answer:
Yes, she was correct.

Explanation:
In the above-given question,
given that,
23 and 23.4.
23 x 23.4 = 538.2.
so she was correct..

Question 25.
Higher Order Thinking Heather clears a rectangular region in her yard for a garden. If the length is a one-digit whole number and the width is 5.5 meters, what is the least possible area? What is the greatest possible area? Explain how you found your answers.

Answer:
The least possible area = 60.5.
the greatest possible area = 126.5.

Explanation:
In the above-given question,
given that,
Heather clears a rectangular region in her yard for a garden.
If the length is a one-digit whole number and the width is 5.5 meters.
the whole number is 11.
11 x 5.5 = 60.5.
the whole number is 23.
23 x 5.5 = 126.5.
so the least possible area = 60.5.

Assessment Practice

Question 26.
Which of the following equations is NOT true?
A. 75 × 3 = 225
B. 75 × 0.3 = 22.5
C. 7.5 × 3 = 2.25
D. 75 × 0.03 = 2.25

Answer:
Option C is not correct.

Explanation:
In the above-given question,
given that,
7.5 x 3 = 22.5.
they have given that
7.5 x 3 = 2.25.
so option C is not correct.

Question 27.
Which of the following equations is NOT true?
A. 50 × 12 = 600
B. 50 × 0.12 = 6
C. 0.5 × 12 = 60
D. 50 × 1.2 = 60

Answer:
Option C is not correct.

Explanation:
In the above-given question,
given that,
0.5 x 12 = 60.
they have given that
0.5 x 12 = 6.
so option C is not correct.

Lesson 4.5 Use Models to Multiply a Decimal and a Decimal

Activity

Solve & Share

A rectangle has an area of 0.24 square meter. What are some possibilities for the length and width of the rectangle? Tell why. Solve this problem any way you choose. You may use hundredths grids if you like.

Can both dimensions be greater than 1 meter? Can both dimensions be less than 1 meter? Show your work in the space below.

Look Back! Generalize How did you use what you know about whole numbers and place value to find the dimensions that worked?

Visual Learning Bridge

Essential Question
How Can You Model Decimal Multiplication?

A.
A farmer has a square field that is 1 mile wide by 1 mile long. Her irrigation system can water the northern 0.5 mile of her field. If her tomatoes are planted in a strip 0.3 mile wide, what is the area of her watered tomatoes?

B.
This problem can be shown on a hundredths grid with each side representing 1 mile. The tomato area is 0.3 mile wide, so shade the first 3 columns red. The watered area is the top 0.5 miles, so shade the top 5 rows blue.

The area of the watered tomatoes is the pink area where the shading overlaps.

C.
This area can be written as a product of decimals.
0.3 × 0.5 = 0.15
There is 0.15 square miles of watered tomatoes.
Because both factors are less than 1, the product is less than either factor.

Convince Me! Be Precise Use the double hundredths grid to model 0.7 × 1.6. What does the length of each side of a hundredths grid represent? Explain how to find the product.

Answer:
Each side of the hundredths grid represents = 2.

Explanation:
In the above-given question,
given that,
0.7 x 1.6.
1.12.
so each side of the hundredths grid represents = 2.

Guided Practice

Do You Understand?

Question 1.
Write a multiplication equation to match the model.

Answer:
0.18 x 0.20 = 0.036.

Explanation:
In the above-given question,
the grid represents the
0.18 and 0.20.
0.18 x 0.20 = 0.036.

Question 2.
Explain why 2.7 is not a reasonable answer for 0.3 × 0.9. What is the correct answer?

Answer:
The correct answer is 0.27.

Explanation:
In the above-given question,
given that,
the two numbers are 0.3 and 0.9.
0.3 x 0.9 = 0.27.
so 2.7 is wrong.

Do You Know How?

In 3 and 4, shade the hundredths grids to find the product.

Question 3.
0.7 × 0.8

Answer:
The product is 0.56.

Explanation:
In the above-given question,
given that,
the two numbers are 0.7 and 0.8.
multiply the numbers.
0.7 x 0.8 = 0.56.
so the product is 0.56.

Question 4.
0.1 × 2.1

Answer:
The product is 0.21.

Explanation:
In the above-given question,
given that,
the two numbers are 0.1 and 2.1.
multiply the numbers.
0.1 x 2.1 = 0.21.
so the product is 0.21.

Independent Practice

In 5-8, shade the hundredths grids to find the product.

Remember that the area where the shading overlaps represents the product.

Question 5.
0.4 × 0.5

Answer:
The product is 0.2.

Explanation:
In the above-given question,
given that,
the two numbers are 0.4 and 0.5.
multiply the numbers.
0.4 x 0.5 = 0.2.
so the product is 0.2.

Question 6.
0.3 × 0.7

Answer:
The product is 0.21.

Explanation:
In the above-given question,
given that,
the two numbers are 0.7 and 0.3.
multiply the numbers.
0.7 x 0.3 = 0.21.
so the product is 0.21.

Question 7.
0.5 × 1.7

Answer:
The product is 0.85.

Explanation:
In the above-given question,
given that,
the two numbers are 0.5 and 1.7.
multiply the numbers.
0.5 x 1.7 = 0.85.
so the product is 0.85.

Question 8.
0.6 × 1.2

Answer:
The product is 0.72.

Explanation:
In the above-given question,
given that,
the two numbers are 0.6 and 1.2.
multiply the numbers.
0.6 x 1.2 = 0.72.
so the product is 0.72.

In 9-16, find the product. You may use grids to help.

Question 9.
0.2 × 0.8

Answer:
The product is 0.16.

Explanation:
In the above-given question,
given that,
the two numbers are 0.2 and 0.8.
multiply the numbers.
0.2 x 0.8 = 0.16.
so the product is 0.16.

Question 10.
2.4 × 0.7

Answer:
The product is 1.68.

Explanation:
In the above-given question,
given that,
the two numbers are 2.4 and 0.7.
multiply the numbers.
0.7 x 2.4 = 1.68.
so the product is 1.68.

Question 11.
3.9 × 0.4

Answer:
The product is 1.56.

Explanation:
In the above-given question,
given that,
the two numbers are 3.9 and 0.4.
multiply the numbers.
3.9 x 0.4 = 1.56.
so the product is 1.56.

Question 12.
0.5 × 0.7

Answer:
The product is 0.35.

Explanation:
In the above-given question,
given that,
the two numbers are 0.7 and 0.5.
multiply the numbers.
0.7 x 0.5 = 0.35.
so the product is 0.35.

Question 13.
0.9 × 0.1

Answer:
The product is 0.09.

Explanation:
In the above-given question,
given that,
the two numbers are 0.9 and 0.1.
multiply the numbers.
0.9 x 0.1 = 0.09.
so the product is 0.09.

Question 14.
0.2 × 1.5

Answer:
The product is 0.3.

Explanation:
In the above-given question,
given that,
the two numbers are 0.2 and 1.5.
multiply the numbers.
0.2 x 1.5 = 0.3.
so the product is 0.3.

Question 15.
0.6 × 0.6

Answer:
The product is 0.36.

Explanation:
In the above-given question,
given that,
the two numbers are 0.6 and 0.6.
multiply the numbers.
0.6 x 0.6 = 0.36.
so the product is 0.36.

Question 16.
2.8 × 0.3

Answer:
The product is 0.84.

Explanation:
In the above-given question,
given that,
the two numbers are 2.8 and 0.3.
multiply the numbers.
2.8 x 0.3 = 0.84.
so the product is 0.84.

Problem Solving

Question 17.
Model with Math Write a multiplication equation to represent this decimal model.

Answer:

Question 18.
Higher Order Thinking explain why 3.4 × 0.5 has only one decimal place in the product.

Answer:
Yes, it has only one decimal point.

Explanation:
In the above-given question,
given that,
3.4 and 0.5.
3.4 x 0.5 = 1.7.
so it has only one decimal place in the product.

Question 19.
Jack’s bookshelf has 6 shelves. Each shelf can hold 12 books. He has already placed 54 books on the shelves. How many more books can the bookshelf hold?

Answer:
The number of books can the bookshelf can hold = 9.

Explanation:
In the above-given question,
given that,
Jack’s bookshelf has 6 shelves.
Each shelf can hold 12 books.
He has already placed 54 books on the shelves.
54 / 6 = 9.
so the number of books can the bookshelf can hold = 9.

Question 20.
Number Sense Write a number that has a 6 in the thousandths place, a 5 in the hundredths place, and a 0 in the tenths place. Then write a number less than your number and a number greater than your number.

Answer:
The number is 65302.

Explanation:
In the above-given question,
given that,
Write a number that has a 6 in the thousandths place,
a 5 in the hundredths place, and a 0 in the tenths place.
the number is 65302.
the number is greater than 65322.
the number is less than 65301.

Question 21.
If you multiply two decimals less than 1, can you predict whether the product will be less than or greater than either of the factors? Explain.

Answer:
It is less than the factors.

Explanation:
In the above-given question,
given that,
If you multiply two decimals less than 1.
0.4 x 0.3 = 0.12.
it is less than the factors.

Question 22.
Judy claims that she can find 0.5 × 2.4 by dividing 2.4 into two equal parts. Is she correct? Draw a decimal model to explain your answer.

Answer:
Yes, she was correct.

Explanation:
In the above-given question,
given that,
0.5 and 2.4.
0.5 x 2.4 = 1.2.
1.2 + 1.2 = 2.4.
so she was correct.

Assessment Practice

Question 23.
Find two numbers that you can multiply to get a product of 0.54. Write the numbers in the box.

Answer:
The numbers are 0.6 and 0.9.

Explanation:
In the above-given question,
given that,
the product is 0.54.
0.6 x 0.9 = 0.54.
so the numbers are 0.6 and 0.9.

Lesson 4.6 Multiply Decimals Using Partial Products
Activity

Solve & Share

Julie has 0.5 of her backyard set up for growing vegetables. Of the vegetable area, 0.4 has bell peppers in it. What part of the backyard contains bell peppers?

You can use appropriate tools, such as a grid, to model decimal multiplication.

Look Back! What do you notice about the factors and their product in the above problem?

Visual Learning Bridge

Essential Question That How Can You Multiply Decimals Using Partial Products?

A.
June walked 1.7 miles in 1 hour. If she walks at the same rate, how far will she walk in 1.5 hours?

You can use what you know about partial products and whole numbers to model multiplication with decimals.

B.
Estimate first.

Since 2 is greater than 1.7 and 1.5, 4 is an overestimate.

C.
Like with whole numbers, you can use an area model to help break the problem into parts.

D.
The partial products make up the different pieces of the area model.

1.7 × 1.5 = 2.55, which is close to the estimate. June will walk 2.55 miles.

Convince Me! Make Sense and Persevere in the example above, how many miles will June walk in 2.8 hours? Estimate first, then compare your answer to the estimate.

Guided Practice

Do You Understand?

Question 1.
Carter is filling 6.5-ounce bottles with salsa. He was able to fill 7.5 bottles. How many ounces of salsa did he have? Draw an area model to show the partial products.

Answer:
The number of ounces

Do You Know How?

In 2-5, estimate first. Then find each product. Check that your answer is reasonable.

Question 2.

Answer:
The product is 38.13.

Explanation:
In the above-given that,
given that,
the numbers are 9.3 and 4.1.
multiply the numbers.
9.3 x 4.1 = 38.13.
so the product is 38.13.

Question 3.

Answer:
The product is 1.92.

Explanation:
In the above-given that,
given that,
the numbers are 3.2 and 0.6.
multiply the numbers.
3.2 x 0.6 = 1.92.
so the product is 1.92.

Question 4.
0.7 × 1.9

Answer:
The product is 1.33.

Explanation:
In the above-given that,
given that,
the numbers are 0.7 and 1.9.
multiply the numbers.
0.7 x 1.9 = 1.33.
so the product is 1.33.

Question 5.
12.6 × 0.2

Answer:
The product is 2.52.

Explanation:
In the above-given that,
given that,
the numbers are 12.6 and 0.2.
multiply the numbers.
12.6 x 0.2 = 2.52
so the product is 2.52.

Independent Practice

Question 6.
Find 7.5 × 1.8 using partial products. Draw an area model to show the partial products.

Answer:
The product is 13.5.

Explanation:
In the above-given that,
given that,
the numbers are 7.5 and 1.8.
multiply the numbers.
7.5 x 1.8 = 13.5.
so the product is 13.5.

In 7-14, estimate first. Then multiply using partial products. Then check that your answer is reasonable.

Question 7.

Answer:
5.2 x 4.6 = 23.92.

Explanation:
In the above-given that,
given that,
the numbers are 5.2 and 4.6.
multiply the numbers.
5.2 x 4.6 = 23.92.
so the product is 23.92.

Question 8.

Answer:
19.1 x 8.5 = 162.35

Explanation:
In the above-given that,
given that,
the numbers are 19.1 and 8.5.
multiply the numbers.
19.1 x 8.5 = 162.35.
so the product is 162.35.

Question 9.

Answer:
0.5 x 4.5 = 2.25.

Explanation:
In the above-given that,
given that,
the numbers are 0.5 and 4.5.
multiply the numbers.
0.5 x 4.5 = 2.25.
so the product is 2.25.

Question 10.

Answer:
8.6 x 0.8 = 6.88.

Explanation:
In the above-given that,
given that,
the numbers are 8.6 and 0.8.
multiply the numbers.
8.6 x 0.8 = 6.88.
so the product is 6.88.

Question 11.
5.5 × 0.6

Answer:
5.5 x 0.6 = 3.3.

Explanation:
In the above-given that,
given that,
the numbers are 5.5 and 0.6.
multiply the numbers.
5.5 x 0.6 = 3.3.
so the product is 3.3.

Question 12.
3.5 × 0.4

Answer:
3.5 x 0.4 = 1.4.

Explanation:
In the above-given that,
given that,
the numbers are 3.5 and 0.4.
multiply the numbers.
3.5 x 0.4 = 1.4.
so the product is 1.4.

Question 13.
6.8 × 7.2

Answer:
6.8 x 7.2 = 48.96.

Explanation:
In the above-given that,
given that,
the numbers are 6.8 and 7.2.
multiply the numbers.
6.8 x 7.2 = 48.96.
so the product is 48.96.

Question 14.
8.3 × 6.4

Answer:
8.3 x 6.4 = 53.12.

Explanation:
In the above-given that,
given that,
the numbers are 8.3 and 6.4.
multiply the numbers.
8.3 x 6.4 = 53.12.
so the product is 53.12.

Problem Solving

Question 15.
enVision® STEM The gravity of Venus is 0.35 times that of Jupiter. What is the gravity of Venus in relation to Earth’s gravity?

Answer:
The gravity of Venus in relation to Earth’s gravity = 0.91.

Explanation:
In the above-given question,
given that,
The gravity of Venus is 0.35 times that of Jupiter.
the gravity of the mercury = 0.39.
the gravity of Neptune = 1.22.
the gravity of Jupiter = 2.6.
2.6 x 0.35 = 0.91.
so the gravity of venus in relation to Earth’s gravity = 0.91.

Question 16.
About how many times as great is Jupiter’s relative surface gravity as Neptune’s relative surface gravity?

Answer:
The number of times as great is Jupiter’s relative surface gravity as Neptune’s relative surface gravity = 1 time.

Explanation:
In the above-given question,
given that,
the gravity of Neptune = 1.22.
the gravity of Jupiter = 2.6.
1.22 – 2.6 = 1.38.
1.38 is equal to 1.

Question 17.
One quart of water weighs about 2.1 pounds. There are 4 quarts in a gallon. How much does a gallon of water weigh?

Answer:
The gallon of water weighs = galloons.

Explanation:
In the above-given question,
given that,
One quart of water weighs about 2.1 pounds.
There are 4 quarts in a gallon.
2.1 x 4 = 9.6.
so the gallon of water weighs = 9.6 galloons.

Question 18.
Isaac bought three packages of nuts. He bought one package of peanuts that weighed 3.07 pounds. He also bought two packages of pecans that weighed 1.46 pounds and 1.5 pounds. Did the peanuts or the pecans weigh more? How much more?

Answer:
The peanuts weigh more.

Explanation:
In the above-given question,
given that,
Isaac bought three packages of nuts.
He bought one package of peanuts that weighed 3.07 pounds.
also bought two packages of pecans that weighed 1.46 pounds and 1.5 pounds.
1.46 + 1.5 = 2.96.
3.07 is greater than 2.96.
so the peanuts weigh more.

Question 19.
Make Sense and Persevere How does estimation help you place the decimal point in a product correctly? Explain your reasoning.

Answer:

Question 20.
Higher Order Thinking The area of Dimitri’s table top is a whole number of square feet. Could the length and width be decimal numbers each with one decimal place? Explain your answer.

Answer:
Yes, the answer is correct.

Explanation:
In the above-given question,
given that,
for example:
2.6 is the length.
3.4 is the width.
area = l x w.
area = 2.6 x 3.4.
area = 8.84.

Assessment Practice

Question 21.
Joy drinks 4.5 bottles of water per day. Each bottle contains 16.5 fluid ounces. How many fluid ounces of water does she drink per day?
A. 20.10 fluid Ounces
B. 64.00 fluid ounces
C. 74.25 fluid Ounces
D. 82.50 fluid Ounces

Answer:
Option C is correct.

Explanation:
In the above-given question,
given that,
Joy drinks 4.5 bottles of water per day.
Each bottle contains 16.5 fluid ounces.
4.5 x 16.5 = 74.25.
so option C is correct.

Question 22.
One square mile equals 2.6 square kilometers. How many square kilometers are in 14.4 square miles?
A. 11.52 square kilometers
B. 17.00 square kilometers
C. 37.44 square kilometers
D. 86.40 square kilometers

Answer:
Option C is correct.

Explanation:
In the above-given question,
given that,
One square mile equals 2.6 square kilometers.
14.4 x 2.6 = 37.44.
so the number of square kilometers are in 14.4 is 37.44 miles.

Lesson 4.7 Use Properties to Multiply Decimals

Activity

Solve & Share
The weight of a small bag of raisins is 0.3 times the weight of a large bag. The large bag weighs 0.8 pound. What is the weight of the small bag? Solve this problem any way you choose.

Answer:
The weight of the small bag = 0.24.

Explanation:
In the above-given question,
given that,
The weight of a small bag of raisins is 0.3 times the weight of a large bag.
The large bag weighs 0.8 pounds.
0.8 x 0.3 = 0.24.
so the weight of the small bag = 0.24.

You can estimate whether the answer is greater than or less than 0.5 pound.

Look Back! Look for Relationships How is solving this problem like finding the product of 3 and 8? How is it different?

Visual Learning Bridge

Essential Question How Can You Use Properties to Multiply Decimals?

A.
Entries for a poster contest must be 0.5 m2 or smaller. Can Jemelle enter her poster?

Use what you know about decimals and properties to multiply 0.6 × 0.9.

B.
Step 1
Rewrite the multiplication expression.
0.6 × 0.9 = (6 × 0.1) × (9 × 0.1)
Rearrange the factors using the Associative and Commutative Properties.
(6 × 0.1) × (9 × 0.1) = (6 × 9) × (0.1 × 0.1)

C.
Step 2
Multiply the whole numbers. Multiply the decimals. Write the product.
(6 × 9) × (0.1 × 0.1) = 54 × 0.01 = 0.54
One tenth of one tenth is one hundredth. 0.1 × 0.1 = 0.01.

0.54 > 0.5,
so Jemelle cannot enter her poster.

Convince Me! Use Structure Tyler explained how he multiplies 0.7 × 0.2. “I multiply 7 × 2 = 14. I know that a tenth times a tenth is a hundredth, so I use hundredths to write the product. The product is 0.14.” Use properties to show that Tyler is correct.

Answer:
Yes, Tailor was correct.

Explanation:
In the above-given question,
given that,
Tyler explained how he multiplies 0.7 × 0.2. “I multiply 7 × 2 = 14.
0.7 x 0.2 = 0.14.
so Tailor was correct.

Another example
A slice of bread has 1.25 grams of fat. How many grams of fat are in 1.5 slices?
1.25 × 1.5 =
= (125 × 0.01) (15 × 0.1)
= (125 × 15) × (0.01 × 0.1)
= 1,875 × 0.001
= 1.875

The picture shows that one tenth of one hundredth is one thousandth. 0.1 × 0.01 = 0.001

There are 1.875 grams of fat in 1.5 slices of bread.

Guided Practice

Do You Understand?

Question 1.
Mason is multiplying (3 × 5) × (0.1 × 0.1). What decimal multiplication problem is he solving?

Answer:
3 x 5 x 0.1 x 0.1 = 0.15.

Explanation:
In the above-given question,
given that,
Mason is multiplying (3 x 5) x (0.1 x 0.1).
15 x (0.1 x 0.1).
15 x 0.01.
0.15.

Question 2.
Complete Mason’s work to find the product.

Answer:
3 x 5 x 0.1 x 0.1 = 0.15.

Explanation:
In the above-given question,
given that,
Mason is multiplying (3 x 5) x (0.1 x 0.1).
15 x (0.1 x 0.1).
15 x 0.01.
0.15.

Do You Know How?

In 3-6, use properties to find each product.

Question 3.
0.3 × 0.7

Answer:
0.3 x 0.7 = 0.21.

Explanation:
In the above-given question,
given that,
the two numbers are 0.3 and 0.7.
0.3 x 0.7.
3 x 0.1 = 0.3.
7 x 0.1 = 0.7.
(3 x 7) x (0.1 x 0.1).
(21) x (0.01).
0.21.

Question 4.
0.63 × 2.8

Answer:
0.63 x 2.8 = 17.64.

Explanation:
In the above-given question,
given that,
the two numbers are 0.63 and 2.8.
0.63 x 2.8.
63 x 0.1 = 0.63.
28 x 0.1 = 0.7.
(63 x 28) x (0.1 x 0.1).
(1764) x (0.01).
17.64.

Question 5.
2.6 × 1.4

Answer:
2.6 x 1.4 = 3.64.

Explanation:
In the above-given question,
given that,
the two numbers are 2.6 and 1.4.
2.6 x 1.4.
2.6 x 0.1 = 2.6.
1.4 x 0.1 = 1.4.
(2.6 x 1.4) x (0.1 x 0.1).
(3.64) x (0.01).
3.64.

Question 6.
4.5 × 0.08

Answer:
4.5 x 0.08 = 3.6.

Explanation:
In the above-given question,
given that,
the two numbers are 0.3 and 0.7.
4.5 x 0.08.
45 x 0.1 = 4.5.
8 x 0.01 = 0.08.
(45 x 8) x (0.1 x 0.01).
(360) x (0.01).
3.6.

Independent Practice

In 7-15, find each product.

Question 7.
0.6 × 0.2

Answer:
The product is 0.12.

Explanation:
In the above-given question,
given that,
the numbers are 0.6 and 0.2.
multiply the two numbers.
0.6 x 0.2 = 0.12.

Question 8.
0.33 × 0.8

Answer:
The product is 0.264.

Explanation:
In the above-given question,
given that,
the numbers are 0.33 and 0.8.
multiply the two numbers.
0.33 x 0.8 = 0.264.

Question 9.
1.7 × 0.22

Answer:
The product is 0.374.

Explanation:
In the above-given question,
given that,
the numbers are 1.7 and 0.22.
multiply the two numbers.
1.7 x 0.22 = 0.374.

Question 10.
1.8 × 0.9

Answer:
The product is 1.62.

Explanation:
In the above-given question,
given that,
the numbers are 0.9 and 1.8.
multiply the two numbers.
0.9 x 1.8 = 1.62.

Question 11.
0.03 × 1.6

Answer:
The product is 0.48.

Explanation:
In the above-given question,
given that,
the numbers are 1.6 and 0.03.
multiply the two numbers.
1.6 x 0.03 = 0.48.

Question 12.
4.2 × 4.2

Answer:
The product is 17.64.

Explanation:
In the above-given question,
given that,
the numbers are 4.2 and 4.2.
multiply the two numbers.
4.2 x 4.2 = 17.64.

Question 13.
11.1 × 0.8

Answer:
The product is 8.88.

Explanation:
In the above-given question,
given that,
the numbers are 0.8 and 11.1.
multiply the two numbers.
0.8 x 11.1 = 8.88.

Question 14.
1.16 × 0.4

Answer:
The product is 0.464.

Explanation:
In the above-given question,
given that,
the numbers are 1.16 and 0.4.
multiply the two numbers.
1.16 x 0.4 = 0.464.

Question 15.
1.6 × 0.01

Answer:
The product is 0.016.

Explanation:
In the above-given question,
given that,
the numbers are 1.6 and 0.01.
multiply the two numbers.
1.6 x 0.01 = 0.016.

Problem Solving

Question 16.
The total rainfall in March was 3.6 inches. In April, the total rainfall was 1.4 times as much. What was the total rainfall in April?

Answer:
The total rainfall in April = 5.04.

Explanation:
In the above-given question,
given that,
The total rainfall in March was 3.6 inches.
In April, the total rainfall was 1.4 times as much.
3.6 x 1.4 = 5.04.
so the total rainfall in April = 5.04.

Question 17.
A newly hatched alligator is 0.5 foot long. An adult alligator is 16.4 times as long. How many feet longer is the adult alligator than the newborn alligator?

Answer:
The longer is the adult alligator than the newborn alligator = 8.2 feet.

Explanation:
In the above-given question,
given that,
A newly hatched alligator is 0.5 feet long.
An adult alligator is 16.4 times as long.
16.4 x 0.5 = 8.2.
so the longer is the adult alligator than the newborn alligator = 8.2 feet.

Question 18.
Make Sense and Persevere The Nature Club held a grasshopper jumping contest. The distance Bugmaster jumped is 1.2 times the distance Green Lightning jumped. The distance Top Hopper jumped is 1.5 times the distance Bugmaster jumped. Complete the table to show the distances Bugmaster and Top Hopper jumped.

Answer:
The Bugmaster and Top Hopper jumped = 1.68 feet and 2.1 feet.

Explanation:
In the above-given question,
given that,
The Nature Club held a grasshopper jumping contest.
The distance Bugmaster jumped is 1.2 times the distance green Lightning jumped.
The distance Top Hopper jumped is 1.5 times the distance Bugmaster jumped.
1.4 x 1.2 = 1.68.
1.4 x 1.5 = 2.1.

Question 19.
Amanda bought a 6-cup bag of shredded cheese for $6.89. She used 2.25 cups to make lasagna and 1.25 cups to make pizza. How much cheese is left?
Is there any information that you don’t need to solve this problem?

Answer:
The quantity of cheese left = $3.39.

Explanation:
In the above-given question,
given that,
Amanda bought a 6-cup bag of shredded cheese for $6.89.
She used 2.25 cups to make lasagna and 1.25 cups to make pizza.
2.25 + 1.25 = 3.5.
6.89 – 3.5 = 3.39.
so the quantity of cheese left = 3.39.

Question 20.
Higher Order Thinking Jodi drew the Eiffel Tower 6.5 inches tall. She thought it was too tall, so she multiplied its height by 0.8. The second drawing was too short, so she multiplied its height by 1.2. Predict whether her last drawing was shorter, the same as, or taller than her first drawing. Check your prediction by finding the height of the last drawing.

Answer:
The taller than her first drawing = 4.5 inches.

Explanation:
In the above-given question,
given that,
Jodi drew the Eiffel Tower 6.5 inches tall.
She thought it was too tall, so she multiplied its height by 0.8.
The second drawing was too short, so she multiplied its height by 1.2.
0.8 + 1.2 = 2.
6.5 – 2 = 4.5.
so the taller than her first drawing = 4.5 inches.

Assessment Practice

Question 21.
Which expression is equivalent to 0.4 × 0.3?
A. (4 × 0.01) × (3 × 0.01)
B. (4 × 10) × (3 × 10)
C. (4 × 0.1) × (3 × 0.1)
D. (4 × 0.1) × (3 × 0.01)

Answer:
Option C is correct.

Explanation:
In the above-given question,
given that,
0.4 x 0.3 = 0.12.
4 x 0.1 = 0.4.
3 x 0.1 = 0.3.
0.4 x 0.3 = 0.12.
so option C is correct.

Question 22.
Which expression is equivalent to 0.71 × 2.8?
A. (71 × 28) × (0.01 × 0.1)
B. (71 × 28) × (0.1 × 0.1)
C. (0.71 × 2.8) × (0.01 × 0.1)
D. (71 × 28) × (100 × 10)

Answer:
Option C is correct.

Explanation:
In the above-given question,
given that,
0.71 x 2.8 = 1.96.
0.71 x 2.8 x 0.01 x 0.1 = 1.96.
so option C is correct.

Lesson 4.8 Use Number Sense to Multiply Decimals

Activity

Solve & Share

Three students in Ms. Cho’s class wrote the following problems on the board. The correct digits in the products are given, but the decimal point isn’t placed yet. Where should the decimal point go in each product?

You can use reasoning to consider the size of each factor when placing the decimal point. Show your work!

Look Back! Generalize If both factors are less than 1, what do you know about their product?

Visual Learning Bridge

Essential Question
How Can You Use Number Sense Question to Multiply Decimals?

A.
You have learned how to estimate when multiplying with decimals. You can also use number sense to reason about the relative size of factors and the product.

You can use number sense to put the decimal point in the correct place.

49.20 × 0.55 = 2706

B.
Think about the relative size of the factors.
Multiplying a number by a decimal less than 1 gives a product less than the other factor. Since 0.55 is less than 1, the product is less than 49.2.
Since 0.55 is about one half, the product is about half of 49.2, or about half of 50. So, the decimal point
1 should be between the 7 and 0.
49.2 × 0.55 = 27.06

C.
Use number sense to reason about the product.
How can you place the decimal point in the product of the equation below?
6.2 × 5.1 is 3162.
Notice that the smallest unit in both factors is a tenth (0.1). Since the product of 0.1 and 0.1 is 0.01, the product of 6.2 and 5.1 will have two decimal places.
So, 6.2 × 5.1 = 31.62.
This makes sense because 6 times 5 equals 30.

Convince Me! Construct Arguments The decimal point is missing in the answer for each of these problems. Use number sense to decide where the decimal point should be placed. Explain your thinking.
54.7 × 0.53 = 28991
54.7 × 5.3 = 28991

Guided Practice

Do You Understand?

Question 1.
Describe the unknown factor.
____ × 5.1 is about 300.

Answer:
The unknown factor is 58.823.

Explanation:
In the above-given question,
given that,
300/5.1 = 58.8.
58.8 x 5.1 = 300.
the unknown factor is 58.823.

Do You Know How?

Question 2.
Number Sense Janelle wrote 23.4 for the product of 7.8 × 0.3. Use number sense to decide if Janelle placed the decimal point in the correct place in the product. If it is incorrect, give the correct product.

Answer:
7.8 x 0.3 = 2.34.

Explanation:
In the above-given question,
given that,
Janelle wrote 23.4 for the product of 7.8 × 0.3.
7.8 x 0.3 = 2.34.
so the product is correct.

Use number sense to decide where the decimal point belongs in the product.

Question 3.
5 × 3.4 = 17

Answer:
5 x 3.4 = 17.

Explanation:
In the above-given question,
given that,
the two numbers are 5 and 3.4.
multiply the two numbers.
5 x 3.4 = 17.

Question 4.
1922 = 3.1 × 6.2

Answer:
3.1 x 6.2 = 19.22.

Explanation:
In the above-given question,
given that,
the two numbers are 3.1 and 6.2.
multiply the two numbers.
3.1 x 6.2 = 19.22.

Question 5.
0.6 × 0.4 = 24

Answer:
0.6 x 0.4 = 0.24.

Explanation:
In the above-given question,
given that,
the two numbers are 0.6 and 0.4.
multiply the two numbers.
0.6 x 0.4 = 0.24.

Independent Practice

In 6-9, the product is shown without the decimal point. Use number sense to place the decimal point correctly.

Question 6.
5.01 × 3 = 1503

Answer:
5.01 x 3 = 15.03.

Explanation:
In the above-given question,
given that,
the two numbers are 5.01 and 3.
multiply the two numbers.
5.01 x 3 = 15.03.

Question 7.
6.22 × 3 = 1866

Answer:
6.22 x 3 = 18.66.

Explanation:
In the above-given question,
given that,
the two numbers are 6.22 and 3.
multiply the two numbers.
6.22 x 3 = 18.66.

Question 8.
81 = 0.9 × 0.9

Answer:
0.9 x 0.9 = 0.81.

Explanation:
In the above-given question,
given that,
the two numbers are 0.9 and 0.9.
multiply the two numbers.
0.9 x 0.9 = 0.81.

Question 9.
1.8 × 1.9 = 342

Answer:
1.8 x 1.9 = 3.42.

Explanation:
In the above-given question,
given that,
the two numbers are 1.8 and 1.9
multiply the two numbers.
1.8 x 1.9 = 3.42.

In 10-15, tell whether or not the decimal point has been placed correctly in the product. If not, rewrite the product with the decimal point correctly placed.

Use number sense or estimation to help you!

Question 10.
12 × 4.8 = 57.6

Answer:
12 x 4.8 = 57.6.

Explanation:
In the above-given question,
given that,
the two numbers are 12 and 4.8.
multiply the two numbers.
12 x 4.8 = 57.6.

Question 11.
5.2 × 6.4 = 3.328

Answer:
5.2 x 6.4 = 33.28.

Explanation:
In the above-given question,
given that,
the two numbers are 5.2 and 6.4.
multiply the two numbers.
5.2 x 6.4 = 33.28.

Question 12.
6.99 × 21 = 14.679

Answer:
6.99 x 21 = 146.79.

Explanation:
In the above-given question,
given that,
the two numbers are 6.99 and 21.
multiply the two numbers.
6.99 x 21 = 146.79.

Question 13.
6.2 = 0.05 × 12.4

Answer:
0.05 x 12.4 = 0.62.

Explanation:
In the above-given question,
given that,
the two numbers are 0.05 and 12.4.
multiply the two numbers.
0.05 x 12.4 = 0.62.

Question 14.
60.84 = 18 × 3.38

Answer:
18 x 3.38 = 60.84.

Explanation:
In the above-given question,
given that,
the two numbers are 18 and 3.38.
multiply the two numbers.
18 x 3.38 = 60.84.

Question 15.
9.01 × 91 = 81.991

Answer:
9.01 x 91 = 819.91.

Explanation:
In the above-given question,
given that,
the two numbers are 9.01 and 91.
multiply the two numbers.
9.01 x 91 = 819.91.

Problem Solving

Question 16.
A pig farmer needs 60 square feet to house a sow. Is the pen pictured to the right large enough? Explain your reasoning

Answer:
The pen is not pictured to the right large enough.

Explanation:
In the above-given question,
given that,
A pig farmer needs 60 square feet to house a sow.
10.5 x 6.4 = 67.2.
so the pen is not pictured to the right large enough.

Question 17.
Critique Reasoning Quincey says that 3 is a good estimate for 3.4 × 0.09. Is he correct? Why?

Answer:
Yes, 3 is a good estimate for 3.4 x 0.09.

Explanation:
In the above-given question,
given that,
3.4 x 0.09 = 0.306.
so 3 is a good estimate for 3.4 x 0.09.

Question 18.
Ron bought 2 DVDs for $12.95 each. He spent $25 on magazines. Did he spend more on DVDs or magazines? How much more? Write equations to show your work.

Answer:
He spends more on DVDs.

Explanation:
In the above-given question,
given that,
Ron bought 2 DVDs for $12.95 each.
He spent $25 on magazines.
he spend more on DVDs.

Question 19.
You can convert gallons to liters by using a factor of 3.79. That is, 1 gallon is about 3.79 liters. About how many liters are in 37 gallons? Is your answer an underestimate or overestimate? Explain.

Answer:
Yes, the answer is an overestimate.

Explanation:
In the above-given question,
given that,
You can convert gallons to liters by using a factor of 3.79.
That is, 1 gallon is about 3.79 liters.
37 x 3.79 = 140.23.
so the answer is overestimated.

Question 20.
Higher Order Thinking Find two factors that would give a product of 0.22.

Answer:
The two factors are 0.11 and 2.

Explanation:
In the above-given question,
given that,
the two factors are 0.11 and 2.
0.11 x 2 = 0.22.
so the factors are 0.11 and 2.

Assessment Practice

Question 21.
Which of the following is the missing factor?
___ × 2.3 = 34.73
A. 0.151
B. 1.51
C. 15.1
D. 151

Answer:
The missing factor is 15.1.

Explanation:
In the above-given question,
given that,
15.1 x 2.3 = 34.73.
34.73 / 2.3 = 15.1.
so the missing factor is 15.1.

Question 22.
Which two factors would give a product of 7.5?
A. 0.3 and 0.25
B. 0.3 and 2.5
C. 3 and 0.25
D. 3 and 2.5

Answer:
Option D is correct.

Explanation:
In the above-given question,
given that,
3 x 2.5 = 7.5.
so option D is correct.

Lesson 4.9 Model with Math

Activity

Problem Solving

Solve&Share
Susan is making sandwiches for a picnic. She needs 1.2 pounds of ham, 1.5 pounds of bologna, and 2 pounds of cheese. How much will she spend in all? Solve this problem any way you choose. Use models to help.

Answer:
The amount he spends in all = 4.7.

Explanation:
In the above-given question,
given that,
Susan is making sandwiches for a picnic.
She needs 1.2 pounds of ham.
1.5 pounds of bologna, and 2 pounds of cheese.
1.2 + 1.5 + 2 = 4.7.
so the amount he spends in all = 4.7.

Thinking Habits
Be a good thinker! These questions can help you.
• How can I use math I know to help solve this problem?
• How can I use pictures, objects, or an equation to represent the problem?
• How can I use numbers, words, and symbols to solve the problem?

Look Back! What math did you use to solve this problem?

Visual Learning Bridge

Essential Question
How Can You Model a Problem with an Equation?

A.
Alex is buying vegetables for dinner. He buys 6 ears of corn, 1.4 pounds of green beans, and 2.5 pounds of potatoes. How much money does he spend?

What do I need to do to solve the problem?
I need to find how much money Alex spends on vegetables.

Here’s my thinking…

B.
How can I model with math?
I can
• use previously learned concepts and skills.
• decide what steps need to be completed to find the final answer
• use an equation to represent and solve this problem.

C.
I will use an equation to represent this situation.
Let t be the total cost.
t = (6 × $0.35) + (1.4 × $1.80) + (2.5 × $0.70)
Multiply with money as you would multiply with decimals.

Now add the subtotals.
$2.10 + $2.52 + $1.75 = $6.37
So, Alex spends $6.37 on vegetables.

Convince Me! Model with Math Beth buys 3.2 pounds of potatoes and gives the clerk a $5 bill. Write an equation that shows how much change she will get back. Explain how your equation represents this problem.

Answer:
The change she will get back = 30.

Explanation:
In the above-given question,
given that,
Beth buys 3.2 pounds of potatoes and gives the clerk a $5 bill.
1 pound = 100.
1 dollar = 70.
3.2 x 100 = 320.
5 x 70 = 350.
350 – 320 = 30.
so the change she will get back = 30.

Guided Practice
Model with Math Jackie downloaded 14 songs priced at $0.99 each and 1 song for $1.29. She had a coupon for $2.50. What was the total amount Jackie paid?

Answer:
The total amount Jackie paid = $2.28.

Explanation:
In the above-given question,
given that,
Jackie downloaded 14 songs priced at $0.99 each and 1 song for $1.29.
she had a coupon for $2.50.
0.99 + 1.29 = 2.28.
so the total amount Jackie paid = $2.28.

You can model with math by writing an equation to show how the quantities in the problem are related.

Question 1.
What do you need to find first?

Answer:
The total amount Jackie paid = $2.28.

Explanation:
In the above-given question,
given that,
Jackie downloaded 14 songs priced at $0.99 each and 1 song for $1.29.
she had a coupon for $2.50.
0.99 + 1.29 = 2.28.
so the total amount Jackie paid = $2.28.

Question 2.
Write an equation to represent the problem.

Answer:
The total amount Jackie paid = $2.28.

Explanation:
In the above-given question,
given that,
Jackie downloaded 14 songs priced at $0.99 each and 1 song for $1.29.
she had a coupon for $2.50.
0.99 + 1.29 = 2.28.
so the total amount Jackie paid = $2.28.

Question 3.
What is the solution to the problem?

Answer:
The total amount Jackie paid = $2.28.

Explanation:
In the above-given question,
given that,
Jackie downloaded 14 songs priced at $0.99 each and 1 song for $1.29.
she had a coupon for $2.50.
0.99 + 1.29 = 2.28.
so the total amount Jackie paid = $2.28.

Independent Practice

Model with Math George bought 2.5 pounds of each type of fruit shown on the sign. What was the total cost of the fruit he bought?

Question 4.
What do you need to find?

Answer:
The total cost of the fruit he bought = $2.635.

Explanation:
In the above-given question,
given that,
George bought 2.5 pounds of each type of fruit shown on the sign.
the cost of Apples is $1.30/lb.
the cost of Grapes is $1.65/lb.
the cost of Bananas is $0.49/lb.
1.30 + 1.65 + 0.49 = 2.635.
so the total cost of the fruit he bought = $2.635.

Question 5.
Write an equation to represent the problem.

Answer:
The total cost of the fruit he bought = $2.635.

Explanation:
In the above-given question,
given that,
George bought 2.5 pounds of each type of fruit shown on the sign.
the cost of Apples is $1.30/lb.
the cost of Grapes is $1.65/lb.
the cost of Bananas is $0.49/lb.
1.30 + 1.65 + 0.49 = 2.635.
so the total cost of the fruit he bought = $2.635.

Question 6.
What is the solution to the problem?

Answer:
The total cost of the fruit he bought = $2.635.

Explanation:
In the above-given question,
given that,
George bought 2.5 pounds of each type of fruit shown on the sign.
the cost of Apples is $1.30/lb.
the cost of Grapes is $1.65/lb.
the cost of Bananas is $0.49/lb.
1.30 + 1.65 + 0.49 = 2.635.
so the total cost of the fruit he bought = $2.635.

Problem Solving

Performance Task
Coin Collection
Tina and Shannon counted the coins in their coin collections. Tina discovered that she had 538 more coins than Shannon. Whose collection is worth more? How much more?

Question 7.
Make Sense and Persevere Do you need all of the information given to solve the problem? Explain.

Answer:
Yes, she was correct.

Explanation:
In the above-given question,
given that,
Tina and Shannon counted the coins in their coin collections.
Tina discovered that she had 538 more coins than Shannon.
917 + 100 + 45 + 19 = 1081.
488 + 23 + 10 + 22 = 543.
1081 – 543 = 538.

Question 8.
Reasoning How is finding the total value of the coins in Tina’s collection similar to finding the total value of coins in Shannon’s collection?

Answer:
The total value of the coins in Tina’s collection is 1081.

Explanation:
In the above-given question,
given that,
Tina and Shannon counted the coins in their coin collections.
Tina discovered that she had 538 more coins than Shannon.
917 + 100 + 45 + 19 = 1081.
488 + 23 + 10 + 22 = 543.
1081 – 543 = 538.

Deciding what steps need to be completed to find the final answer can help you model with math.

Question 9.
Model with Math Write and solve an equation to represent the total value of the coins in Tina’s collection. Then write and solve an equation to represent the total value of the coins in Shannon’s collection.

Answer:
The total value of the coins in Shanno’s collection = 543.

Explanation:
In the above-given question,
given that,
Tina and Shannon counted the coins in their coin collections.
Tina discovered that she had 538 more coins than Shannon.
917 + 100 + 45 + 19 = 1081.
488 + 23 + 10 + 22 = 543.
1081 – 543 = 538.

Question 10.
Be Precise Whose collection is worth more? How much more? Show your work.

Answer:
Tina’s collection is worth more.

Explanation:
In the above-given question,
given that,
Tina and Shannon counted the coins in their coin collections.
Tina discovered that she had 538 more coins than Shannon.
917 + 100 + 45 + 19 = 1081.
488 + 23 + 10 + 22 = 543.
1081 – 543 = 538.

Topic 4 Fluency Practice

Activity

Find a match

Work with a partner. Point to a clue.
Read the clue.
Look below the clues to find a match. Write the clue letter in the box above the match.
Find a match for every clue.

Clues

Answer:
51 x 63 = 3213.
10 x 10 = 100.
42 x 11 = 462.
20 x 12 = 240.
15 x 17 = 255.
40 x 23 = 920.
331 x 29 = 9599.
25 x 16 = 400.

Explanation:
In the above-given question,
given that,
the product 240 is equal to A.
the product 100 is equal to B.
the product 462 is equal to C.
the product 255 is equal to D.
the product 400 is equal to E.
product 9 is equal to F.
product 3 is equal to G.
product 9 is equal to H.

Topic 4 Vocabulary Review

Glossory

Word List
• compatible numbers
• estimate
• exponent
• overestimate
• partial products
• power of 10
• product
• underestimate

Write always, sometimes, or never.

Question 1.
Multiplying a decimal by 104 shifts the decimal point 4 places to the right.

Answer:
104 x 2.1 = 218.4.

Explanation:
In the above-given question,
given that,
104 x 2.1 = 218.4.
we will use compatible numbers.

Question 2.
The product of a whole number and a decimal number is a whole number.

Answer:
The product of a whole number and a decimal number is a whole number.

Explanation:
In the above-given question,
given that,
for example:
if we divide the whole number by decimal number.
we will get decimal numbers.
so the product of a whole number and a decimal number is a whole number.

Question 3.
The product of two decimal numbers less than 1 is greater than either factor.

Answer:
0.1 x 0.2 = 0.02.

Explanation:
In the above-given question,
given that,
0.1 x 0.2.
0.02.
the two decimal numbers is less than 1.

Question 4.
The product of a number multiplied by 0.1 is 10 times as much as multiplying the same number by 0.01.

Answer:
0.1 x 10 = 1.

Explanation:
In the above-given question,
given that,
The product of a number multiplied by 0.1.
0.1 x 10 = 1.
but they have given the 0.01.
so the expression was wrong.

Cross out the numbers that are NOT powers of 10.

Question 5.
10⁶ 40 × 10³ 1 × 0.001 0.55 0.001

Answer:
The numbers that are not powers of 10 are 0.55 and 0.001.

Explanation:
In the above-given question,
given that,
the numbers are 10⁶ 40 × 10³ 1 × 0.001 0.55 0.001.
0.55 and 0.001 are not the powers of 10.
so the numbers that are not powers of 10 are 0.55 and 0.001.

Draw a line from each number in Column A to the same value in Column B. Column A

Answer:
7.2 x 1000 = 7.2 x 10 x 10 x 10.
0.38 x 10000 = 3800.
240 x 0.03 = 7.2.
3.8 x 0.1 = 0.38.
0.08 x 0.9 = 0.072.

Explanation:
In the above-given question,
given that,
the numbers are 7.2 x 1000.
7.2 x 1000 = 7.2 x 10 x 10 x 10.
0.38 x 10000 = 3800.
240 x 0.03 = 7.2.
3.8 x 0.1 = 0.38.
0.08 x 0.9 = 0.072.

Use Vocabulary in Writing

Question 11.
The digits in the product of 0.48 and a decimal number between 350 and 400 are 182136. Explain how to correctly place the decimal point without knowing the other factor. Then place the decimal point in the product.

Answer:
The missing number is 379.

Explanation:
In the above-given question,
given that,
The digits in the product of 0.48 and a decimal number between 350 and 400 are 182136.
182136/0.48 = 379.
so the number between 350 and 400 is 379.

Topic 4 Reteaching

Set A
pages 129-132, 133-136

Use the patterns in this table to find 8.56 × 10 and 0.36 × 100.

Remember you can use rounding or compatible numbers to estimate.

Find each product.

Question 1.
10 × 4.5

Answer:
10 x 4.5 = 45.

Explanation:
In the above-given question,
given that,
the two numbers are 10 and 4.5.
multiply the numbers.
10 x 4.5 = 45.

Question 2.
10³ × 3.67

Answer:
3.67 x 1000 = 3670.

Explanation:
In the above-given question,
given that,
the two numbers are 3.67 and 1000.
multiply the numbers.
3.67 x 1000 = 3670.

Question 3.
100 × 4.5

Answer:
100 x 4.5 = 450.

Explanation:
In the above-given question,
given that,
the two numbers are 100 and 4.5.
multiply the numbers.
100 x 4.5 = 450.

Question 4.
0.008 × 10²

Answer:
0.08 x 100 = 8.

Explanation:
In the above-given question,
given that,
the two numbers are 100 and 0.08.
multiply the numbers.
0.08 x 100 = 8.

Estimate each product.

Question 5.
0.38 × 99

Answer:
0.38 x 99 = 3.42.

Explanation:
In the above-given question,
given that,
the two numbers are 0.38 and 99.
multiply the numbers.
0.38 x 99 = 3.42.

Question 6.
8 × 56.7

Answer:
8 x 56.7 = 453.6.

Explanation:
In the above-given question,
given that,
the two numbers are 8 and 56.7.
multiply the numbers.
8 x 56.7 = 453.6.

Question 7.
11 × 4.89

Answer:
11 x 4.89 = 53.79.

Explanation:
In the above-given question,
given that,
the two numbers are 11 and 4.8.
multiply the numbers.
11 x 4.89 = 53.79.

Question 8.
24 × 3.9

Answer:
24 x 3.9 = 93.6.

Explanation:
In the above-given question,
given that,
the two numbers are 24 and 3.9.
multiply the numbers.
24 x 3.9 = 93.6.

Set B
pages 137-140, 141-144
Find 3 × 0.15.
Model the multiplication as repeated addition with place-value blocks.

So, 3 × 0.15 = 0.45.

Remember to estimate by rounding or using compatible numbers to check the reasonableness of the answer.

Find each product. Use place-value blocks as necessary

Question 1.
50 × 3.67

Answer:
50 x 3.67 = 183.5.

Explanation:
In the above-given question,
given that,
the two numbers are 50 and 3.67.
multiply the numbers.
50 x 3.67 = 183.5.

Question 2.
5.86 × 5

Answer:
5.86 x 5 = 29.3.

Explanation:
In the above-given question,
given that,
the two numbers are 5 and 5.86.
multiply the numbers.
5.86 x 5 = 29.3.

Question 3.
14 × 9.67

Answer:
14 x 9.67 = 135.38.

Explanation:
In the above-given question,
given that,
the two numbers are 14 and 9.67.
multiply the numbers.
14 x 9.67 = 135.38.

Question 4.
8 × 56.7

Answer:
8 x 56.7 = 453.6.

Explanation:
In the above-given question,
given that,
the two numbers are 8 and 56.7.
multiply the numbers.
8 x 56.7 = 453.6.

Question 5.
11 × 0.06

Answer:
11 x 0.06 = 0.66.

Explanation:
In the above-given question,
given that,
the two numbers are 11 and 0.06.
multiply the numbers.
11 x 0.06 = 0.66.

Question 6.
2.03 × 6

Answer:
2.03 x 6 = 12.18.

Explanation:
In the above-given question,
given that,
the two numbers are 2.03 and 6.
multiply the numbers.
2.03 x 6 = 12.18.

Question 7.
25 × 1.63

Answer:
25 x 1.63 = 40.75.

Explanation:
In the above-given question,
given that,
the two numbers are 25 and 1.63.
multiply the numbers.
25 x 1.63 = 40.75.

Question 8.
5.62 × 75

Answer:
5.62 x 75 = 421.5.

Explanation:
In the above-given question,
given that,
the two numbers are 5.62 and 75.
multiply the numbers.
5.62 x 75 = 421.5.

Set C
pages 145-148, 149-152

Find 0.2 × 1.7.
Model using hundredths grids.

The product is where the shading overlaps. So, 0.2 × 1.7 = 0.34.

Find each product.

Question 1.
1.3 × 0.4

Answer:
The product is 0.52.

Explanation:
In the above-given question,
given that,
the two numbers are 1.3 and 0.4.
multiply the numbers.
1.3 x 0.4 = 0.52.
so the product is 0.52.

Question 2.
5.8 × 5.2

Answer:
The product is 30.16.

Explanation:
In the above-given question,
given that,
the two numbers are 5.8 and 5.2.
multiply the numbers.
5.8 x 5.2 = 30.16.
so the product is 30.16.

Question 3.
8.3 × 10.7

Answer:
The product is 30.16.

Explanation:
In the above-given question,
given that,
the two numbers are 5.8 and 5.2.
multiply the numbers.
5.8 x 5.2 = 30.16.
so the product is 30.16.

Question 4.
3.4 × 0.7

Answer:
The product is 2.38.

Explanation:
In the above-given question,
given that,
the two numbers are 3.4 and 0.7.
multiply the numbers.
3.4 x 0.7 = 2.38.
so the product is 2.38.

Question 5.
2.4 × 3.6

Answer:
The product is 8.64.

Explanation:
In the above-given question,
given that,
the two numbers are 2.4 and 3.6.
multiply the numbers.
2.4 x 3.6 = 8.64.
so the product is 8.64.

Question 6.
9.7 × 11.2

Answer:
The product is 108.64.

Explanation:
In the above-given question,
given that,
the two numbers are 9.7 and 11.2.
multiply the numbers.
9.7 x 11.2 = 108.64.
so the product is 108.64.

Question 7.
1.5 × 0.6

Answer:
The product is 0.9.

Explanation:
In the above-given question,
given that,
the two numbers are 1.5 and 0.6.
multiply the numbers.
1.5 x 0.6 = 0.9.
so the product is 0.9.

Question 8.
67.5 × 9.2

Answer:
The product is 621.

Explanation:
In the above-given question,
given that,
the two numbers are 67.5 and 9.2.
multiply the numbers.
67.5 x 9.2 = 621.
so the product is 621.

Set D
pages 153-156 Use properties to find 0.8 × 0.4.

Rewrite each decimal as a product of a whole number and a decimal. Next, use the Associative and Commutative Properties to rearrange the factors.
0.8 × 0.4 =
= (8 × 0.1) × (4 × 0.1)
= (8 × 4) (0.1 × 0.1)
= 32 × 0.01
= 0.32
So, 0.8 × 0.4 = 0.32

Remember if two factors less than one are multiplied, their product is less than either factor.

Use properties to find each product.

Question 1.
0.6 × 0.3

Answer:
0.6 x 0.3 = 18.

Explanation:
In the above-given question,
given that,
the two numbers are 0.6 and 0.3.
6 x 0.1 = 0.6.
3 x 0.1 = 0.7.
6 x 3 = 18.
18 x 0.1

Question 2.
2.5 × 0.7

Answer:
2.5 x 0.7 = 1.75.

Explanation:
In the above-given question,
given that,
the two numbers are 2.5 and 0.7.
25 x 0.1 = 2.5.
7 x 0.1 = 0.7.
25 x 7 x 0.01 = 1.75.

Question 3.
0.04 × 1.9

Answer:
0.04 x 1.9 = 0.076.

Explanation:
In the above-given question,
given that,
the two numbers are 0.04 and 1.9.
0.04 x 1.9 = 0.076.

Question 4.
0.23 × 0.8

Answer:
0.23 x 0.8 = 0.184.

Explanation:
In the above-given question,
given that,
23 x 0.1 = 0.23.
8 x 0.1 = 0.8.
23 x 8 = 184.
0.1 x 0.1 = 0.01.
0.23 x 0.8 = 0.184.

Question 5.
0.1 × 8.2

Answer:
0.1 x 8.2 = 0.82.

Explanation:
In the above-given question,
given that,
1 x 0.1 = 0.1.
82 x 0.1 = 0.82.
1 x 82 = 82.
0.1 x 0.1 = 0.01.
0.1 x 8.2 = 0.82.

Question 6.
5.7 × 3.6

Answer:
5.7 x 3.6 = 20.52.

Explanation:
In the above-given question,
given that,
57 x 0.1 = 5.7.
36 x 0.1 = 3.6.
57 x 36 x 0.01 = 20.52.

Question 7.
4.2 × 6.5

Answer:
4.2 x 6.5 = 27.3.

Explanation:
In the above-given question,
given that,
42 x 0.1 = 4.2.
65 x 0.1 = 6.5.
4.2 x 6.5 = 27.3.

Question 8.
9.11 × 0.3

Answer:
9.11 x 0.3 = 2.733.

Explanation:
In the above-given question,
given that,
911 x 0.01 = 9.11.
3 x 0.1 = 0.3.
9.11 x 0.3 = 2.733.

Set E
pages 157-160
The decimal is missing in the product below. Use number sense to place the decimal point correctly.
43.5 × 1.7 = 7395

Since 1.7 is greater than 1, the product will be greater than 43.5. Since 43.5 is about 40 and 1.7 is about 2, the answer will be about 80. The decimal point should be between the 3 and the 9.
So, 43.5 × 1.7 = 73.95.

Remember that it may be Reteaching helpful to compare each factor to 1 in order to determine the relative size of the product.

The decimal point is missing in each product. Use number sense to place the decimal point correctly.

Question 1.
4 × 0.21 = 84

Answer:
4 x 0.21 = 0.84.

Explanation:
In the above-given question,
given that,
the two numbers are 4 and 0.21.
multiply the numbers.
4 x 0.21 = 0.84.

Question 2.
4.5 × 6.2 = 279
Answer:

Question 3.
7 × 21.6 = 1512
Answer:

Question 4.
6.4 × 3.2 = 2048
Answer:

Question 5.
31.5 × 0.01 = 315
Answer:

Question 6.
1.4 × 52.3 = 7322
Answer:

Question 7.
0.12 × 0.9 = 108
Answer:

Question 8.
12.5 × 163.2 = 2040
Answer:

Set F
pages 161-164
Think about these questions to help you model with math.

Thinking Habits

• How can I use math I know to help solve this problem?
• How can I use numbers, words, and symbols to solve the problem?
• How can I use pictures, objects, or an equation to represent the problem?

Remember that you can write an equation to show how the quantities in a problem are related.
Mr. Jennings made the stained glass window below with the dimensions shown. What is the total area of the window?

Question 1.
What do you need to find first?
Answer:

Question 2.
Write an equation to model the problem. Then solve the problem.
Answer:

Patti went to the bakery. She bought a loaf of bread for $3.49, 6 muffins that cost $1.25 each, and a bottle of juice for $1.79. She gave the cashier a $20 bill. How much change should Patti receive?

Question 3.
What do you need to find first?

Answer:
The change should Patti receive = $13.47.

Explanation:
In the above-given question,
given that,
Patti went to the bakery.
She bought a loaf of bread for $3.49, 6 muffins that cost $1.25 each, and a bottle of juice for $1.79.
She gave the cashier a $20 bill.
3.49 + 1.25 + 1.79 =
20 – 6.53 = 13.47.
so the change should patti receive = $13.47.

Question 4.
Answer the question. Write equations to show your work.

Answer:
The change should Patti receive = $13.47.

Explanation:
In the above-given question,
given that,
Patti went to the bakery.
She bought a loaf of bread for $3.49, 6 muffins that cost $1.25 each, and a bottle of juice for $1.79.
She gave the cashier a $20 bill.
3.49 + 1.25 + 1.79 =
20 – 6.53 = 13.47.
so the change should patti receive = $13.47

Topic 4 Assessment Practice

Question 1.
A credit card is 0.76 mm thick. How thick is a stack of 103 credit cards? Explain.

Answer:
The thick is a stack of 103 credit cards = 78.28 mm thick.

Explanation:
In the above-given question,
given that,
A credit card is 0.76 mm thick.
103 x 0.76 = 78.28.
so the thick is a stack of 78.28 mm.

Question 2.
Leo has 59 bricks each measuring 0.19 m long. He lines up the bricks to make a row.
A. Estimate the length of Leo’s row of bricks. Write an equation to model your work.
B. Find the actual length of the row of bricks.

Answer:
The actual length of the row of bricks = 11.21 m.

Explanation:
In the above-given question,
given that,
Leo has 59 bricks each measuring 0.19 m long.
He lines up the bricks to make a row.
59 x 0.19 = 11.21 m.
so the actual length of the row of bricks = 11.21 m.

Question 3.
Susan colored in the decimal grid shown below. Write an expression that shows the area she colored. Then evaluate the expression.

Answer:
The area she colored = 0.24.

Explanation:
In the above-given question,
given that,
l = 0.8 and w = 0.3.
0.3 x 0.8 = 0.24.
so the area she colored = 0.24.

Question 4.
Match each expression on the left with its product.

Answer:
5 x 0.08 = 0.4.
0.5 x 0.08 = 0.04.
50 x 0.8 = 40.
5 x 0.8 = 4.

Explanation:
In the above-given question,
given that,
match each expression with its product.
5 x 0.08 = 0.4.
0.5 x 0.08 = 0.04.
50 x 0.8 = 40.
5 x 0.8 = 4.

Question 5.
Michelle bought 4.6 yards of fabric. Each yard of fabric cost $4.95. How much did Michelle spend on fabric? Enter your answer in the box.

Answer:
The Michelle spend on fabric = $22.77.

Explanation:
In the above-given question,
given that,
Michelle bought 4.6 yards of fabric.
Each yard of fabric costs $4.95.
4.6 x $4.95 = 22.77.
so the Michelle spend on fabric = $22.77.

Question 6.
Choose all the expressions that are equal to 0.75 × 0.5.

Answer:
The expressions are 5/10 x 75/100 and 75/100 x 5/10.

Explanation:
In the above-given question,
given that,
the expressions are 5/10 x 75/10, 5/10 x 75/100, 5/100 x 75/100, 50/100 x 75/100, and 75/100 x 5/10.
0.75 x 0.5 = 0.375.
5/10 x 75/10 = 0.5 x 7.5 = 3.75.
5/10 x 75/100 = 0.5 x 0.75 = 0.375.
5/100 x 75/100 = 0.05 x 0.75 = 0.0375.
50/100 x 75/100 = 0.75 x 0.05 = 0.0375.
75/100 x 5/10 = 0.75 x 0.5 = 0.375.

Question 7.
Select each equation that the number 10² makes true.
0.031 × = 31
0.501 × = 501
04.08 × = 408
0.97 × = 97
55 × = 550

Answer:
55 x 10 = 550.

Explanation:
In the above-given question that,
given that,
0.031 x 10 = 0.31.
0.501 x 10 = 5.01.
4.08 x 10 = 40.8.
0.97 x 10 = 9.7.
55 x 10 = 550.

Question 8.
Nadia drew a square in her notebook. Each side measured 2.5 centimeters.

A. What is the perimeter of Nadia’s square? Write an equation to model your work.
B. What is the area of Nadia’s square? Write an equation to model your work.

Answer:
A. The perimeter of Nadia’s square = 10 cm.
B. the area of square = 6.25 sq cm.

Explanation:
In the above-given question,
given that,
the side of the square is 2.5 cm.
perimeter of square = 4 x side.
4 x 2.5 = 10 cm.
area of square = side x side.
2.5 x 2.5 = 6.25.
so the area of the square is 6.25 cm.

Question 9.
Match each expression on the left with its product.

Answer:
3.04 x 100 = 304.
0.304 x 10000 = 3040.
3.04 x 10 = 30.4.
0.304 x 10 = 3.04.

Explanation:
In the above-given question,
given that,
match each expression with its product.
3.04 x 100 = 304.
0.304 x 10000 = 3040.
3.04 x 10 = 30.4.
0.304 x 10 = 3.04.

Question 10.
Select all the expressions that are equal to 0.09 × 0.4.

Answer:
That epressions are A and B.

Explanation:
In the above-given question,
given that,
0.09 x 0.4 = 0.036.
9/100 x 4/10 = 0.09 x 0.4 = 0.036.
4/10 x 9/100 = 0.09 x 0.4 = 0.036.
4/100 x 9/100 = 0.9 x 0.4 = 0.36.
40/100 x 9/100 = 0.4 x 0.09 = 0.036.
so the expressions are true is A and B.

Question 11.
One glass of lemonade has 115 calories. How many calories are in 3.5 glasses of lemonade? Write an equation to model your work.

Answer:
The number of calories in 3.5 glasses of lemonade = 402.5.

Explanation:
In the above-given question,
given that,
One glass of lemonade has 115 calories.
3.5 x 115 = 402.5.
so the number of calories in 3.5 glasses of lemonade = 402.5.

Question 12.
Natalie likes to mail lots of postcards when she goes on vacation. In 2014, the cost of a postcard stamp was $0.34.
A. Natalie buys 10 postcard stamps. What is the total cost of the stamps? Explain.
B. Natalie and her friends decide to buy 100 postcard stamps. What will be the total cost of the stamps? Explain.

Answer:
A. The total cost of 10 postcard stamps = 3.4.
B. The total cost of 100 postcard stamps = 34.

Explanation:
In the above-given question,
given that,
Natalie likes to mail lots of postcards when she goes on vacation.
In 2014, the cost of a postcard stamp was $0.34.
10 x 0.34 = 3.4.
100 x 0.34 = 34.
so the total cost of 10 postcard stamps = 3.4.
The total cost of 100 postcard stamps = 34.

Question 13.
Without doing the multiplication, match each expression on the left with the correct product. Use number sense to help you.

Answer:
8.32 x 1.15 = 9.568.
5.78 x 0.35 = 2.023.
6.12 x 4.85 = 29.682.
7.95 x 2.48 = 19.716.

Explanation:
In the above-given question,
given that,
match each expression with its product.
8.32 x 1.15 = 9.568.
5.78 x 0.35 = 2.023.
6.12 x 4.85 = 29.682.
7.95 x 2.48 = 19.716.

Question 14.
Derrick runs 2.25 miles each day. How many total miles will he have run after 10 days? Explain.

Answer:
The total number of miles will he have run after 10 days = 22.5 miles.

Explanation:
In the above-given question,
given that,
Derrick runs 2.25 miles each day.
2.25 x 10 = 22.5 miles.
so the total number of miles will he have run after 10 days = 22.5 miles.

Question 15.
Select each equation that the decimal 0.65 makes true.
10² × = 65
10⁴ × = 650
10¹× = 6.5
10³× = 65
10¹ × = 65

Answer:
The equation that are true is 100 x 0.65 = 65, 10 x 0.65 = 6.5.

Explanation:
In the above-given question,
given that,
100 x 0.65 = 65.
10000 x 0.65 = 6500.
10 x 0.65 = 6.5.
1000 x 0.65 = 650.
10 x 0.65 = 6.5.

Question 16.
A farmer plants 0.4 of a field with wheat. The field is 2.3 acres.
A. Shade the grids to model the multiplication.

B. How many acres are planted with wheat? Use an equation and the model to explain.

Answer:
The number of acres is planted with wheat = 5.75 acres.

Explanation:
In the above-given question,
given that,
A farmer plants 0.4 of a field with wheat.
The field is 2.3 acres.
2.3 / 0.4 = 5.75.
so the number of acres is planted with wheat = 5.75 acres.

Question 17.
Bradley walks 0.65 mile each Friday to his friend’s house. He takes a different route home that is 1.2 miles. How many miles will Bradley walk to his friend’s house and back in a year? Show your work. Reminder: there are 52 weeks in a year.

Answer:
The number of miles will Bradley walk to his friend’s house and back in a year = 0.78 miles.

Explanation:
In the above-given question,
given that,
Bradley walks 0.65 miles each Friday to his friend’s house.
He takes a different route home that is 1.2 miles.
0.65 x 1.2 = 0.78.
so the number of miles will Bradley walk to his friend’s house and back in a year = 0.78 miles.

Question 18.
Alyssa paints 3 walls blue. Each of the walls is 8.3 feet tall and 7.5 feet wide.
A. Round the length and width to the nearest whole number. Then estimate the area that Alyssa will paint. Write equations to model your work.
B. Find the exact area. Write equations to model your work.
C. Compare your estimate to the exact answer. Why is your answer reasonable?

Answer:
The exact area = 47.4.

Explanation:
In the above-given question,
given that,
Alyssa paints 3 walls blue. Each of the walls is 8.3 feet tall and 7.5 feet wide.
8.3 + 7.5 = 15.8.
15.8 x 3 = 47.4.
so the exact area is 47.4.

Question 19.
Leticia and Jamal go to a bakery.

A. Jamal wants to buy 6 muffins. How much will this cost? Write an equation to model your work.
B. Leticia wants to buy 12 bagels. She uses partial products to find her total. She says “$16.80 is close to my estimate of 12 × $1 = $12, so this total is reasonable.” Do you agree with her? Explain your reasoning

Answer:
A. The cost of 6 muffins = $1.99.
B. No, it is not reasonable.

Explanation:
In the above-given question,
given that,
Leticia and Jamal go to a bakery.
the cost of the bagel is $0.95.
the cost of the Muffin is $1.99.
1.99 x 6 = 11.94.
0.95 x 12 = 11.4.

Question 20.
The area of one fabric square is 4.85 square inches. What is the area of a quilt made with 10² fabric squares? Explain.

Answer:
The area of the quilt made with 100 fabric squares = 485 in.

Explanation:
In the above-given question,
given that,
The area of one fabric square is 4.85 square inches.
4.85 x 100 = 485.
so the area of the quilt made with 100 fabric squares = 485 sq in.

Question 21.
Jen bought 3.72 pounds of apples at a farmer’s market. Andrea bought 4 times as much as Jen.

A. Which expression represents the problem? Use the bar diagram to help.
A. 3.72 × 4
B. 3.72 × 1
C. 3.72 ÷ 4.
D. 3.72 ÷ 1

Answer:
Option C is the correct answer.

Explanation:
In the above-given question,
given that,
Jen bought 3.72 pounds of apples at a farmer’s market.
Andrea bought 4 times as much as Jen.
3.72 / 4 = 0.93.
so option C is the correct answer.

B. How many pounds of apples did Andrea buy?

Answer:
The number of pounds of apples did Andrea buy = 0.93 pounds.

Explanation:
In the above-given question,
given that,
Jen bought 3.72 pounds of apples at a farmer’s market.
Andrea bought 4 times as much as Jen.
3.72 / 4 = 0.93.
so the number of pounds of apples did Andrea buy = 0.93.

Topic 4 Performance Task

An exchange rate is how much of one country’s currency (money) you would get for another country’s currency. The table below shows the recent exchange rate between the U.S. dollar and the currency of some other countries.
For example, if you had $2 (U.S.) dollars, you could exchange that for 26.48 Mexican pesos:
2 × 13.24 = 26.48.

Question 1.
Jade has $10, Julio has $100, and Anna has $1,000. How many Japanese yen would each person get in exchange for their dollars?

Answer:
The Japanese yen would each person get in exchange for their dollars = $1.0182.

Explanation:
In the above-given question,
given that,
Jade has $10, Julio has $100, and Anna has $1,000.
Japanese yen has 101.82.
101.82/10 = 10.182.
101.82/100 = 1.0182.
101.82/1000 = 0.10182.

Question 2.
Ivana has $5.50. Luc has 1.4 times as much money as Ivana.

Part A
How many Indian rupees would Ivana get?
Part B
How many dollars does Luc have? How many rupees can he exchange for that? Round your answer to the nearest hundredth.

Answer:
The number of rupees can he exchange for that = $3.92.

Explanation:
In the above-given question,
given that,
Ivana has $5.50. Luc has 1.4 times as much money as Ivana.
5.50/1.4 = 3.92.
so the number of rupees can he exchange for that = $3.92.

Question 3.
Use estimation to solve. Marcus has $250. About how many Venezuelan bolívares could he obtain? Is your estimate an overestimate or an underestimate? Explain.

Answer:
The number of Venezuelan bolivares could he obtain = $38.

Explanation:
In the above-given question,
given that,
Marcus has $250.
250/6.59 = 37.9.
so the number of Venezuelan bolivares could he obtain = $37.9.

Question 4.
Jorge is traveling to Kenya to photograph wildlife. How many Kenyan shillings will he receive for $500? His friends give him a $50 gift certificate right before he leaves. How many Kenyan shillings can he exchange for that amount?

Answer:
The Kenyan shillings can be exchanged for that amount = 10.

Explanation:
In the above-given question,
given that,
Jorge is traveling to Kenya to photograph wildlife.
His friends give him a $50 gift certificate right before he leaves.
500/50 = 10.
so the Kenyan shillings can be exchanged for that amount = 10.

Question 5.
Kofi is taking a trip to Mexico to see Mayan and Aztec pyramids. He wants to exchange $300 before arriving there. How many Mexican pesos can he receive? Write and solve an equation to show your work.

Answer:
The number of Mexican pesos can he receive = 0.044 pesos.

Explanation:
In the above-given question,
given that,
Kofi is taking a trip to Mexico to see the Mayan and Aztec pyramids.
He wants to exchange $300 before arriving there.
13.24/300 = 0.044.
so the number of Mexican pesos can he receive = 0.044.

Question 6.
Mary is planning a vacation to Europe. The exchange rate for the U.S. dollar to the Euro is 0.73.
Part A
Sketch place-value blocks to model how many Euros Mary will get for 3 U.S. dollars.
Part B
How many Euros can Mary get for $3? Write an equation to show your work.
Part C
How does the model help you find the product?

Answer:
The number of Euros can Mary get for $3 = $0.24.

Explanation:
In the above-given question,
given that,
Mary is planning a vacation to Europe.
The exchange rate for the U.S. dollar to the Euro is 0.73.
0.73/3 = 0.24.
so the number of Euros can Mary get for $3 = $0.24