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Practice with the help of enVision Math Common Core Grade 8 Answer Key Topic 1 Real Numbers regularly and improve your accuracy in solving questions.

enVision Math Common Core 8th Grade Answers Key Topic 1 Real Numbers

enVision STEM Project

Did You Know?
Natural resources are materials that occur in nature, such as water, fossil fuels, wood, and minerals. Natural resources not only meet basic human needs, but also support industry and economy.

Minerals are used in the manufacturing of all types of common objects, including cell phones, computers, light bulbs, and medicines.

Water, oil, and forests are some of the natural resources that are in danger of someday being depleted.
70% of available fresh water is used in agriculture…

…and 10% for human consumption.

Each person in the United States needs over 48,000 pounds of minerals each year.

About 18 million acres of forest are lost to deforestation each year.

Solar power, wind power and other renewable energy sources are helping to lessen the dependency on oil and fossil fuels.

Fossil fuels are expected to supply almost 80% of world energy use through 2040.

Your Task: Going, Going, Gone?

Natural resource depletion is an important issue facing the world. Suppose a natural resource is being depleted at the rate of 1.333% per year. If there were 300 million tons of this resource in 2005, and there are no new discoveries, how much will be left in the year 2045? You and your classmates will explore the depletion of this resource over time.

Answer:
It is given that
A natural resource is being depleted at the rate of 1.333% per year and there were 300 million tons of this resource in 2005, and there are no new discoveries
So,
The number of resources left in 2045 = The number of resources present in 2005 – The decrease of the number of resources from 2005 to 2045
So,
The number of resources left in 2045 = 300 million – 1.33% of 300 million × (2045 – 2005)
The number of resources left in 2045 = 300 million – 159.6 million
The number of resources left in 2045 = 140.4 million
Hence, from the above,
We can conclude that the number of resources left in 2045 is: 140.4 million

Topic 1 GET READY!

Review What You Know!

Vocabulary

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

Question 1.
A(n) ____ is a decimal that ends in repeating zeros.
Answer:
A “Terminating decimal” is a decimal that ends in repeating zeros
Hence, from the above,
We can conclude that the best term from the box for this definition is “Terminating decimal”

Question 2.
A(n) ____ is a decimal in which a digit or digits repeat
Answer:
A “Repeating decimal” is a decimal in which a digit or digits repeat
Hence, from the above,
We can conclude that the best term from the box for this definition is “Repeating decimal”

Question 3.
A(n) ____ is either a counting number, the opposite of a counting number, or zero
Answer:
An “Integer” is either a counting number, the opposite of a counting number, or zero
Hence, from the above,
We can conclude that the best term from the box for this definition is “Integer”

Question 4.
A(n) ___ is a number that can be used to describe a part of a whole, a part of a set, a location on a number line, or a division of whole numbers.
Answer:
A “Fraction” is a number that can be used to describe a part of a whole, a part of a set, a location on a number line, or a division of whole numbers.
Hence, from the above,
We can conclude that the best term from the box for this definition is “Fraction”

Terminating and Repeating Decimals

Determine whether each decimal is terminating or repeating.

Question 5.
5.692
Answer:
The given decimal is: 5.692
We know that,
A terminating decimal is usually defined as a decimal number that contains a finite number of digits after the decimal point
Hence, from the above,
We can conclude that the given decimal is a terminating decimal

Question 6.
-0.222222…
Answer:
The given decimal is -0.222222….
We know that,
A repeating decimal or recurring decimal is the decimal representation of a number whose digits are periodic and the infinitely repeated portion is not zero
Hence, from the above,
We can conclude that the given decimal is a repeating decimal

Question 7.
7.0001
Answer:
The given decimal is: 7.0001
We know that,
A terminating decimal is usually defined as a decimal number that contains a finite number of digits after the decimal point
Hence, from the above,
We can conclude that the given decimal is a terminating decimal

Question 8.
7.2\(\overline{8}\)
Answer:
The given decimal is: 7.2\(\overline{8}\)
We know that,
A repeating decimal or recurring decimal is the decimal representation of a number whose digits are periodic and the infinitely repeated portion is not zero
Hence, from the above,
We can conclude that the given decimal is a repeating decimal

Question 9.
1.\(\overline{178}\)
Answer:
The given decimal is: 1.\(\overline{178}\)
We know that,
A repeating decimal or recurring decimal is the decimal representation of a number whose digits are periodic and the infinitely repeated portion is not zero
Hence, from the above,
We can conclude that the given decimal is a repeating decimal

Question 10.
-4.03479
Answer:
The given decimal is: -4.03479
We know that,
A terminating decimal is usually defined as a decimal number that contains a finite number of digits after the decimal point
Hence, from the above,
We can conclude that the given decimal is a terminating decimal

Multiplying Integers

Find each product.

Question 11.
2.2
Answer:
The given expression is: 2 × 2
Hence,
2 × 2 = 4

Question 12.
-5. (-5)
Answer:
The given expression is: -5 × -5
We know that,
– × – = +
Hence,
-5 × -5 = 25

Question 13.
7.7
Answer:
The given expression is: 7 × 7
Hence,
7 × 7 = 49

Question 14.
-6 ∙ (-6) ∙ (-6)
Answer:
The given expression is: -6 × -6 × -6
We know that,
– × – × –
= + × –
= –
Hence,
-6 × -6 × -6
= 36 × -6
= -216

Question 15.
10 ∙ 10 ∙ 10
Answer:
The given expression is: 10 × 10 × 10
Hence,
10 × 10 × 10 = 1,000

Question 16.
-9 ∙ (-9) ∙ (-9)
Answer:
The given expression is: -9 × -9 × -9
We know that,
– × – × –
= + × –
= –
Hence,
-9 × -9 × -9
= 81 × -9
= -729

Simplifying Expressions
Simplify each expression.

Question 17.
(4 ∙ 10) + (5 ∙ 100)
Answer:
The given expression is:
(4 × 10) + (5 × 100)
So,
(4 × 10) + (5 × 100)
= 40 + 500
= 540
Hence, from the above,
We can conclude that the value of the expression is: 540

Question 18.
(2100) + (7.10)
Answer:
The given expression is:
2100 + (7 × 10)
So,
2100 + (7 × 10)
= 2100 + 70
= 2170
Hence, from the above,
We can conclude that the value of the given expression is: 2170

Question 19.
(6 · 100) – (1 · 10)
Answer:
The given expression is:
(6 × 100) – (1 × 10)
So,
(6 × 100) – (1 × 10)
= 600 – 10
= 590
Hence, from the above,
We can conclude that the value of the given expression is: 590

Question 20.
(9 ∙ 1,000) + (4 ∙ 10)
Answer:
The given expression is:
(9 × 1,000) + (4 × 10)
So,
(9 × 1,000) + (4 × 10)
= 9,000 + 40
= 9,040
Hence, from the above,
We can conclude that the value of the given expression is: 9,040

Question 21.
(3 · 1,000) – (2 ∙ 100)
Answer:
The given expression is:
(3 × 1,000) – (2 × 100)
So,
(3 × 1,000) – (2 × 100)
= 3,000 – 200
= 2,800
Hence, from the above,
We can conclude that the value of the given expression is: 2,800

Question 22.
(2 ∙ 10) + (7 · 100)
Answer:
The given expression is:
(2 × 10) + (7 × 100)
So,
(2 × 10) + (7 × 100)
= 20 + 700
= 720
Hence, from the above,
we can conclude that the value of the given expression is: 720

Language Development

Fill in the word map with new terms, definitions, and supporting examples or illustrations.

Topic 1 PICK A PROJECT

PROJECT 1A
Who is your favorite poet, and why?
PROJECT: WRITE A POEM

PROJECT 1B
If you moved to a tiny house, what would you bring with you?
PROJECT: DESIGN A TINY HOUSE

PROJECT 1C
If you could travel anywhere in space, where would you go?
PROJECT: PLAN A TOUR OF THE MILKY WAY

PROJECT 1D
Why do you think people tell stories around a campfire?
PROJECT: TELL A FOLK STORY

Lesson 1.1 Rational Numbers as Decimals

Solve & Discuss It!

Jaylon has a wrench labeled 0.1875 inches and bolts labeled in fractions of an inch. Which size bolt will fit best with the wrench? Explain.

Answer:
It is given that
Jaylon has a wrench labeled 0.1875 inches and bolts labeled in fractions of an inch.
Now,
We know that,
The bolt will be fit in a wrench only when
The size of the bolt (inches) = The size of the wrench (inches)
Now,
The representation of the bolts in the decimal numbers is:
\(\frac{3}{8}\) = 0.375 inches
\(\frac{1}{8}\) = 0.046 inches
\(\frac{3}{16}\) = 0.1875 inches
\(\frac{1}{4}\) = 0.25 inches
Hence, from the above,
We can conclude that the bolt which has the size \(\frac{3}{16}\) inches will fit best with the wrench

Reasoning
How can you write these numbers in the same form?
Answer:
The representation of the sizes of bolts in the decimal form is by using the properties of place values
We know that,
A terminating decimal can be written as a fraction by using properties of place value.
Example:
3.75 = three and seventy-five hundredths or \(\frac{375}{100}\), which is equal to the improper fraction

Focus on math practices
Reasoning Why is it useful to write a rational number as a fraction or as a decimal?
Answer:
Rational numbers are whole numbers, fractions, and decimals – the numbers we use in our daily lives. They can be written as a ratio of two integers. … The definition says that a number is rational if you can write it in the form \(\frac{a}{b}\) where a and b are integers, and b is not zero.

? Essential Question
How can you write repeating decimals as fractions?
Answer:
Because repeating decimals are rational numbers, you can write them in fraction form.
STEP 1
Assign a variable to represent the repeating decimal.
STEP 2
Write an equation: variable = decimal.
STEP 3
Multiply each side of the equation by 10^d, where d is the number of repeating digits in the repeating decimal.
STEP 4
Subtract equivalent expressions of the variable and the repeating decimal from each side of the equation.
STEP 5
Solve for the variable. Write an equivalent fraction so that the numerator and denominator are integers, if necessary.

Try It!

In another baseball division, one team had a winning percentage of 0.444…. What fraction of their games did this team win?
The team won their games.

Answer:
It is given that
In another baseball division, one team had a winning percentage of 0.444…
Since only 1 number is repeated,
The repeating decimal can be written as 0.\(\overline{4}\)
Now,
According to the steps of converting a repeating decimal to a fraction (Essential Question),
Step 1:
Let x = 0.\(\overline{4}\)
Step 2:
Multiply with 10 on both sides since only 1 number is repeating
So,
10x = 10 (0.\(\overline{4}\))
10x = 4.\(\overline{4}\)
Step 3:
10x – x = 4.\(\overline{4}\) – 0.\(\overline{4}\)
9x = 4
x = \(\frac{4}{9}\)
Hence, from the above,
We can conclude that the team won \(\frac{4}{9}\) of their games

Convince Me!
How do you know what power of ten to multiply by in the second step at the right?
Answer:
Let x be the repeating portion.
Multiply this equation by a power of 10 to move the repeating digits immediately to the left of the decimal point (in other words, to eliminate any zeros preceding the repeating digits).

Try It!
Write the repeating decimal 0.63333… as a fraction.
Answer:
The given repeating decimal is: 0.6333333
Now,
According to the steps of converting a repeating decimal to a fraction (Essential Question),
Step 1:
Let x = 0.6\(\overline{3}\)
Step 2:
Multiply with 10 on both sides since only 1 number is repeating
So,
10x = 10 (0.6\(\overline{3}\))
10x = 6.\(\overline{3}\)
Step 3:
10x – x = 6.\(\overline{3}\) – 0.6\(\overline{3}\)
9x = 6.33 – 0.63
9x = 5.7
Divide by 9 into both sides
So,
\(\frac{9}{9}\)x = \(\frac{5.7}{9}\)
x = \(\frac{57}{90}\)
Hence, from the above,
We can conclude that the value of the given repeating decimal in the fraction form is: \(\frac{57}{90}\)

Try It!
Write the repeating decimal 4.1363636… as a fraction.
Answer:
The given repeating decimal is 4.1363636…..
Now,
According to the steps of converting a repeating decimal to a fraction (Essential Question),
Step 1:
Let x = 4.1\(\overline{36}\)
Step 2:
Multiply with 100 on both sides since 2 numbers are repeating
So,
100x = 100 (4.1\(\overline{36}\))
100x = 41.\(\overline{36}\)
Step 3:
100x – x = 41.\(\overline{36}\) – 4.1\(\overline{36}\)
99x = 413.636 – 4.136
99x = 409.5
Divide by 99 into both sides
So,
\(\frac{99}{99}\)x = \(\frac{409.5}{99}\)
x = \(\frac{4095}{990}\)
Hence, from the above,
We can conclude that the value of the given repeating decimal in the fraction form is: \(\frac{4095}{990}\)

KEY CONCEPT

Because repeating decimals are rational numbers, you can write them in fraction form.
STEP 1
Assign a variable to represent the repeating decimal.
STEP 2
Write an equation: variable = decimal.
STEP 3
Multiply each side of the equation by 10^d, where d is the number of repeating digits in the repeating decimal.
STEP 4
Subtract equivalent expressions of the variable and the repeating decimal from each side of the equation.
STEP 5
Solve for the variable. Write an equivalent fraction so that the numerator and denominator are integers, if necessary.

Do You Understand?

Question 1.
? Essential Question
How can you write repeating decimals as fractions?
Answer:
Because repeating decimals are rational numbers, you can write them in fraction form.
STEP 1
Assign a variable to represent the repeating decimal.
STEP 2
Write an equation: variable = decimal.
STEP 3
Multiply each side of the equation by 10^d, where d is the number of repeating digits in the repeating decimal.
STEP 4
Subtract equivalent expressions of the variable and the repeating decimal from each side of the equation.
STEP 5
Solve for the variable. Write an equivalent fraction so that the numerator and denominator are integers, if necessary.

Question 2.
Use Structure Why do you multiply by a power of 10 when writing a repeating decimal as a rational number?
Answer:
The idea is to multiply by some number (10, 100, 1000, etc.) so that when we subtract the original number from the multiple, the repeating part cancels out.

Question 3.
Be Precise How do you decide by which power of 10 to multiply an equation when writing a decimal with repeating digits as a fraction?
Answer:
The idea is to multiply by some number (10, 100, 1000, etc.) so that when we subtract the original number from the multiple, the repeating part cancels out.

Do You Know How?

Question 4.
A survey reported that 63.63% of moviegoers prefer action films. This percent represents a repeating decimal. Write it as a fraction.
Answer:
It is given that
A survey reported that 63.63% of moviegoers prefer action films. This percent represents a repeating decimal
So,
The given repeating decimal is 63.6363…..%
Now,
According to the steps of converting a repeating decimal to a fraction (Essential Question),
Step 1:
Let x = 63.\(\overline{63}\)
Step 2:
Multiply with 100 on both sides since 2 numbers are repeating
So,
100x = 100 (63.\(\overline{63}\))
100x = 6363.\(\overline{63}\)
Step 3:
100x – x = 6363.\(\overline{63}\) – 63.\(\overline{63}\)
99x = 6,300
Divide by 99 into both sides
So,
\(\frac{99}{99}\)x = \(\frac{6,300}{99}\)
x = \(\frac{6,300}{99}\)
Hence, from the above,
We can conclude that the value of the given repeating decimal in the fraction form is: \(\frac{6,300}{99}\)

Question 5.
A student estimates the weight of astronauts on the Moon by multiplying their weight by the decimal 0.16666… What fraction can be used for the same estimation?

Answer:
It is given that
A student estimates the weight of astronauts on the Moon by multiplying their weight by the decimal 0.16666…
Now,

Hence, from the above,
We can conclude that the fraction that can be used for the same estimation is: \(\frac{1}{6}\)

Question 6.
Write 2.3181818… as a mixed number.
Answer:
The given repeating decimal is 2.3181818…
Now,
According to the steps of converting a repeating decimal to a fraction (Essential Question),
Step 1:
Let x = 2.3\(\overline{18}\)
Step 2:
Multiply with 100 on both sides since 2 numbers are repeating
So,
100x = 100 (2.3\(\overline{18}\))
100x = 23.\(\overline{18}\)
Step 3:
100x – x = 23.\(\overline{18}\) – 2.3\(\overline{18}\)
99x = 231.818 – 2.318
99x = 229.5
Divide by 99 into both sides
So,
\(\frac{99}{99}\)x = \(\frac{229.5}{99}\)
x = \(\frac{2295}{990}\)
x = \(\frac{51}{22}\)
So,
The representation of the above fraction in the mixed form is: 2\(\frac{7}{22}\)
Hence, from the above,
We can conclude that the value of the given repeating decimal in the mixed fraction form is: 2\(\frac{7}{22}\)

Practice & Problem Solving

Leveled Practice In 7 and 8, write the decimal as a fraction or mixed number.

Question 7.
Write the number 0.21212121… as a fraction.
Let x =
100x =
100x – x = –
99x =
x =
So 0.2121… is equal to
Answer:
The given repeating decimal is: 0.212121….
Now,

Hence, from the above,
We can conclude that the value of the repeating decimal in the fraction form is: \(\frac{21}{99}\)

Question 8.
Write 3.7 as a mixed number.
Let x =
10x =
9x =
x =
So 3.\(\overline{7}\) is equal to
Answer:
The given repeating decimal is: 3.\(\overline{7}\)
Now,

Hence, from the above,
We can conclude that the representation of the given repeating decimal in the mixed fraction form is: 3\(\frac{7}{9}\)

Question 9.
Write the number shown on the scale as a fraction.

Answer
The number that is shown on the scale is: 0.233333
Now,
From the: above number,
We can observe that the number is a repeating decimal
Now,

So,
The simplified form of \(\frac{2.1}{9}\) is: \(\frac{7}{30}\)
Hence, from the above,
We can conclude that the representation of the number that is shown on the scale as a fraction is: \(\frac{7}{30}\)

Question 10.
Tomas asked 15 students whether summer break should be longer. He used his calculator to divide the number of students who said yes by the total number of students. His calculator showed the result as 0.9333…
a. Write this number as a fraction.
Answer:
The given repeating decimal is 0.93333…
Now,

So,
The simplified form of \(\frac{8.4}{9}\) is: \(\frac{14}{15}\)
Hence, from the above,
We can conclude that the representation of the repeating number in the form of the fraction is: \(\frac{14}{15}\)

b. How many students said that summer break should be longer?
Answer:
It is given that
Tomas asked 15 students whether summer break should be longer. He used his calculator to divide the number of students who said yes by the total number of students.
So,
The number of students that said summer break should be longer = \(\frac{The number of students that said yes that summer break is longer}{The total number of students}
Now,
From part (a),
The fraction form of his calculated result from part (a) is: 14 / 15
Hence, from the above,
We can conclude that the number of students that said the summer break should be longer is: 14 students

Question 11.
Write 0.[latex]\overline{87}\) as a fraction.
Answer:
The given repeating decimal is 0.\(\overline{87}\)
Now,

Hence, from the above,
We can conclude that the representation of the repeating decimal in the fraction form in the simplest form is: \(\frac{29}{33}\)

Question 12.
Write 0.\(\overline{8}\) as a fraction.
Answer:
The given repeating decimal is 0.\(\overline{8}\)
Now,

Hence, from the above,
We can conclude that the representation of the given repeating decimal in the fraction form is: \(\frac{8}{9}\)

Question 13.
Write 1.\(\overline{48}\) as a mixed number.
Answer:
The given repeating decimal number is 1.\(\overline{48}\)
Now,

Hence, from the above,
We can conclude that the representation of the given repeating decimal in the mixed fraction form is: 1\(\frac{16}{33}\)

Question 14.
Write 0.\(\overline{6}\) as a fraction.
Answer:
The given repeating decimal is 0.\(\overline{6}\)
Now,

Hence, from the above,
We can conclude that the representation of the given repeating decimal in the fraction form is: \(\frac{2}{3}\)

Question 15.
A manufacturer determines that the cost of making a computer component is $2.161616. Write the cost as a fraction and as a mixed number.

Answer:
It is given that
A manufacturer determines that the cost of making a computer component is $2.161616
So,
The given repeating decimal is 2.161616…
Now,

Hence, from the above,
We can conclude that
The cost of a computer component in a fraction form is: \(\frac{214}{99}\)
The cost of a computer component in a mixed fraction form is: 2\(\frac{16}{99}\)

Question 16.
Reasoning When writing a repeating decimal as a fraction, does the number of repeating digits you use matter? Explain.
Answer:
No. Even if the number of different digits in the cycle is 1 or 1 million, the method of finding the fraction is the same

Question 17.
Higher Order Thinking When writing a repeating decimal as a fraction, why does the fraction always have only 9s or 9s and 0s as digits in the denominator?
Answer:
When writing a repeating decimal as a fraction, the fraction always has only 9s or 9s and 0s as digits in the denominator because we are talking here about a geometric series and they are decimals, so the right side i.e., after the decimal point, the digits are in tenths, hundredths and so on

Assessment Practice

Question 18.
Which decimal is equivalent to \(\frac{188}{11}\)?
A. 17.\(\overline{09}\)
B. 17.0\(\overline{09}\)
C. 17.\(\overline{1709}\)
D. 17.\(\overline{1709}\)0
Answer:
The given fraction is: \(\frac{188}{11}\)
So,
The representation of the given fraction in the decimal form is:

we know that,
\(\frac{1}{11}\) = 0.090909…..
So,
\(\frac{188}{11}\) = 17.090909….
= 17.\(\overline{09}\)
Hence, from the above,
We can conclude that option A matches the representation of the repeating decimal for the given fraction

Question 19.
Choose the repeating decimal that is equal to the fraction on the left.

Answer:
Follow the process that is mentioned below to solve the given repeating decimals in the fraction form
Now,
STEP 1
Assign a variable to represent the repeating decimal.
STEP 2
Write an equation: variable = decimal.
STEP 3
Multiply each side of the equation by 10^d, where d is the number of repeating digits in the repeating decimal.
STEP 4
Subtract equivalent expressions of the variable and the repeating decimal from each side of the equation.
STEP 5
Solve for the variable. Write an equivalent fraction so that the numerator and denominator are integers, if necessary.
Hence,

Lesson 1.2 Understand Irrational Numbers

Explain It!
Sofia wrote a decimal as a fraction. Her classmate Nora says that her method and answer are not correct. Sofia disagrees and says that this is the method she learned.

A. Construct Arguments Is Nora or Sofia correct? Explain your reasoning.
Answer:
The given number is 0.121121112111112…..
Now,
From the given number,
We can observe that the given decimal is not a repeating decimal because there are other numbers other than the repeating numbers in the given decimal or a terminating decimal because the decimal is not finite
So,
Since the given decimal is not a repeating decimal,
The method that we used to convert the repeating decimal into a fraction is not applicable
Hence, from the above,
We can conclude that Nora is correct

B. Use Structure What is a nonterminating decimal number that can not be written as a fraction.
Answer:
A non-terminating, non-repeating decimal is a decimal number that continues endlessly, with no group of digits repeating endlessly. Decimals of this type cannot be represented as fractions, and as a result, are irrational numbers

Focus on math practices
Construct Arguments is 0.12112111211112… a rational number? Explain.
Answer:
0.12112111211112… can’t be represented in the form of \(\frac{p}{q}\) and it has non terminating non-repeating decimal expansion
Hence, from the above,
We can conclude that 0.12112111211112… is not a rational number

? Essential Question
How is an irrational number different from a rational number?
Answer:
Numbers that can be expressed in \(\frac{a}{b}\) or fraction form are rational numbers where a is an integer and b is a non-zero integer and the irrational numbers are the numbers that cannot be written in \(\frac{a}{b}\) form

Try It!
Classify each number as rational or irrational.


Answer:
The given numbers are:

Now,
We know that,
Numbers that can be expressed in \(\frac{a}{b}\) or fraction form are rational numbers where a is an integer and b is a non-zero integer and the irrational numbers are the numbers that cannot be written in \(\frac{a}{b}\) form
Hence,
The representation of the rational and irrational numbers are:

Convince Me!
Construct Arguments Jen classifies the number 4.567 as irrational because it does not repeat. Is Jen correct? Explain.
Answer:
The given decimal is: 4.567
We know that,
A rational number is a number that can be written in the form of \(\frac{a}{b}\)
A terminating decimal has the finite number of digits without repeating and it is also a rational number
So,
We can observe that we can write 4.567 as a rational number
Hence, from the above,
We can conclude that Jen is not correct

Try It!
Classify each number as rational or irrational and explain.
A) \(\frac{2}{3}\)
B) \(\sqrt{25}\)
C) -0.7\(\overline{5}\)
D) \(\sqrt{2}\)
E) 7,548,123
Answer:
The given numbers are:
A) \(\frac{2}{3}\)
B) \(\sqrt{25}\)
C) -0.7\(\overline{5}\)
D) \(\sqrt{2}\)
E) 7,548,123
Now,
We know that,
A “Rational number” is a number that can be written in the form of \(\frac{a}{b}\)
Ex: Terminating decimals, perfect squares, etc
An “Irrational number” is a number that can not be written in the form of \(\frac{a}{b}\)
Ex: Non-terminating decimals
So,
From the given numbers,
Rational numbers ——> A, B, E
Irrational numbers ——> C, D

KEY CONCEPT
Numbers that are not rational are called irrational numbers.

Do You Understand?

Question 1.
? Essential Question How is an irrational number different from a rational number?
Answer:
Numbers that can be expressed in \(\frac{a}{b}\) or fraction form are rational numbers where a is an integer and b is a non-zero integer and the irrational numbers are the numbers that cannot be written in \(\frac{a}{b}\) form

Question 2.
Reasoning How can you tell whether a square root of a whole number is rational or irrational?
Answer:
If the square root of an integer is itself an integer (Ex: √4 = 2), then by definition it is rational – If the square root is not an integer (Ex: √2 = 1.41414…), then it must be irrational. Put another way the only integers for which the square root of an integer can be rational is if is a perfect square – that is where x is an integer

Question 3.
Construct Arguments Could a number ever be both rational and irrational? Explain.
Answer:
No. A rational number is a number that can be expressed as the quotient of two integers. An irrational number is a number that cannot be expressed as a quotient of two integers. So if a number is either rational or irrational, it cannot also be the other.

Do You Know How?

Question 4.
Is the number 65.4349224… rational or irrational? Explain.
Answer:
The given number is 65.4349224…
From the given number,
We can observe that the given number is a non-repeating and non-terminating decimal number
Hence, from the above,
We can conclude that the given number is an irrational number

Question 5.
Is the number \(\sqrt{2,500}\) rational or irrational? Explain.
Answer:
The given number is: \(\sqrt{2,500}\)
We know that,
A perfect square number is a rational number
So,
\(\sqrt{2,500}\) = 50
Hence, from the above,
We can conclude that the given number is a rational number

Question 6.
Classify each number as rational or irrational.

Answer:
We know that,
A “Rational number” is a number that can be written in the form of \(\frac{a}{b}\)
Ex: Perfect squares
An “Irrational number” is a number that can not be written in the form of \(\frac{a}{b}\)
Ex: Non-terminating decimal numbers
Hence,
The representation of the rational and irrational numbers from the given numbers are:

Practice & Problem Solving

Question 7.
Is 5.787787778… a rational or irrational number? Explain.
Answer:
The given number is 5.787787778…
From the given number,
We can observe that the given number is a non-repeating and non-terminating decimal number
Hence, from the above,
We can conclude that the given number is an irrational number

Question 8.
Is \(\sqrt{42}\) rational or irrational? Explain.
Answer:
The given number is \(\sqrt{42}\)
From the given number,
We can observe that the given number is not a perfect square
Hence, from the above,
We can conclude that the given number is an irrational number

Question 9.
A teacher places seven cards, lettered A-G, on a table. Which cards show irrational numbers?

Answer:
The given cards are:

We know that,
An “Irrational number” is a number that can not be written in the form of \(\frac{a}{b}\)
Hence, from the above,
We can conclude that from the given cards,
The irrational numbers are:
A) π
B) 8.25635…,
C) 6.\(\overline{31}\)

Question 10.
Circle the irrational number in the list below.
A) 7.\(\overline{27}\)
B) \(\frac{5}{9}\)
C) \(\sqrt{15}\)
D) \(\sqrt{196}\)
Answer:
The given numbers are:
A) 7.\(\overline{27}\)
B) \(\frac{5}{9}\)
C) \(\sqrt{15}\)
D) \(\sqrt{196}\)
Now,
We know that,
A “Rational number” is a number that can be written in the form of \(\frac{a}{b}\)
Ex: Perfect squares
An “Irrational number” is a number that can not be written in the form of \(\frac{a}{b}\)
Ex: Non-terminating decimal numbers
Hence, from the above,
We can conclude that
From the given numbers,
The irrational numbers are A) and C)

Question 11.
Lisa writes the following list of numbers.
5.737737773…, 26, \(\sqrt{45}\), –\(\frac{3}{2}\), 0, 9
Answer:
The given numbers are:
A) 5.7377377737… B) 26  C) \(\sqrt{45}\)
D) –\(\frac{3}{2}\) E) 0 F) 9
Now,
We know that,
A “Rational number” is a number that can be written in the form of \(\frac{a}{b}\)
Ex: Perfect squares
An “Irrational number” is a number that can not be written in the form of \(\frac{a}{b}\)
Ex: Non-terminating decimal numbers

a. Which numbers are rational?
Answer:
From the given numbers,
The rational numbers are: B, D, E, and F

b. Which numbers are irrational?
Answer:
From the given numbers,
The irrational numbers are: A and C

Question 12.
Construct Arguments Deena says that 9.565565556… is a rational number because it has a repeating pattern. Do you agree? Explain.
Answer:
The given number is 9.565565556…
From the given number,
We can observe that the number is a non-repeating and a non-terminating decimal
So,
The given number is an irrational number
Hence, from the above,
We can conclude that we don’t have to agree with Deena

Question 13.
Is \(\sqrt{1,815}\) rational? Explain.
Answer:
The given number is: \(\sqrt{1,815}\)
We know that,
A “Rational number” is a number that can be written in the form of \(\frac{a}{b}\)
Ex: Perfect squares
Now,
From the given square root,
We can observe that it won’t form a perfect square
Hence, from the above,
We can conclude that the given number is an irrational number

Question 14.
Is the decimal form of \(\frac{13}{3}\) Explain.
Answer:
The given number is: \(\frac{13}{3}\)
We know that,
\(\frac{13}{3}\) = 4.3333…..
We know that,
An “Irrational number” is a number that can not be written in the form of \(\frac{a}{b}\)
Hence, from the above,
We can conclude that the decimal form of \(\frac{13}{3}\) is an irrational number

Question 15.
Write the side length of the square rug as a square root. Is the side length a rational or irrational number? Explain.

Answer:
The given figure is:

From the given figure,
We can observe that the given figure depicts the shape of a square
Now,
Let the side length of a square be x
We know that,
Area = (Side length)²
x² = 100
x = \(\sqrt{100}\)
We know that,
A “Perfect square” is a rational number
Hence, from the above,
We can conclude that a side length is a rational number

Question 16.
Reasoning The numbers 2.888… and 2.999… are both rational numbers. What is an irrational number that is between the two rational numbers?
Answer:
A rational number is a number which is can be represented as the quotient of two numbers without having any remainder i.e., having remainder 0. For example 2.45, 2, 3 etc.
An irrational number has a non-zero remainder and has a nonterminating quotient.
Hence,
The numbers between 2.888… and 2.999… are 2.8889………, 2.8890…….., 2.8891…… etc

Question 17.
Higher Order Thinking You are given the expressions \(\sqrt{76+n}\) and \(\sqrt{2 n+26}\). What is the smallest value of n that will make each number rational?
Answer:
The given expressions are: \(\sqrt{76+n}\) and \(\sqrt{2 n+26}\)
Now,
To find the smallest value of n so that each expression will be a rational number,
\(\sqrt{76+n}\) = \(\sqrt{2 n+26}\)
Squaring on both sides
So,
76 + n = 2n + 26
2n – n = 76 – 26
n = 50
Hence, from the above,
We can conclude that the smallest value of n so that the given expressions will become a rational number is: 5

Assessment Practice

Question 18.
Which numbers are rational?
I. 1.1111111…
II. 1.567
III. 1.101101110…
A. II and III
B. III only
C. II only
D. I and II
E I only
F. None of the above
Answer:
The given numbers are:
I. 1.1111111…
II. 1.567
III. 1.101101110…
We know that,
A “Rational number” is a number that can be written in the form of \(\frac{a}{b}\)
So,
From the given numbers,
1 and 2 are the rational numbers
Hence, from the above,
we can conclude that option D matches with the given situation

Question 19.
Determine whether the following numbers are rational or irrational.

Answer:
The representation of the given numbers as rational and irrational numbers is:

Lesson 1.3 Compare and Order Real Numbers

Solve & Discuss It!

Courtney and Malik are buying a rug to fit in a 50-square-foot space. Which rug should they purchase? Explain.

Answer:
It is given that
Courtney and Malik are buying a rug to fit in a 50-square-foot space
Now,
The given figure is:

Now,
From the above figure,
We can observe that
The rugs are in different shapes i.e., a square, a circle, and a rectangle
Now,
The area of a square rug = The side length of a square rug × The side length of a square rug
= 7 × 7
= 49 ft²
Now,
The area of a circular rug = π × \(\frac{Diameter of a circular rug²}{4}\)
= 3.14 × \(\frac{8 ×8}{4}\)
= 3.14 ×16
= 50.24 ft²
Now,
The area of a rectangular rug = Length × Width
= 6 × 8\(\frac{1}{2}\)
= 6 × \(\frac{17}{2}\)
= \(\frac{6 ×17}{2}\)
= 51 ft²
Now,
When we compare the area of the rugs,
The area of the square rug is less than 50 ft²
Hence, from the above,
We can conclude that
Courtney and Malik should buy the square rug

Focus on math practices
Make Sense and Persevere How did you decide which rug Courtney and Malik should purchase?
Answer:
It is given that
Courtney and Malik are buying a rug to fit in a 50 ft² space
So,
To fit in a 50 ft² space,
The area of any type of rug should be less than 50 ft²
Now,
From the above problem,
We can observe that
The area of the square rug is the only area that is less than 50 ft²
Hence, from the above,
We can conclude that
Courtney and Mali should purchase the rugs based on the areas of the rugs

?Essential Question
How can you compare and order rational and irrational numbers?
Answer:
In the given numbers, one of them is rational while other one is irrational. To make the comparison, let us first make the given irrational number into rational number and then carry out the comparison. So, let us square both the given numbers

Try It!
Between which two whole numbers is \(\sqrt{12}\)?

Answer:
The given number is: \(\sqrt{12}\)
Now,

Hence, from the above,
We can conclude that
\(\sqrt{12}\) is in between 3 and 4

Convince Me!
Which of the two numbers is a better estimate for \(\sqrt{12}\)? Explain.
Answer:
The given number is: \(\sqrt{12}\)
Now,
We know that,
12 will be 3² and 4²
So,
\(\sqrt{12}\) will be between 3 and 4
Now,
When we observe the numbers between 3 and 4
The value of \(\frac{12}\) will be near to 3.4
Hence, from the above,
We can conclude that
The two numbers that are better estimate for \(\sqrt{12}\) is: 3 and 4

Try It!
Compare and order the following numbers:
\(\sqrt{11}\), 2\(\frac{1}{4}\), -2.5, 3.\(\overline{6}\), -3.97621 …
Answer:
The given numbers are:
\(\sqrt{11}\), 2\(\frac{1}{4}\), -2.5, 3.\(\overline{6}\), -3.97621 …
So,
\(\sqrt{11}\) ≅ 3.3
2\(\frac{1}{4}\) = 2.25
3.\(\overline{6}\) = 3.666……
So,
The order of the numbers from the least to the greatest is:
-3.97621…….. < -2.5 < 2.25 < \(\sqrt{11}\) < 3.\(\overline{6}\)

KEY CONCEPT
To compare rational and irrational numbers, you must first find rational approximations of the irrational numbers. You can approximate irrational numbers using perfect squares or by rounding.

Do You Understand?

Question 1.
? Essential Question How can you compare and order rational and irrational numbers?
Answer:
In the given numbers, one of them is rational while other one is irrational. To make the comparison, let us first make the given irrational number into rational number and then carry out the comparison. So, let us square both the given numbers

Question 2.
Reasoning
The “leech” is a technical term for the slanted edge of a sail. Is the length of the leech shown closer to 5 meters or 6 meters? Explain.

Answer:
It is given that
The “leech” is a technical term for the slanted edge of a sail
Now,
The given figure is:

Now,
From the given figure,
We can observe that
The length of the leech is: \(\sqrt{30}\) meters
Now,
We know that,
5² < 30 < 6²
5 < \(\sqrt{30}\) < 6
Now,
We know that,
\(\sqrt{30}\) ≅ 5.4
So,
\(\sqrt{30}\) is close to 5
Hence, from the above,
We can conclude that
The length of the leech shown above is close to 5 meters

Question 3.
Construct Arguments which is a better approximation of \(\sqrt{20}\), 4.5 or 4.47? Explain.
Answer:
The given number is: \(\sqrt{20}\)
Now,
We know that,
4² < 20 < 5²
4 < \(\sqrt{20}\) < 5
Now,
We know that,
4.5² = 20.25
So,
The value of \(\sqrt{20}\) is close to 4.4
Hence, from the above,
We can conclude that
The better approximation of \(\sqrt{20}\) is: 4.4

Do You Know How?

Question 4.
Approximate \(\sqrt{39}\) to the nearest whole number.
Answer:
The given number is: \(\sqrt{39}\)
Now,
We know that,
6² < 39 < 7²
6 < \(\sqrt{39}\) < 7
Now,
We know that,
6.5² = 42.25
So,
The value of \(\sqrt{39}\) is close to 6.2
So,
The value of \(\sqrt{39}\) is closes to 6 that is the nearest whole number
Hence, from the above,
We can conclude that
The better approximation of \(\sqrt{39}\) that is the closest to the whole number is: 6

Question 5.
Approximate \(\sqrt{18}\) to the nearest tenth and plot the number on a number line.

Answer:
The given number is: \(\sqrt{18}\)
Now,
We know that,
4² < 18 < 5²
4 < \(\sqrt{18}\) < 5
Now,
We know that,
4.5² = 20.25
4.2² = 17.69
So,
\(\sqrt{18}\) is close to 4.2
Hence, from the above,
We can conclude that
The representation of the approximate value of \(\sqrt{18}\) on the given number line is:

The approximate value of \(\sqrt{18}\) is: 4.2

Question 6.
Compare 5.7145… and \(\sqrt{29}\). Show your work.
Answer:
The given numbers are: 5.7145…… and \(\sqrt{29}\)
Now,
We know that,
5² < 29 < 6²
So,
5 < \(\sqrt{29}\) < 6
Now,
We know that,
5.5² = 30.25
5.3² = 28.09
So,
The approximate value of \(\sqrt{29}\) is: 5.3
Now,
When we compare the given numbers,
We can observ ethat
5.7145….. > 5.3
Hence, from the above,
We can conclude that
The order of the given numbers is:
5.7145……….. > \(\sqrt{29}\)

Question 7.
Compare and order the following numbers
5.2, -5.\(\overline{6}\), 3\(\frac{9}{10}\), \(\sqrt{21}\)
Answer:
The given numbers are:
5.2, -5.\(\overline{6}\), 3\(\frac{9}{10}\), \(\sqrt{21}\)
Now,
We know that,
-5.\(\overline{6}\) = -5.666666…..
3\(\frac{9}{10}\) = 3.9
\(\sqrt{21}\) ≅ 4.58
So,
The order of the given numbers from the least to the greatest is:
-5.\(\overline{6}\) < 3\(\frac{9}{10}\) < \(\sqrt{21}\) < 5.2
Hence, from the above,
We can conclude that
The order of the given numbers from the least to the greatest is:
-5.\(\overline{6}\) < 3\(\frac{9}{10}\) < \(\sqrt{21}\) < 5.2

Practice & Problem Solving

Question 8.
Leveled Practice Find the rational approximation of \(\sqrt{15}\).
a. Approximate using perfect squares.
< 15 <
= < \(\sqrt{15}\) <
< \(\sqrt{15}\) <
Answer:
The given number is: \(\sqrt{15}\)
Now,
We know that,
By using the approximation using the perfect squares,

Hence, from the above,
We can conclude that
The approximate numbers that are between \(\sqrt{15}\) are: 3 and 4

b. Locate and plot \(\sqrt{15}\) on a number line. Find a better approximation using decimals.

Answer:
From Part (a),
We know that,
The approximate numbers that are between \(\sqrt{15}\) are: 3 and 4
Now,
We know that,
3.5 ² = 12.25
Now,

So,
The approximate number that is the closest to \(\sqrt{15}\) is: 3.8
Hence, from the above,
We can conclude that
The representation of the approximation of \(\sqrt{15}\) in the given number line is:

The approximate number that is close to \(\sqrt{15}\) is: 3.8

Question 9.
Compare – 1.96312… and –\(\sqrt{5}\). Show your work.
Answer:
The given numbers are: -1.96312…… and –\(\sqrt{5}\)
Now,
We know that,
2² < 5 < 3²
2 < \(\sqrt{5}\) < 3
Now,
We know that,
2.5² = 6.25
2.2² = 4.84
So,
The approximate value of –\(\sqrt{5}\) is: -2.2
So,
The order of the given numbers from the least to the greatest is:
-1.96312……. > –\(\sqrt{5}\)
Hence, from the above,
We can conclude that
The order of the given numbers from the least to the greatest is:
-1.96312……. > –\(\sqrt{5}\)

Question 10.
Does \(\frac{1}{6}\), -3, \(\sqrt{7}\), –\(\frac{6}{5}\), or 4.5 come first when the numbers are listed from least to greatest? Explain.
Answer:
The given numbers are: \(\frac{1}{6}\), -3, \(\sqrt{7}\), –\(\frac{6}{5}\), and 4.5
Now,
\(\frac{1}{6}\) = 0.166
\(\sqrt{7}\) = 2.64
–\(\frac{6}{5}\) = -1.2
So,
The order of the given numbers from the least to the greatest is:
-3 < –\(\frac{6}{5}\) < \(\frac{1}{6}\) < \(\sqrt{7}\) < 4.5
Hence, from the above,
We can conclude that
“-3” will come first when the given numbers will be arranged from the least to the greatest

Question 11.
A museum director wants to hang the painting on a wall. To the nearest foot, how tall does the wall need to be?

Answer:
It is given that
A museum director wants to hang the painting on a wall
Now,
The given figure is:

Now,
From the given figure,
We can obsere that
The painting on a wall is about \(\sqrt{90}\) ft
Now,
We know that,
9² < 90 < 10²
9 < \(\sqrt{90}\) < 10
Now,
We know that,
9.5² = 90.25
So,
The approximate value of \(\sqrt{90}\) is: 9.4 ft
Hence, from the above,
We can conclude that
The height of the wall needed to hang a painting is about 9.4 ft

Question 12.
Dina has several small clay pots. She wants to display them in order of height, from shortest to tallest. What will be the order of the pots?

Answer:
It is given that
Dina has several small clay pots. She wants to display them in order of height, from shortest to tallest
Now,
The given heights are:
\(\sqrt{8}\), 2\(\frac{1}{3}\), \(\sqrt{5}\), and 2.5
Now,
We now that,
\(\sqrt{8}\) ≅ 2.82 in.
2\(\frac{1}{3}\) = 2.033 in.
\(\sqrt{5}\) ≅ 2.23 in.
So,
The order of the heights from the shortest to the tallest is:
2\(\frac{1}{3}\) in. < \(\sqrt{5}\) in. < 2.5 in. < \(\sqrt{8}\) in.
Hence, from the above,
We can conclude that
The order of the pots is:
2\(\frac{1}{3}\) in. < \(\sqrt{5}\) in. < 2.5 in. < \(\sqrt{8}\) in.

Question 13.
Rosie is comparing \(\sqrt{7}\) and 3.44444…. She says that \(\sqrt{7}\) > 3.44444… because \(\sqrt{7}\) = 3.5.
a. What is the correct comparison?
Answer:
It is given that
Rosie is comparing \(\sqrt{7}\) and 3.44444…. She says that \(\sqrt{7}\) > 3.44444… because \(\sqrt{7}\) = 3.5.
Now,
We know that,
2² < 7 < 3²
2 < \(\sqrt{7}\) < 3
Now,
We know that,
2.6² = 6.76
2.7² = 7.29
So,
The approximate value of \(\sqrt{7}\) is: 2.6
So,
The order of the given numbers is:
\(\sqrt{7}\) < 3.44444…..
Hence, from the above,
We can conclude that
The correct comparison of the given numbers is:
\(\sqrt{7}\) < 3.44444…..

b. Critique Reasoning What mistake did Rosie likely make?
Answer:
The given numbers are: \(\sqrt{7}\) and 3.44444…..
Now,
It is also given that
\(\sqrt{7}\) = 3.5
But, 3.5 = \(\frac{7}{2}\)
Hence, from the above,
We can conclude that
The mistake Rosie likely make is:
She considered \(\sqrt{7}\) = \(\frac{7}{2}\)

Question 14.
Model with Math Approximate – √23 to the nearest tenth. Draw the point on the number line.

Answer:
The given number is: –\(\sqrt{23}\)
Now,
We know that,
4² < 23 < 5²
4 < \(\sqrt{23}\) < 5
Now,
We know that,
4.5² = 20.25
4.7² = 22.09
So,
The approximate value of –\(\sqrt{23}\) is: -4.7
Hence,
The representtaion of the approximate value of –\(\sqrt{23}\) on the given number line is:

Question 15.
Higher Order Thinking The length of a rectangle is twice the width. The area of the rectangle is 90 square units. Note that you can divide the rectangle into two squares.

a. Which irrational number represents the length of each side of the squares?
Answer:
It is given that
The length of a rectangle is twice the width. The area of the rectangle is 90 square units. Note that you can divide the rectangle into two squares.
Now,
The given figure is:

Now,
According to the given information,
The area of each square = \(\frac{90}{2}\)
= 45 square units
Now,
We know that,
The area of a square = Side²
So,
Side of a squre = \(\sqrt{The area of a square}\)
So,
The side length of each square = \(\sqrt{45}\) units
Hence, from the above,
We can conclude that
The irrational number that represents the length of each side of the squares is: \(\sqrt{45}\) units

b. Estimate the length and width of the rectangle.
Answer:
It is given that
The length of a rectangle is twice the width. The area of the rectangle is 90 square units. Note that you can divide the rectangle into two squares.
Now,
The given figure is:

Now,
Let the width of the rectangle be x units
So,
The length of the rectangle = 2 (Width) = 2x units
Now,
We know that,
The length of a rectangle = Length × Width
So,
According to the given information,
90 = 2x × x
90 = 2x²
x² = \(\frac{90}{2}\)
x = \(\sqrt{45}\)
Hence, from the above,
We can conclude that
The length of the rectangle is: 2\(\sqrt{45}\) units
The width of the rectangle is: \(\sqrt{45}\) units

Assessment Practice

Question 16.
Which list shows the numbers in order from least to greatest?
A. -4, –\(\frac{9}{4}\), \(\frac{1}{2}\), 3.7, \(\sqrt{5}\)
B. -4, –\(\frac{9}{4}\), \(\frac{1}{2}\), \(\sqrt{5}\), 3.7
C. –\(\frac{9}{4}\), \(\frac{1}{2}\), 3.7, \(\sqrt{5}\), -4
D. –\(\frac{9}{4}\), -4, \(\frac{1}{2}\), 3.7, \(\sqrt{5}\)
Answer:
The given list of numbers are:
-4, –\(\frac{9}{4}\), \(\frac{1}{2}\), 3.7, \(\sqrt{5}\)
Now,
We know that,
–\(\frac{9}{4}\) = -2.25
\(\frac{1}{2}\) = 0.5
\(\sqrt{5}\) ≅ 2.23
So,
The order of the given list of numbers from the least to the greatest is:
-4 < –\(\frac{9}{4}\) < \(\frac{1}{2}\) < \(\sqrt{5}\) < 3.7
Hence, from the above,
We can conclude that
The list that shows the numbers from the least to the greatest is:

Question 17.
The area of a square poster is 31 square inches. Find the length of one side of the poster. Explain.

PART A
To the nearest whole inch
Answer:
It is given that
The area of a square poster is 31 square inches
Now,
We know that,
The area of a square = Side²
Side = \(\sqrt{Area of a square}\)
So,
According to the given information,
The length of one side of the poster = \(\sqrt{31}\) inches
Now,
We know that,
5² < 31 < 6²
5 < \(\sqrt{31}\) < 6
Now,
We know that,
5.5² = 30.25
5.6² = 31.36
So,
The approximate value of \(\sqrt{31}\) is: 5.5
Hence, from the above,
We can conclude that
The length of one side of the poster to the nearest whole inch is: 6 inches

PART B
To the nearest tenth of an inch
Answer:
It is given that
The area of a square poster is 31 square inches
Now,
We know that,
The area of a square = Side²
Side = \(\sqrt{Area of a square}\)
So,
According to the given information,
The length of one side of the poster = \(\sqrt{31}\) inches
Now,
We know that,
5² < 31 < 6²
5 < \(\sqrt{31}\) < 6
Now,
We know that,
5.5² = 30.25
5.6² = 31.36
So,
The approximate value of \(\sqrt{31}\) is: 5.5
Hence, from the above,
We can conclude that
The length of one side of the poster to the nearest tenth of an inch is: 5.5 inches

Lesson 1.4 Evaluate Square Roots and Cube Roots

Solve & Discuss It!

ACTIVITY

Matt and his dad are building a tree house. They buy enough flooring material to cover an area of 36 square feet. What are all possible dimensions of the floor?

Answer:
It is given that
Matt and his dad are building a tree house. They buy enough flooring material to cover an area of 36 square feet
Now,
To find the length and width of the floor, find the multiples of 36
So,
The multiples of 36 are:
36 = 1 × 36, 2 × 18, 3 × 12, 4 × 9, 9 ×4, 12 × 3, 18 × 2, 36 × 1
So,
All the possible dimensions of the floor are:
1 × 36, 2 × 18, 3 × 12, 4 × 9, 9 ×4, 12 × 3, 18 × 2, 36 × 1
Hence, from the above,
We can conclude that
All the possible dimensions of the floor are:
1 × 36, 2 × 18, 3 × 12, 4 × 9, 9 ×4, 12 × 3, 18 × 2, 36 × 1

Look for Relationships
Can different floor dimensions result in the same area?
Answer:
From the above problem,
We can observe that
All the possible dimensions of the floor are:
1 × 36, 2 × 18, 3 × 12, 4 × 9, 9 ×4, 12 × 3, 18 × 2, 36 × 1
Now,
When we find the area by using all the different dimensions of the floor,
We can observe that the area of the floor is the same
Hence, from the above
We can conclude that
The different floor dimensions result in the same area

Focus on math practices
Reasoning Why is there only one set of dimensions for a square floor when there are more sets for a rectangular floor? Are all the dimensions reasonable? Explain.
Answer:
From the above problem,
We can observe that
All the possible dimensions of the floor are:
1 × 36, 2 × 18, 3 × 12, 4 × 9, 9 ×4, 12 × 3, 18 × 2, 36 × 1
Now,
We know that,
A square has the same side lengths
A square has the same parallel side lengths
Hence,
The square floor has only one set of dimensions whereas the rectangular floor has more sets and all the dimensions will be reasonable

? Essential Question
How do you evaluate cube roots and square roots?
Answer:
Let the number be: p
Now,
The square of a number is: p²
The cube of a number is: p³
The square root of a number is: \(\sqrt{p}\)
The cube root of a number is: \(\sqrt[3]{p}\)

Try It!
A cube-shaped art sculpture has a volume of 64 cubic feet. What is the length of each edge of the cube?
The length of each edge is feet.

Answer:
It is given that
A cube-shaped art sculpture has a volume of 64 cubic feet.
Now,
We know that,
A cube has all the same side lengths
Now,
Let the side length of a cube be: s
So,
The volume of a cube (V) = s³
So,
Side = \(\sqrt[3]{V}\)
Now,

So,

Hence, from the above,
We can conclude that
The length of each edge of the cube is: 4 feet

Convince Me!
How can you find the cube root of 64?
Answer:
Let the number be: p
Now,
The cube root of a number is: \(\sqrt[3]{p}\)
So,
The cube root of 64 = \(\sqrt[3]{64}\)
= \(\sqrt[3]{4 × 4 × 4}\)
= 4
Hence, from the above,
We can conclude that
The cube root of 64 is: 4

Try It!

Evaluate.
a. \(\sqrt[3]{27}\)
Answer:
The given number is: \(\sqrt[3]{27}\)
Now,

Hence, from the above,
We can conclude that
The cube root of the given number is: 3

b. \(\sqrt{25}\)
Answer:
The given number is: \(\sqrt{25}\)
Now,

Hence, from the above,
We can conclude that
The square root of the given number is: 5

c. \(\sqrt{81}\)
Answer:
The given number is: \(\sqrt{81}\)
Now,

Hence, from the above,
We can conclude that
The square root of the given number is: 9

d. \(\sqrt[3]{1}\)
Answer:
The given number is: \(\sqrt[3]{1}\)
Now,

Hence, from the above,
We can conclude that
The cube root of the given number is: 1

Try It!

Emily wants to buy a tablecloth to cover a square card table. She knows the tabletop has an area of 9 square feet. What are the minimum dimensions of the tablecloth Emily needs?

Emily should buy a tablecloth that measures at least
feet by feet.
Answer:
It is given that
Emily wants to buy a tablecloth to cover a square card table. She knows the tabletop has an area of 9 square feet.
Now,
We know that,
The area of a square = Side²
So,
According to the given information,
The area of a square table = Side²
Side² = 9
Now,

So,

Hence, from the above,
We can conclude that
The minimum dimensions of the table cloth Emily needs is: 3 feet × 3 feet

KEY CONCEPT

The cube root of a number is a number whose cube is equal to that number.

Cubing a number and taking the cube root of the number are inverse operations.

The square root of a number is a number whose square is equal to that number.

Squaring a number and taking the square root of the number are inverse operations.

Do You Understand?

Question 1.
? Essential Question How do you evaluate cube roots and square roots?
Answer:
Let the number be: p
Now,
The square of a number is: p²
The cube of a number is: p³
The square root of a number is: \(\sqrt{p}\)
The cube root of a number is: \(\sqrt[3]{p}\)

Question 2.
Generalize A certain number is both a perfect square and a perfect cube. Will its square root and its cube root always be different numbers? Explain.
Answer:
We know that,
A perfect square is a number whose square root is an integer; and a perfect cube is a number whose cube root is an integer.
A number that is a perfect square and perfect cube will not always have different numbers as its square root and cube root.

Question 3.
Critique Reasoning A cube-shaped box has a volume of 27 cubic inches. Bethany says each side of the cube measures 9 inches because 9 × 3 = 27. Is Bethany correct? Explain your reasoning.
Answer:
It is given that
A cube-shaped box has a volume of 27 cubic inches. Bethany says each side of the cube measures 9 inches because 9 × 3 = 27
Now,
We know that,
A “Cube” has the equal side lengths
Now,
We know that,
The volume of a cube (V) = Side³
So,
Side = \(\sqrt[3]{V}\)
Now,
For the volume of 27 cubic inches,
Side = \(\sqrt[3]{27}\)
= \(\sqrt[3]{3 × 3 ×3}\)
= 3 inches
So,
Each side of the cube measures 3 inches
Hence, from the above,
We can conclude that
Bethany is not correct

Do You Know How?

Question 4.
A cube has a volume of 8 cubic inches. What is the length of each edge of the cube?
Answer:
It is given that
A cube has a volume of 8 cubic inches
Now,
We know that,
A “Cube” has the equal side lengths
Now,
We know that,
The volume of a cube (V) = Side³
So,
Side = \(\sqrt[3]{V}\)
Now,
For the volume of 8 cubic inches,
Side = \(\sqrt[3]{8}\)
= \(\sqrt[3]{2 × 2 ×2}\)
= 2 inches
Hence, from the above,
We can conclude that
The length of each edge of the given cube is: 2 inches

Question 5.
Below is a model of the infield of a baseball stadium. How long is each side of the infield?

Answer:
It is given that
Below is a model of the infield of a baseball stadium
Now,
The given figure is:

Now,
From the given figure,
We can observe that
The infield is in the form of a square
Now,
We know that,
The area of a square = Side²
So,
Each Side of the infield =\(\sqrt{The area of the infield}\)
= \(\sqrt{81}\)
Now,

Hence, from the above,
We can conclude that
The length of each side of the infield is: 9 inches

Question 6.
Julio cubes a number and then takes the cube root of the result. He ends up with 20. What number did Julio start with?
Answer:
It is given that
Julio cubes a number and then takes the cube root of the result. He ends up with 20
Now,
Let us say that the number is x.
So
The first step that Julio do is to cube the number so that the number will become: x³
Then,
The next he did was take cube root of the number, so that the result will become: \(\sqrt[3]{x³}\)
Now,
By solving the above expression,
\(\sqrt[3]{x³}\) = x
Now,
It is given that
The end result is 20
So,
x = 20
So,
Julio started and ended with the same number which is 20.
Hence, from the above,
We can conclude that
The number did Julio start with is: 20

Practice & Problem Solving

Leveled Practice
In 7 and 8, evaluate the cube root or square root.

Question 7.
Relate the volume of the cube to the length

Answer:
The given figure is:

Now,
We know that,
The volume of a cube (V) = Side³
So,
Side = \(\sqrt[3]{V}\)
So,
Side = \(\sqrt[3]{8}\)
Now,

Hence, from the above,
We can conclude that
The length of each edge of the cube is: 2 cm

Question 8.
Relate the area of the square to the length of each edge.

Answer:
The given figure is:

Now,
We know that,
The area of a square (V) = Side²
So,
Side = \(\sqrt{V}\)
So,
Side = \(\sqrt{16}\)
Now,

Hence, from the above,
We can conclude that
The length of each side of the square is: 4 cm

Question 9.
Would you classify the number 169 as a perfect square, a perfect cube, both, or neither? Explain.
Answer:
The given number is: 169
Now,
We know that,
A perfect cube is a number that can be expressed as the product of three equal integers
A perfect square is a number that can be expressed as the product of two equal integers
Now,
169 can be written as:
169 = 13 × 13
Hence, from the above,
We can conclude that
169 would be classified as a perfect square

Question 10.
The volume of a cube is 512 cubic inches. What is the length of each side of the cube?
Answer:
It is given that
The volume of a cube is 512 cubic inches
Now,
We know that,
The volume of a cube (V) = Side³
So,
Side = \(\sqrt[3]{V}\)
So,
Side = \(\sqrt[3]{512}\)
So,

Hence, from the above,
We can conclude that
The length of each edge of the cube is: 8 inches

Question 11.
A square technology chip has an area of 25 square cm. How long is each side of the chip?
Answer:
It is given that
A square technology chip has an area of 25 square cm.
Now,
We know that,
The area of a square = Side²
So,
Side = \(\sqrt{The area of a square}\)
So,
Side = \(\sqrt{25}\)
So,

Hence, from the above,
We can conclude that
The length of each side of the chip is: 5 cm

Question 12.
Would you classify the number 200 as a perfect square, a perfect cube, both, or neither? Explain.
Answer:
The given number is: 200
Now,
We know that,
A perfect cube is a number that can be expressed as the product of three equal integers
A perfect square is a number that can be expressed as the product of two equal integers
Now,
200 can be written as:
200 = 100 × 2
= 10 × 10 × 2
Hence, from the above,
We can conclude that
200 would not be classified neither as a perfect square nor a perfect cube

Question 13.
A company is making building blocks. What is the length of each side of the block?

Answer:
It is given that
A company is making building blocks
Now,
The given figure is:

Now,
From the given figure,
We can observe that
The building blocks is in the form of a cube
Now,
We know that,
The volume of a cube (V) = Side³
So,
Side = \(\sqrt[3]{V}\)
So,
Side = \(\sqrt[3]{1}\)
Now,

Hence, from the above,
We can conclude that
The length of each side of the block is: 1 ft

Question 14.
Mrs. Drew wants to build a square sandbox with an area of 121 square feet. What is the total length of wood Mrs. Drew needs to make the sides of the sandbox?
Answer:
It is given that
Mrs. Drew wants to build a square sandbox with an area of 121 square feet
Now,
We know that,
The area of a square (A) = Side²
So,
Side = \(\sqrt{A}\)
So,
Side = \(\sqrt{121}\)
Now,

So,
The side of the sandbox is: 11 feet
Now,
To find the total length of wood Mrs. Drew needs to make the sides of the sandbox = 4 × (The length of the side of the sandbox)
= 4 × 11
= 44 feet
Hence, from the above,
We can conclude that
The total length of wood Mrs. Drew needs to make the sides of the sandbox is: 44 feet

Question 15.
Construct Arguments Diego says that if you cube the number 4 and then take the cube root of the result, you end up with 8. Is Diego correct? Explain.
Answer:
It is given that
Diego says that if you cube the number 4 and then take the cube root of the result, you end up with 8
Now,
According to the given information,
Step 1:
4³ = 4 × 4 × 4
= 64
Step 2:
\(\sqrt[3]{64}\)
= \(\sqrt[3]{4 × 4 ×4}\)
= 4
But,
It is given that
The end result is 8 and we got 4
Hence,f rom the above,
We can conclude that
Diego is not correct

Question 16.
Higher Order Thinking Talia is packing a moving box. She has a square-framed poster with an area of 9 square feet. The cube-shaped box has a volume of 30 cubic feet. Will the poster lie flat in the box? Explain.

Answer:
It is given that
Talia is packing a moving box. She has a square-framed poster with an area of 9 square feet. The cube-shaped box has a volume of 30 cubic feet
Now,
The given figure is:

Now,
To make the square-framed poster fit into a cube-shaped box,
The side of square-framed poster < The side of each edge of the cube-shaped box
Now,
We know that,
The area of a square = Side²
The volume of a cube = Side³
So,
\(\sqrt{9}\) = 3 feet
\(\sqrt[3]{30}\) = 3.10 feet
So,
3 < 3.10
So,
The side of square-framed poster < The side of each edge of the cube-shaped box
Hence, from the above,
We can conclude that
The poster lie flat in the box

Assessment Practice

Question 17.
Which expression has the greatest value?
A. \(\sqrt{49}\) . 2
B. \(\sqrt{49}\) – \(\sqrt{16}\)
C. \(\sqrt{25}\) + \(\sqrt{16}\)
D. \(\sqrt{25}\).3
Answer:
The given expressions are:
a.
The given expression is: \(\sqrt{49}\) . 2
Now,
We know that,
\(\sqrt{49}\) = 7
So,
The value of the given expression is: 14
b.
The given expression is:
\(\sqrt{49}\) – \(\sqrt{16}\)
Now,
We know that,
\(\sqrt{49}\) = 7
\(\sqrt{16}\) = 4
So,
The value of the given expression is: 3
c.
The given expression is:
\(\sqrt{25}\) + \(\sqrt{16}\)
Now,
We know that,
\(\sqrt{25}\) = 5
\(\sqrt{16}\) = 4
So,
The value of the given expression is: 9
d.
The given expression is:
\(\sqrt{25}\).3
Now,
We know that,
\(\sqrt{25}\) = 5
So,
The value of the given expression is: 15
Hence, from the above,
We can conclude that
The expression that has the greatest value is:

Question 18.
A toy has various shaped objects that a child can push through matching holes. The area of the square hole is 8 square cm. The volume of a cube-shaped block is 64 cubic cm
PART A
Which edge length can you find? Explain.
Answer:
It is given that
A toy has various shaped objects that a child can push through matching holes. The area of the square hole is 8 square cm. The volume of a cube-shaped block is 64 cubic cm
Now,
We know that,
The area of a square (A) = Side²
The volume of a cube (V) = Side³
So,
\(\sqrt{8}\) = 2.82 cm
\(\sqrt[3]{64}\) = 4 cm
Hence, from the above,
We can conclude that
The value of the edge lengths you found are:
The side of a square-shaped hole is: 2.82 cm
The side of a cube-shaped block is: 4 cm

PART B
Will the block fit in the square hole? Explain.
Answer:
Now,
From Part A,
We can observe that
The side of a square-shaped hole is: 2.82 cm
The side of a cube-shaped block is: 4 cm
Now,
For the block to fit in the square hole,
The side of the block < The side of the hole
But,
4 cm > 2.82 cm
Hence, from the above,
We can conclude that
The block will not fit in the square hole

Lesson 1.5 Solve Equations Using Square Roots and Cube Roots

Solve & Discuss It!

Janine can use up to 150 one-inch blocks to build a solid, cube-shaped model. What are the dimensions of the possible models that she can build? How many blocks would Janine use for each model? Explain.

Answer:
It is given that
Janine can use up to 150 one-inch blocks to build a solid, cube-shaped model
Now,
The given figure is:

Now,
We know that,
The volume of a cube = Length × Width × Height
Now,
To find the dimensions of the possible models that Janine can model,
We have to find the multiples of 150 in terms of three
So,
150 = 25 × 6
150 = 5 × 5 × 6
So,
The total number of blocks Janine would use for each model = The sum of the above three multiples of 150
= 5 + 5 + 6
= 16 blocks
Hence, from the above,
We can conclude that
The dimensions of the possible model that Janine can build is: 5 × 5 × 6
The total number of blocks Janine would use for each model is: 16 blocks

Look for Relationships
How are the dimensions of a solid related to its volume?
Answer:
The volume, V , of any rectangular solid is the product of the length, width, and height. We could also write the formula for volume of a rectangular solid in terms of the area of the base. The area of the base, B , is equal to length × Width.

Focus on math practices
Reasoning Janine wants to build a model using \(\frac{1}{2}\)-inch cubes. How many \(\frac{1}{2}\)-inch cubes would she use to build a solid, cube-shaped model with side lengths of 4 inches? Show your work.
Answer:
It is given that
Janine wants to build a model using \(\frac{1}{2}\)-inch cubes and a cube-shaped model with side lengths of 4 inches
Now,
According to the given information,
The number of \(\frac{1}{2}\)-inch cubes would Janine used to build a solid = \(\frac{1}{2}\) × 8 blocks
Hence, from the above,
We can conclude that
The number of \(\frac{1}{2}\)-inch cubes would she use to build a solid, cube-shaped model with side lengths of 4 inches is: 8 blocks

? Essential Question
How can you solve equations with squares and cubes?
Answer:
The steps to solve equations with squares are:
Step 1:
Divide all terms by a (the coefficient of x²).
Step 2:
Move the number term (c/a) to the right side of the equation.
Step 3:
Complete the square on the left side of the equation and balance this by adding the same value to the right side of the equation.
The steps to solve equations with cubes are:
Step 1:
Divide all terms by a (the coefficient of x^³).
Step 2:
Move the number term (\(\frac{d}{a}\)) to the right side of the equation.
Step 3:
Complete the square on the left side of the equation and balance this by adding the same value to the right side of the equation.

Try It!

What is the side length, s, of the square below?

Each side of the square measures meters.
Answer:
The given figure is:

Now,
From the given figure,
We can observe that
It is a square
Now,
We know that,
The area of a square = Side²
Now,

So,

Hence, from the above,
We can conclude that
Each side of the square measures 10 meters

Convince Me!
Why are there two possible solutions to the equation s² = 100? Explain why only one of the solutions is valid in this situation.
Answer:
The given equation is: s²
Now,
We know that,
The square of a positive number or a negative number is always positive and the square root of a number must always be positive
So,
s² = 100
s = \(\sqrt{100}\)
s = ± 10
Now,
We know that,
The side of any figure will always be positive
Hence, from the above,
We can conclude that
Only one of the solutions is valid in this situation because of the property of sides of the geometrical figures

Try It!

Solve x³ = 64.
Answer:
The given equation is:
x³ = 64
So,
x = \(\sqrt[3]{64}\)
x = \(\sqrt[3]{4 × 4 × 4}\)
x = 4
Hence, from the above,
We can conclude that
The value of x for the given equation is: 4

Try It!

a. Solve a³ = 11.
Answer:
The given equation is:
a³ = 11
So,
a = \(\sqrt[3]{11}\)
Now,
We know that,
The cube of a number will always be positive
Hence, from the above,
We can conclude that
The possible solution for the given equation is: \(\sqrt[3]{11}\)

b. Solve c² = 27.
Answer:
The given equation is:
c² = 27
So,
c = ±\(\sqrt{27}\)
Hence, from the above,
We can conclude that
The possible solutions for the given equation is: \(\sqrt{27}\), –\(\sqrt{27}\)

KEY CONCEPT
You can use square roots to solve equations involving squares.
x² = a
\(\sqrt{x^{2}}\) = \(\sqrt{a}\)
x = + \(\sqrt{a}\), –\(\sqrt{a}\)

You can use cube roots to solve equations involving cubes. x2 = b Vx3 = xb
x³ = b
\(\sqrt[3]{x^{3}}\) = \(\sqrt[3]{b}\)
x = \(\sqrt[3]{b}\)

Do You Understand?

Question 1.
? Essential Question
How can you solve equations with squares and cubes?
Answer:
The steps to solve equations with squares are:
Step 1:
Divide all terms by a (the coefficient of x²).
Step 2:
Move the number term (c/a) to the right side of the equation.
Step 3:
Complete the square on the left side of the equation and balance this by adding the same value to the right side of the equation.
The steps to solve equations with cubes are:
Step 1:
Divide all terms by a (the coefficient of x^³).
Step 2:
Move the number term (\(\frac{d}{a}\)) to the right side of the equation.
Step 3:
Complete the square on the left side of the equation and balance this by adding the same value to the right side of the equation.

Question 2.
Be Precise Suri solved the equation x² = 49 and found that x = 7. What error did Suri make?
Answer:
It is given that
Suri solved the equation x² = 49 and found that x = 7.
Now,
The given equation is:
x² = 49
Now,
We know that,
The square of a number is always positive but the square root of a number can either be positive or negative
So,
x = \(\sqrt{49}\)
x = ±7
x = 7, -7
Hence, from the above,
We can conclude that
The error did Suri made is that he did not consider the negative square root of 49

Question 3.
Construct Arguments There is an error in the work shown below. Explain the error and provide a correct solution.
x³ = 125
\(\sqrt[3]{x^{3}}\) = \(\sqrt[3]{125}\)
x = 5 and x = -5
Answer:
The given equation is:
x³ = 125
Now,
We know that,
The cube root of a number will always be positive
So,
\(\sqrt[3]{x^{3}}\) = \(\sqrt[3]{125}\)
x = \(\sqrt[3]{x^{3}}\) = \(\sqrt[3]{5 × 5 × 5}\)
x = 5
Hence, from the above,
We can conclude that
The error is not considering that the cube root of a number will always be positive
The correct solution is:
x = 5

Question 4.
Why are the solutions to x² = 17 irrational?
Answer:
The given equation is:
x² = 17
So,
x = ±\(\sqrt{17}\)
Now,
We know that,
The perfect square will be an integer and we know that an integer is a rational number
Now,
When we look at 17,
It is not a perfect square
Hence, from the above,
We can conclude that
The solutions of the given equation are irrational

Do You Know How?

Question 5.
If a cube has a volume of 27 cubic cm, what is the length of each edge? Use the volume formula, V = s³, and show your work.
Answer:
It is given that
A cube has a volume of 27 cubic cm
Now,
We know that,
The volume of a cube (V) = Side³
So,
Side = \(\sqrt[3]{V}\)
Now,
According to the given information,
Side = \(\sqrt[3]{27}\)
Side = \(\sqrt[3]{3 × 3 × 3}\)
Side = 3 cm
Hence, from the above,
We can conclude that
The length of each edge of the give cube is: 3 cm

Question 6.
Darius is building a square launch pad for a rocket project. If the area of the launch pad is 121 square cm, what is its side length? Use the area formula, A = s², and show your work.

Answer:
It is given that
Darius is building a square launch pad for a rocket project. If the area of the launch pad is 121 square cm,
Now,
We know that,
The area of a square (A) = Side²
So,
Side = \(\sqrt{A}\)
Now,
According to the given information,
Side = \(\sqrt{121}\)
Side = \(\sqrt{11 × 11}\)
Side = 11 cm
Hence, from the above,
We can conclude that
The side length of the launch pad is: 1 cm

Question 7.
Solve the equation x³ = -215.
Answer:
The given equation is:
x³ = -215
Now,
We know that,
The square root won’t accept negative values but the cube root will accept both positive and negative values
So,
x = \(\sqrt[3]{-215}\)
Now,

Hence, from the above,
We can conclude that
The solution for the given equation is: -5.99

Practice & Problem Solving

Leveled Practice
In 8 and 9, solve.

Question 8.
z² = 1

Answer:
The given equation is:
z² = 1
Now,

Hence, from the above,
We can conclude that

Question 9.
a³ = 216

Answer:
The given equation is:
a³ = 216
Now,

Hence, from the above,
We can conclude that
The solution for the given equation is: 6

Question 10.
Solve v² = 47.
Answer:
The give equation is:
v² = 47
Now,
\(\sqrt{v²}\) = \(\sqrt{47}\)
v = ±\(\sqrt{47}\)
Now,

Hence, from the above,
We can conclude that
The solutions for the given equation are: 6.85, and -6.85

Question 11.
The area of a square photo is 9 square inches. How long is each side of the photo?
Answer:
It is given that
The area of a square photo is 9 square inches
Now,
We know that,
The area of a square (A) = Side² (s)
So,
s² = 9
\(\sqrt{s²}\) = \(\sqrt{9}\)
s = 3 inches [Because the length of the side will never be negative]
Hence, from the above,
We can conclude that
The length of each side of the photo is: 3 inches

Question 12.
Solve the equation y² = 81.
Answer:
The given equation is:
y² = 81
Now,
\(\sqrt{y²}\) = ±\(\sqrt{81}\)
y = ±9
Hence, from the above,
We can conclude that
The solutions for the given equation are: 9, and -9

Question 13.
Solve the equation w³ = 1,000.
Answer:
The given equation is:
w³ = 1,000
Now,
\(\sqrt[3]{w³}\) = \(\sqrt[3]{1,000}\)
w = \(\sqrt[3]{10 × 10 × 10}\)
w = 10
Hence, from the above,
We can conclude that
The solution for the given equation is: 10

Question 14.
The area of a square garden is shown. How long is each side of the garden?

Answer:
It is given that
The area of a square garden is shown.
Now,
The given figure is:

Now,
From the given figure,
We can observe that
The area of the square garden is: 121 ft²
Now,
We know that,
The area of a square (A) = s²
So,
According to the given information,
s² = 121
\(\sqrt{s²}\) = \(\sqrt{121}\)
s = 11 ft [Since the length of the side will never be negative]
Hence, from the above,
We can conclude that
The length of each side of the garden is: 11 ft

Question 15.
Solve b² = 77.
Answer:
The given equation is:
b² = 77
Now,
\(\sqrt{b²}\) = ±\(\sqrt{77}\)
Now,

Hence, from the above,
We can conclude that
The solutions for the givene quation are: 8.77, and -8.77

Question 16.
Find the value of c in the equation c³ = 1,728.
Answer:
The given equation is:
c³ = 1,728
Now,
\(\sqrt[3]{c³}\) = \(\sqrt[3]{1,728}\)
c = 12
Hence, from the above,
We can conclude that
The value of c for the given equation is: 12

Question 17.
Solve the equation v³ = 12.
Answer:
The given equation is:
v³ = 12
Now,
\(\sqrt[3]{v³}\) = \(\sqrt[3]{12}\)
Now,

Hence, from the above,
We can conclude that
The solution for the given equation is: 2.28

Question 18.
Higher Order Thinking Explain why
\(\sqrt[3]{-\frac{8}{27}}\) is –\(\frac{2}{3}\)
Answer:
The given equation is:
\(\sqrt[3]{-\frac{8}{27}}\)
Now,
We know that,
\(\sqrt[3]{\frac{a}{b}}\) = \(\frac{\sqrt[3]{a}}{\sqrt[3]{b}}\)
So,
\(\sqrt[3]{-\frac{8}{27}}\) = –\(\frac{\sqrt[8]{a}}{\sqrt[3]{27}}\)
= –\(\frac{\sqrt[2 × 2 ×2]{a}}{\sqrt[3]{3 ×3 × 3}}\)
= –\(\frac{2}{3}\)
Hence, from the above,
We can conclude that
\(\sqrt[3]{-\frac{8}{27}}\) is –\(\frac{2}{3}\) due to the below property of Exponents:
\(\sqrt[3]{\frac{a}{b}}\) = \(\frac{\sqrt[3]{a}}{\sqrt[3]{b}}\)

Question 19.
Critique Reasoning Manolo says that the solution of the equation g² = 36 is g = 6 because 6 × 6 = 36. Is Manolo’s reasoning complete? Explain.
Answer:
It is given that
Manolo says that the solution of the equation g² = 36 is g = 6 because 6 × 6 = 36
Now,
The given equation is:
g² = 36
Now,
We know that,
The square of a number will always be positive but the square root of a number will either be positive or negative
So,
\(\sqrt{g²}\) = ±\(\sqrt{36}\)
g = ±6
So,
The solutions for the given equation are: 6, -6
Hence, from the above,
We can conclude that
Manolo’s reasoning is not complete

Question 20.
Evaluate \(\sqrt[3]{-512}\).
a. Write your answer as an integer.
Answer:
The given equation is:
\(\sqrt[3]{-512}\)
Now,
\(\sqrt[3]{-512}\) = \(\sqrt[3]{(-8) × (-8) × (-8)}\)
= -8
Hence, from the above,
We can conclude that
The value of \(\sqrt[3]{-512}\) as an integer is: -8

b. Explain how you can check that your result is correct.
Answer:
The given equation is:
\(\sqrt[3]{-512}\)
Now,
From part (a),
We get the value of the given equation is: 8
Now,
\(\sqrt[3]{(-8) × (-8) × (-8)}\)
= \(\sqrt[3]{64 × (-8)}\)
= \(\sqrt[3]{-512}\)
Hence, from the above,
We can conclude that
The result is correct because the givene quation and the result are the same

Question 21.
Yael has a square-shaped garage with 228 square feet of floor space. She plans to build an addition that will increase the floor space by 50%. What will be the length, to the nearest tenth, of one side of the new garage?

Answer:
It is given that
Yael has a square-shaped garage with 228 square feet of floor space. She plans to build an addition that will increase the floor space by 50%.
Now,
According to the given information,
50% of 228 = \(\frac{50}{100}\) × 228
= \(\frac{50 × 228}{100}\)
= 114 square feet
So,
The area of the new garage = 228 + 114
= 342 square feet
Now,
We know that,
The area of a square (A) = Side (s)²
So,
s² = 342
\(\sqrt{s²}\) = \(\sqrt{342}\)
Now,

Hence, from the above,
We can conclude that
The length of one side of the new garage is: 18.5 feet

Assessment Practice

Question 22.
The Traverses are adding a new room to their house. The room will be a cube with a volume of 6,859 cubic feet. They are going to put in hardwood floors, which costs $10 per square foot. How much will the hardwood floors cost?
Answer:
It is given that
The Traverses are adding a new room to their house. The room will be a cube with a volume of 6,859 cubic feet. They are going to put in hardwood floors, which costs $10 per square foot
Now,
We know that,
The volume of a cube (V) = s³
So,
According to the given information,
s³ = 6,859
\(\sqrt[3]{s³}\) = \(\sqrt[3]{6,859}\)
s = 19 feet
So,
The length of each edge of the new room is: 19 feet
Now,
To find the total cost of hardwood floors, find the perimeter of the room and multiply the result with the cost per square foot
Now,
We know that,
The perimeter of a cube = 6s
So,
The perimeter of the cube (p) = 6 × 19
= 114 feet
Now,
The total cost of hardwood floors = 114 × $10
= $1,140
Hence, from the above,
We can conclude that
The total cost of hardwood floors is: $1,140

Question 23.
While packing for their cross-country move, the Chen family uses a crate that has the shape of a cube.
PART A
If the crate has the volume V = 64 cubic feet, what is the length of one edge?
It is given that
While packing for their cross-country move, the Chen family uses a crate that has the shape of a cube and the crate has the volume V = 64 cubic feet
Now,
We know that,
The volume of a cube (V) = s³
So,
According to the given information,
s³ = 64
\(\sqrt[3]{s³}\) = \(\sqrt[3]{64}\)
s = 4 feet
Hence, from the above,
We can conclude that
The length of each edge of the crate is: 4 feet

PART B
The Chens want to pack a large, framed painting. If the framed painting has the shape of a square with an area of 12 square feet, will the painting fit flat against a side of the crate? Explain.
Answer:
It is given that
The Chens want to pack a large, framed painting. If the framed painting has the shape of a square with an area of 12 square feet
Now,
We know that,
The area of a square (A) = s²
So,
According to the given information,
s² = 12
Now,
\(\sqrt{s²}\) = \(\sqrt{12}\)
Now,

So,
According to the given information,
The side of the crate > The side of the painting
4 > 2.28
Hence, from the above,
We can conclude that
The painting will fit flat against a side of the crate

Topic 1 MID-TOPIC CHECKPOINT

Question 1.
Vocabulary How can you show that a number is a rational number? Lesson 1.2
Answer:
A rational number is a number that can be written as a ratio. That means it can be written as a fraction, in which both the numerator (the number on top) and the denominator (the number on the bottom) are whole numbers

Question 2.
Which shows 0.2\(\overline{3}\) as a fraction? Lesson 1.1
A. \(\frac{2}{33}\)
B. \(\frac{7}{33}\)
C. \(\frac{23}{99}\)
D. \(\frac{7}{30}\)
Answer:
The given expression is: 0.2\(\overline{3}\)
Hence, from the above,
We can conclude that
The options that show 0.2\(\overline{3}[/latex as a fraction is:

Question 3.
Approximate [latex]\sqrt{8}\) to the nearest hundredth. Show your work. Lesson 1.3
Answer:
The given expression is: \(\sqrt{8}\)
Now,

Hence, from the above,
We can conclude that
The approximate value of the given expression to the nearest hundredth is: 2.82

Question 4.
Solve the equation m² = 14. Lesson 1.5
Answer:
The given equation is:
m² = 14
Now,
\(\sqrt{m²}\) = ±\(\sqrt{14}\)
Now,

Hence, from the above,
We can conclude that

The solutions for the given equation are: 3.74, -3.74

Question 5.
A fish tank is in the shape of a cube. Its volume is 125 ft³. What is the area of one face of the tank? Lessons 1.4 and 1.5
Answer:
It is given that
A fish tank is in the shape of a cube. Its volume is 125 ft³.
Now,
We know that,
The volume of a cube (V) = s³
So,
s³= 125
Now,
\(\sqrt[3]{s³}\) = \(\sqrt[3]{125}\)
s = \(\sqrt[3]{5 × 5 × 5}\)
s = 5 ft
Now,
We know that,
The surface area of a cube = 4s²
So,
The area of one face of the tank = 4 × 5²
= 4 × 25
= 100 ft²
Hence, from the above,
We can conclude that
The area of one face of the tank is: 100 ft²

Question 6.
Write 1.\(\overline{12}\) as a mixed number. Show your work. Lesson 1.1
Answer:
The given expression is: 1.\(\overline{12}\)
Now,
The representation of the given expression in the form of a fraction is: \(\frac{28}{25}\)
Now,
The representation of \(\frac{28}{25}\) into a mixed number is: 1\(\frac{3}{25}\)
Hence, from the above,
We can conclude that
The representation of 1.\(\overline{12}\) as a mixed number is: 1\(\frac{3}{25}\)
How well did you do on the mid-topic checkpoint? Fill in the stars.

Topic 1 MID-TOPIC PERFORMANCE TASK

Six members of the math club are forming two teams for a contest. The teams will be determined by having each student draw a number from a box.

PART A

The table shows the results of the draw. The students who drew rational numbers will form the team called the Tigers. The students who drew irrational numbers will form the team called the Lions. List the members of each team.
Answer:
It is given that
Six members of the math club are forming two teams for a contest. The teams will be determined by having each student draw a number from a box.
Now,
The given table is:

Now,
We know that,
The numbers that can be written in the form of \(\frac{p}{q}\) are “Rational numbers”
The numbers that can not be written in the form of \(\frac{p}{q}\) are “Irrational numbers”
Now,
From the given table,
The list of rational numbers are:
6.\(\overline{34}\), \(\sqrt{36}\), 6.3\(\overline{4}\)
The list of Irrational numbers are:
\(\sqrt{38}\), 6.343443444….,, \(\sqrt{34}\)
Hence, from the above,
We can conclude that
The students present in the Tigers Team are:
6.\(\overline{34}\), \(\sqrt{36}\), 6.3\(\overline{4}\)
The students present in the Lions Team are:
\(\sqrt{38}\), 6.343443444….,, \(\sqrt{34}\)

PART B

The student on each team who drew the greatest number will be the captain of that team. Who will be the captain of the Tigers? Show your work.
Answer:
From Part A,
We can observe that
The students present in the Tigers Team are:
6.\(\overline{34}\), \(\sqrt{36}\), 6.3\(\overline{4}\)
Now,
From the above list of numbers,
We can observe that
6.\(\overline{34}\) is the greater number
Hence, from the above,
We can conclude that
The captain of the Tigers Team is: Anya

PART C

Who will be the captain of the Lions? Show your work.
Answer:
The students present in the Lions Team are:
\(\sqrt{38}\), 6.343443444….,, \(\sqrt{34}\)
Now,
From the above list of numbers,
We can observe that
6.343443444…., is the greater number
Hence, from the above,
We can conclude that
The captain of the Lions Team is: Ryan

Lesson 1.6 Use Properties of Integer Exponents

Solve & Discuss It!

One band’s streaming video concert to benefit a global charity costs $1.00 to view.
The first day, the concert got 2,187 views. The second day, it got about three times as many views. On the third day, it got 3 times as many views as on the second day. If the trend continues, how much money will the band raise on Day 7?

Answer:
It is given that
One band’s streaming video concert to benefit a global charity costs $1.00 to view.
The first day, the concert got 2,187 views. The second day, it got about three times as many views. On the third day, it got 3 times as many views as on the second day
So,
According to the given information,
The amount of money got on the first day of concert = (The total number of views) × $1
= 2,187 × $1
= $2,187
The amount of money got on the second day of concert = (The amount of money got on the first day of concert) × 3
= $2,187 × 3
= $6,561
The amount of money got on the third day of concert = (The amount of money got on the second day of concert) × 3
= $6,561 × 3
= $19,683
The amount of money got on the fourth day of concert = (The amount of money got on the third day of concert) × 3
= $19,683 × 3
= $59,049
The amount of money got on the fifth day of concert = (The amount of money got on the fourth day of concert) × 3
= $59,049 × 3
= $1,77,147
The amount of money got on the sixth day of concert = (The amount of money got on the fifth day of concert) × 3
= $1,77,147 × 3
= $5,31,441
The amount of money got on the seventh day of concert = (The amount of money got on the sixth day of concert) × 3
= $5,31,441 × 3
= $15,94,323
Hence, from the above,
We can conclude that
The amount of money the Band raise on Day 7 is: $15,94,323

Focus on math practices
Use Structure Use prime factorization to write an expression equivalent to the amount of money raised by the band on the last day of the week.
Answer:
From the above,
We can observe that
The amount of money got on the last day of the wee is: $15,94,323
Now,
By using the Prime factorisation method,

Hence, from the above,
We can conclude that
The prime factorisation of 15,94,323 is: 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3

? Essential Question
How do properties of integer exponents help you write equivalent expressions?
Answer:
The properties of integer exponents can be used to write equivalent expressions by combining numeric or algebraic expressions that have a common base, distributing exponents to products and quotients, and simplifying powers of powers.

Convince Me!
Explain why the Product of Powers Property makes mathematical sense.
Answer:
The Power of a Product rule states that a term raised to a power is equal to the product of its factors raised to the same power

Try It!
Write equivalent expressions using the properties of exponents.

a. (7³)²
Answer:
The given expression is: (7³)²
Now,
We know that,
The “Power of Powers Property” states that to find the power of a power, multiply the exponents
So,
(7³)² = 7^{3 × 2}
= 7⁶
Hence, from the above,
We can conclude that
The value of the given expression is: 7⁶

b. (4⁵)³
Answer:
The given expression is: (4⁵)³
Now,
We know that,
The “Power of Powers Property” states that to find the power of a power, multiply the exponents
So,
(4⁵)³ = 4^{3 × 5}
= 4¹⁵
Hence, from the above,
We can conclude that
The value of the given expression is: 4¹⁵

c. 9⁴ × 8⁴
Answer:
The given expression is: 9⁴ × 8⁴
Now,
We know that,
By using the “Power of Products Property”, when multiplying two exponential expressions with the same exponent and different powers, multiply the bases and keep the exponents the same
So,
9⁴ × 8⁴ = (9 × 8)⁴
= 72⁴
Hence, from the above,
We can conclude that
The value of the given expression is: 72⁴

d. 8⁹ ÷ 8³
Answer:
The given expression is: 8⁹ ÷ 8³
Now,
We know that,
The “Quotient of powers Property” states that when dividing two exponential expressions with the same base, subtract the exponents
So,
8⁹ ÷ 8³ = 8^{9 – 3}
= 8⁶
Hence, from the above,
We can conclude that
The value of the given expression is: 8⁶

KEY CONCEPT

Use these properties when simplifying expressions with exponents (when a, m, and n ≠ 0).

Do You Understand?

Question 1.
Essential Question How do properties of integer exponents help you write equivalent expressions?
Answer:
The properties of integer exponents can be used to write equivalent expressions by combining numeric or algebraic expressions that have a common base, distributing exponents to products and quotients, and simplifying powers of powers.

Question 2.
Look for Relationships If you are writing an equivalent expression for 2³ × 2⁴, how many times would you write 2 as a factor?
Answer:
It is given that
you are writing an equivalent expression for 2³ × 2⁴
Now,
The given expression is: 2³ × 2⁴
Now,
We know that,
The “Power of Powers Property” states that to find the power of a power, multiply the exponents
So,
2³ × 2⁴ = 2^{3 + 4}
= 2⁷
= 2 × 2 × 2 × 2 × 2 × 2 × 2
Hence, from the above,
We can conclude that
We would write 2 as a factor 7 times

Question 3.
Construct Arguments Kristen wrote 5⁸ as an expression equivalent to (5²)⁴. Her math partner writes 5⁶. Who is correct?
Answer:
It is given that
Kristen wrote 5⁸ as an expression equivalent to (5²)⁴. Her math partner writes 5⁶
Now,
The given expression is: 5⁸
Now,
We know that,
(a^m)ⁿ = a^mn
Now,
According to the above Property,
5⁸ = 5^{4 × 2}
= (5⁴)²
= (5²)⁴
Hence, from the above,
We can conclude that
Kristen is correct

Question 4.
Critique Reasoning Tyler says that an equivalent expression for 2³ × 5³ is 10⁹. Is he correct? Explain.
Answer:
It is given that
Tyler says that an equivalent expression for 2³ × 5³ is 10⁹
Now,
The given expression is: 2³ × 5³
Now,
We know that,
a^m × b^m = (a × b)^m
So,
2³ × 5³ = (2 × 5)³
= 10³
So,
The equivalent expression for 2³ × 5³ is: 10³
Hence, from the above,
We can conclude that
Tyler is not correct

Do You Know How?

Question 5.
Write an equivalent expression for 7¹² × 7⁴.
Answer:
The given expression is: 7¹² × 7⁴
Now,
We know that,
a^m × aⁿ = a^{m + n}
So,
7¹² × 7⁴ = (7)^{12 + 4}
= 7¹⁶
Hence, from the above,
We can conclude that
The equivalent expression for the given expression is: 7¹⁶

Question 6.
Write an equivalent expression for (8²)⁴.
Answer:
The given expression is: (8²)⁴
Now,
We know that,
(a^m)ⁿ = a^mn
So,
(8²)⁴ = 8^{2 × 4}
= 8⁸
Hence, from the above,
We can conclude that
The equivalent expression for the given expression is: 8⁸

Question 7.
A billboard has the given dimensions.

Using exponents, write two equivalent expressions for the area of the rectangle.
Answer:
It is given that
A billboard has the given dimensions.
Now,
The given figure is:

Now,
From the given figure,
We can observe that
The billboard is in the form of a rectangle.
Now,
From the given figure,
We can observe that
The length of the billboard is: 7² ft
The width of the billboard is: 4² ft
Now,
We know that,
We know that,
a^m × b^m = (a × b)^m
So,
The area of the rectangle (A) = 7² × 4²
= (7 × 4)²
= 28² ft²
Hence, from the above,
We can conclude that
The two equivalent expressions for the area of the rectangle is: 7² × 4² and 28² ft²

Question 8.
Write an equivalent expression for 18⁹ – 18⁴.
Answer:
The given expression is: 18⁹ – 18⁴
Now,
18⁹ – 18⁴ = 18⁴ (18⁵ – 1)
Hence, from the above,
We can conclude that
The equivalent expression for the given expression is: 18⁴ (18⁵ – 1)

Practice & Problem Solving

Leveled Practice
In 9-12, use the properties of exponents to write an equivalent expression for each given expression.

Question 9.
2⁸ × 2⁴

Answer:
The given expression is: 2⁸ × 2⁴
Now,

Hence, from the above,
We can conclude that
The equivalent expression for the given expression is: 2¹²

Question 10.
\(\frac{8^{7}}{8^{3}}\)

Answer:
The given expression is: \(\frac{8⁷}{8³}\)
Now,
We know that,
\(\frac{a^m}{aⁿ}\) = a^{m – n}
So,

Hence, from the above,
We can conclude that
The equivalent expression for the given expression is: 8⁴

Question 11.
(3⁴)⁵

Answer:
The given expression is: (3⁴)⁵
Now,
We know that,
(a^m)ⁿ = a^mn
Now,

Hence, from the above,
We can conclude that
The equivalent expression for thegiven expression is: 3²⁰

Question 12.
3⁹ × 2⁹

Answer:
The given expression is: 3⁹ × 2⁹
Now,
We know that,
a^m × b^m = (a × b)^m
Now,

Hence, from the above,
We can conclude that
The equivalent expression for the given expression is: (3 ×  2)⁹

Question 13.
a. How do you multiply powers that have the same base?
Answer:
If the two exponential expressions have the same base and different exponents and both are multiplying each other, then add the exponents by keeping the bases same

b. How do you divide powers that have the same base?
Answer:
If the two exponential expressions have the same base and different exponents and both are dividing each other, then subtract the exponents by keeping the bases same

c. How do you find the power of a power?
Answer:
When an exponential expression contains the power of a power, we will multiply both the powers by keeping the base constant

d. How do you multiply powers with different bases but the same exponent?
Answer:
If the two exponential expressions have the same exponent and different bases and both are multiplying each other, then multiply the bases by keeping the exponents same

Question 14.
Which expressions are equivalent to 2¹¹?
Select all that apply.
\(\frac{2^{23}}{2^{12}}\)
2⁷ ∙ 2⁴
\(\frac{2^{9}}{2^{2}}\)
2² ∙ 2⁹
Answer:
The given expression is: 2¹¹
Hence, from the above,
We can conclude that
The expressions that are equivalent to 2¹¹ are:

In 15-18, use the properties of exponents to write an equivalent expression for each given expression.

Question 15.
(4⁴)³
Answer:
The given expression is: (4⁴)³
Now,
We know that,
(a^m)ⁿ = a^{m × n}
So,
(4⁴)³ = 4^{4 × 3}
= 4¹²
Hence, from the above,
We can conclude that
(4⁴)³ = 4¹²

Question 16.
\(\frac{3^{12}}{3^{3}}\)
Answer:
The given expression is: \(\frac{3^{12}}{3^{3}}\)
Now,
We know that,
\(\frac{a^m}{aⁿ}\) = a^{m – n}
So,
\(\frac{3^{12}}{3^{3}}\) = 3^{12 – 3}
= 3⁹
Hence, from the above,
We can conclude that
\(\frac{3^{12}}{3^{3}}\) = 3⁹

Question 17.
4⁵ × 4²
Answer:
The given expression is: 4⁵ × 4²
Now,
We know that,
a^m × aⁿ = a^{m + n}
So,
4⁵ × 4² = 4^{5 + 2}
= 4⁷
Hence, from the above,
We can conclude that
4⁵ × 4² = 4⁷

Question 18.
6⁴ × 2⁴
Answer:
The given expression is: 6⁴ × 2⁴
Now,
We know that,
a^m × b^m = (a × b)^m
So,
6⁴ × 2⁴ = (6 × 2)⁴
= 12⁴
Hence, from the above,
We can conclude that
6⁴ × 2⁴ = 12⁴

Question 19.
Critique Reasoning Alberto incorrectly stated that \(\frac{5^{7}}{5^{4}}\) = 1³. What was Alberto’s error? Explain your reasoning and find the correct answer.
Answer:
It is given that
Alberto incorrectly stated that \(\frac{5^{7}}{5^{4}}\) = 1³
Now,
The given expression is: \(\frac{5^{7}}{5^{4}}\)
Now,
We know that,
\(\frac{a^m}{aⁿ}\) = a^{m – n>}
So,
\(\frac{5^{7}}{5^{4}}\) = 5 ^{7 – 4}
= 5³
So,
From the above,
We can observe that
Alberto applied the “Quotient of Powers Property” incorrectly
Hence, from the above,
We can conclude that
Alberto’s error is: Alberto applied the “Quotient of Powers Property” incorrectly

Question 20.
Is the expression 8 × 8⁵ equivalent to (8 × 8)⁵? Explain.
Answer:

Question 21.
Is the expression (3²)^-3 equivalent to (3³)^-2? Explain.
Answer:

Question 22.
Is the expression 3² ∙ 3^-3 equivalent to 3³ ∙ 3^-2? Explain.
Answer:

Question 23.
Model with Math What is the width of the rectangle written as an exponential expression?

Answer:

Question 24.
Simplify the expression \(\left(\left(\frac{1}{2}\right)^{3}\right)^{3}\).
Answer:

Question 25.
Higher Order Thinking Use a property of exponents to write (3b)⁵ as a product of powers.
Answer:

Assessment Practice

Question 26.
Select all the expressions equivalent to 4⁵ ∙ 4¹⁰
4⁵ + 4¹⁰
4³ ∙ 4⁵
4³ ∙ 4¹²
4³ + 4¹²
4¹⁸ – 4³
4¹⁵

Question 27.
Your teacher asks the class to evaluate the expression (2³)¹. Your classmate gives an incorrect answer of 16.
PART A
Evaluate the expression.
PART B
What was the likely error?
A. Your classmate divided the exponents.
B. Your classmate multiplied the exponents.
C. Your classmate added the exponents.
D. Your classmate subtracted the exponents.
Answer:

Lesson 1.7 More Properties of Integer Exponents

ACTIVITY

Explore It!

Calvin and Mike do sit-ups when they work out. They start with 64 sit-ups for the first set and do half as many each subsequent set.

Look for Relationships
Determine whether the relationship shown for Set 1 is also true for Sets 2-5.

A. What representation can you use to show the relationship between the set number and the number of sit-ups?

B. What conclusion can you make about the relationship between the number of sit-ups in each set?

Focus on math practices
Use Structure How could you determine the number of sit-up sets Calvin and Mike do?

? Essential Question
What do the Zero Exponent and Negative Exponent Properties mean?

Try It!

Evaluate
a. (-7)⁰
b. (43)⁰
c. 1⁰
d. (0.5⁰)

Convince Me!
Why is 2(7⁰) = 2?

Try It!

Write each expression using positive exponents.
a. 8^-2
b. 2^-4
c. 3^-5
Answer:

Try It!

Write each expression using positive exponents.

a. \(\frac{1}{5^{-3}}\)

b. \(\frac{1}{2^{-6}}\)

KEY CONCEPT
Use these additional properties when simplifying or generating equivalent expressions with exponents (when a ≠ 0 and n ≠ 0).
Zero Exponent Property
a⁰ = 1

Negative Exponent Property
a^-n = \(\frac{1}{a^{n}}\)

Do You Understand?

Question 1.
Essential Question What do the Zero Exponent and Negative Exponent Properties mean?

Question 2.
Reasoning In the expression 9^-12, what does the negative exponent mean?

Question 3.
Reasoning in the expression 3(2⁰), what is the order of operations? Explain how you would evaluate the expression.

Do You Know How?

Question 4.
Simplify 1,999,999⁰.
Answer:

Question 5.
a. Write 7^-6 using a positive exponent.
b. Rewrite \(\frac{1}{10^{-3}}\) using a positive exponent.
Answer:

Question 6.
Evaluate 27x^oy^-2 for x = 4 and y = 3.
Answer:

Practice & Problem Solving

Leveled Practice In 7-8, complete each table to find the value of a nonzero number raised to the power of 0.

Question 7.

Answer:

Question 8.

Answer:

Question 9.
Given: (-3.2)⁰
a. Simplify the given expression.
b. Write two expressions equivalent to the given expression. Explain why the three expressions are equivalent.
Answer:

Question 10.
Simplify each expression for x = 6.
a. 12x⁰(x^-4)
b. 14(x^-2)

In 11 and 12, compare the values using >,<, or =.

Question 11.
3^-2 1
Answer:

Question 12.
\(\left(\frac{1}{4}\right)^{0}\) 1

In 13 and 14, rewrite each expression using a positive exponent.

Question 13.
9^-4
Answer:

Question 14.
\(\frac{1}{2^{-6}}\)
Answer:

Question 15.
Given: 9y⁰
a. Simplify the expression for y = 3.
b. Construct Arguments Will the value of the given expression vary depending on y? Explain.
Answer:

Question 16.
Simplify each expression for x = 4.
a. -5x^-4
b. 7x^-3
Answer:

Question 17.
Evaluate each pair of expressions.
a. (-3)^-8 and -3^-8
b. (-3)^-9 and -3^-9
Answer:

Question 18.
Be Precise To win a math game, Lamar has to pick a card with an expression that has a value greater than 1. The card Lamar chooses reads \(\left(\frac{1}{2}\right)^{-4}\). Does Lamar win the game? Explain.

Answer:

Question 19.
Simplify the expression. Assume that x is nonzero. Your answer should have only positive exponents. x^-10 ∙ x⁶
Answer:

Question 20.
Higher Order Thinking
a. Is the value of the expression \(\left(\frac{1}{4^{-3}}\right)^{-2}\) greater than 1, equal to 1, or less than 1?
b. If the value of the expression is greater than 1, show how you can change one sign to make the value less than 1. If the value is less than 1, show how you can change one sign to make the value greater than 1. If the value is equal to 1, show how you can make one change to make the value not equal to 1.

Assessment Practice

Question 21.
Which expressions are equal to 5^-3? Select all
that apply.
125
125^-1
5³
\(\frac{1}{5^{3}}\)
\(\frac{1}{125}\)
Answer:

Question 22.
Which expressions have a value less than 1 when x = 4? Select all that apply.
\(\left(\frac{3}{x^{2}}\right)^{0}\)
\(\frac{x^{0}}{3^{2}}\)
\(\frac{1}{6^{-x}}\)
\(\frac{1}{x^{-3}}\)
3x^-4

Lesson 1.8 Use Powers of 10 to Estimate Quantities

ACTIVITY

Explain It!

Keegan and Jeff did some research and found that there are approximately 7,492,000,000,000,000,000 grains of sand on Earth. Jeff says that it is about 7 × 10¹⁵ grains of sand. Keegan says that this is about 7 × 10¹⁸ grains of sand.

A. How might Jeff have determined his estimate? How might Keegan have determined his estimate?

B. Whose estimate, Jeff’s or Keegan’s, is more logical? Explain.

Focus on math practices
Be Precise Do you think the two estimates are close in value? Explain your reasoning.

? Essential Question
when would you use a power of 10 to estimate a quantity?

Try It!

Light travels 299,792,458 meters per second. Sound travels at 332 meters per second. Use a power of 10 to compare the speed of light to the speed of sound.
299,792,458 rounded to the greatest place value is
There are zeros in the rounded number.
The estimated speed of light is × 10 meters per second.
3 × 10 > 3 × 10, so the speed of light is faster than the speed of sound.

322 rounded to the greatest place value is
There are zeros in the rounded number. The estimated speed of sound is × 10 meters per second.

Convince Me!
Country A has a population of 1,238,682,005 and Country B has a population of 1,106,487,394. How would you compare these populations?

Try It!

There are approximately 1,020,000,000 cars in the world. The number of cars in the United States is approximately 239,800,000.
Compare the number of cars in the world to that in the United States.

KEY CONCEPT

You can estimate a very large or very small number by rounding the number to its greatest place value, and then writing that number as a single digit times a power of 10.

Do You Understand?

Question 1.
? Essential Question when would you use powers of 10 to estimate a quantity?
Answer:

Question 2.
Construct Arguments Kim writes an estimate for the number 0.00436 as 4 × 10³. Explain why this cannot be correct.
Answer:

Question 3.
Be Precise Raquel estimated 304,900,000,000 as 3 × 10⁸. What error did she make?
Answer:

Do You Know How?

Question 4.
Use a single digit times a power of 10 to estimate the height of Mt. Everest to the nearest ten thousand feet.

Answer:

Question 5.
A scientist records the mass of a proton as 0.0000000000000000000000016726231 gram. Use a single digit times a power of 10 to estimate the mass.
Answer:

Question 6.
The tanks at the Georgia Aquarium hold approximately 8.4 × 10⁶ gallons of water. The tanks at the Audubon Aquarium of the Americas hold about 400,000 gallons of water. Use a single digit times a power of 10 to estimate how many times greater the amount of water is at the Georgia Aquarium.
Answer:

Practice & Problem Solving

Leveled Practice in 7-9, use powers of 10 to estimate quantities.

Question 7.
A city has a population of 2,549,786 people. Estimate this population to the nearest million. Express your answer as the product of a single digit and a power of 10. Rounded to the nearest million, the population is about
Written as the product of a single digit and a power of ten, this number is
Answer:

Question 8.
Use a single digit times a power of 10 to estimate the number 0.00002468. Rounded to the nearest hundred thousandth, the number is about
Written as a single digit times a power of ten, the estimate is
Answer:

Question 9.
The approximate circumferences of Earth and Saturn are shown. How many times greater is the circumference of Saturn than the circumference of Earth?
The circumference of Saturn is


Saturn’s circumference is about times greater than the circumference of Earth.
Answer:

Question 10.
Estimate 0.037854921 to the nearest hundredth. Express your answer as a single digit times a power of ten.
Answer:

Question 11.
Compare the numbers 6 × 10^-6 and 2 × 10^-8.
a. Which number has the greater value?
b. Which number has the lesser value?
c. How many times greater is the greater number?
Answer:

Question 12.
Taylor made $43,785 last year. Use a single digit times a power of ten to express this value rounded to the nearest ten thousand.
Answer:

Question 13.
The length of plant cell A is 8 × 10^-5 meter. The length of plant cell B is 0.000004 meter. How many times greater is plant cell A’s length than plant cell B’s length?
Answer:

Question 14.
Critique Reasoning The diameter of one species of bacteria is shown. Bonnie approximates this measure as 3 × 10^-11 meter. Is she correct? Explain.

Answer:

Question 15.
The populations of Cities A and B are 2.6 × 10⁵ and 1,560,000, respectively. The population of City C is twice the population of City B. The population of City C is how many times the population of City A?
Answer:

Assessment Practice

Question 16.
Earth is approximately 5 × 10⁹ years old. For which of these ages could this be an approximation?
A. 4,762,100,000 years
B. 48,000,000,000 years
C. 4.45 × 10⁹ years
D. 4.249999999 × 10⁹ years
Answer:

Question 17.
PART A
Express 0.000000298 as a single digit times a power of ten rounded to the nearest ten millionth.

PART B
Explain how negative powers of 10 can be helpful when writing and comparing small numbers.
Answer:

Lesson 1.9 Understand Scientific Notation

ACTIVITY

Solve & Discuss It!

Scientists often write very large or very small numbers using exponents. How might a scientist write the number shown using exponents?

Use Structure
How can you use your knowledge of powers of 10 to rewrite the number?

Focus on math practices
Look for Relationships What does the exponent in 1015 tell you about the value of the number?

? Essential Question
What is scientific notation and why is it used?

Try It!
The height of Angel Falls, the tallest waterfall in the world, is 3,212 feet. How do you write this number in scientific notation?

Convince Me!
Why do very large numbers have positive exponents when written in scientific notation? Explain.

Try It!
A common mechanical pencil lead measures about 0.005 meter in diameter. How can you express this measurement using scientific notation ?

Try It!

Write the numbers in standard form.
a. 9.225 × 10¹⁸
b. 6.3 × 10^-8
Answer:

KEY CONCEPT
Scientific notation is a way to write very large numbers or very small numbers. Scientists use scientific notation as a more efficient and convenient way of writing such numbers.
A number in scientific notation is the product of two factors. The first factor must be greater than or equal to 1 and less than 10. The second factor is a power of 10.

To write a number in scientific notation in standard form, multiply the decimal number by the power of 10.

Do You Understand?

Question 1.
?Essential Question What is scientific notation and why is it used?
Answer:

Question 2.
Critique Reasoning Taylor states that 2,800,000 in scientific notation is 2.8 × 10^-6 because the number has six places to the right of the 2. Is Taylor’s reasoning correct?
Answer:

Question 3.
Construct Arguments Sam will write 0.000032 in scientific notation. Sam thinks that the exponent of 10 will be positive. Do you agree? Construct an argument to support your response.
Answer:

Do You Know How?

Question 4.
Express 586,400,000 in scientific notation.
Answer:

Question 5.
The genetic information of almost every living thing is stored in a tiny strand called DNA. Human DNA is 3.4 × 10^-8 meter long. Write the length in standard form.

Answer:

Question 6.
The largest virus known to man is the Megavirus, which measures 0.00000044 meter across. Express this number in scientific notation.
Answer:

Question 7.
How would you write the number displayed on the calculator screen in standard form?

Answer:

Practice & Problem Solving

Leveled Practice In 8 and 9, write the numbers in the correct format.

Question 8.
The Sun is 1.5 × 10⁸ kilometers from Earth. 1.5 × 10⁸ is written as in standard form.
Answer:

Question 9.
Brenna wants an easier way to write 0.0000000000000000587.
0.0000000000000000587 is written as × 10 in scientific notation.
Answer:

Question 10.
Is 23 × 10^-8 written in scientific notation? Justify your response.
Answer:

Question 11.
Is 8.6 × 10⁷ written in scientific notation? Justify your response.
Answer:

Question 12.
Simone evaluates an expression using her calculator. The calculator display is shown at the right. Express the number in standard form.

Answer:

Question 13.
Express the number 0.00001038 in scientific notation.
Answer:

Question 14.
Express the number 80,000 in scientific notation.
Answer:

Question 15.
Peter evaluates an expression using his calculator. The calculator display is shown at the right. Express the number in standard form.

Answer:

Question 16.
a. What should you do first to write 5.871 × 10^-7 in standard form?
b. Express the number in standard form
Answer:

Question 17.
Express 2.58 × 10^-2 in standard form.
Answer:

Question 18.
At a certain point, the Grand Canyon is approximately 1,600,000 centimeters across. Express this number in scientific notation.

Answer:

Question 19.
The length of a bacterial cell is 5.2 × 10^-6 meter. Express the length of the cell in standard form.
Answer:

Question 20.
Higher Order Thinking Express the distance 4,300,000 meters using scientific notation in meters, and then in millimeters.
Answer:

Assessment Practice

Question 21.
Which of the following numbers are written in scientific notation?
12 × 10⁶
12
6.89 × 10⁶
6.89
0.4
4 × 10^-1
Answer:

Question 22.
Jeana’s calculator display shows the number to the right.
PART A
Express this number in scientific notation.

PART B

Express this number in standard form.

3-Act Mathematical Modeling: Hard-Working Organs

3-ACT MATH

ACT 1

Question 1.
After watching the video, what is the first question that comes to mind?
Answer:

Question 2.
Write the Main Question you will answer.
Answer:

Question 3.
Construct Arguments Predict an answer to this Main Question. Explain your prediction.

Answer:

Question 4.
On the number line below, write a number that is too small to be the answer. Write a number that is too large.

Answer:

Question 5.
Plot your prediction on the same number line.

АСТ 2

Question 6.
What information in this situation would be helpful to know? How would you use that information?
Answer:

Question 7.
Use Appropriate Tools What tools can you use to solve the problem? Explain how you would use them strategically.

Answer:

Question 8.
Model with Math Represent the situation using mathematics. Use your representation to answer the Main Question.
Answer:

Question 9.
What is your answer to the Main Question? Is it greater or less than your prediction? Explain why.

Answer:

ACT 3

Question 10.
Write the answer you saw in the video.
Answer;

Question 11.
Reasoning Does your answer match the answer in the video? If not, what are some reasons that would explain the difference?
Answer:

Question 12.
Make Sense and Persevere Would you change your model now that you know the answer? Explain.

Answer:

ACT 3

Reflect

Question 13.
Model with Math Explain how you used a mathematical model to represent the situation. How did the model help you answer the Main Question?
Answer:

Question 14.
Generalize What pattern did you notice in your calculations? How did that pattern help you solve the problem?
Answer:

SEQUEL

Question 15.
Use Structure How many times does a heart beat in a lifetime? Use your solution to the Main Question to help you solve.

Answer:

Lesson 1.10 Operations with Numbers in Scientific Notation

Solve & Discuss It!

The homecoming committee wants to fly an aerial banner over the football game. The banner is 1,280 inches long and 780 inches tall. How many different ways can the area of the banner be expressed?

Focus on math practices
Be Precise Which of the solutions is easiest to manipulate?

? Essential Question
How does using scientific notation help when computing with very large or very small numbers?

Try It!

The planet Venus is on average 2.5 × 10⁷ kilometers from Earth. The planet Mars is on average 2.25 × 10⁸ kilometers from Earth. When Venus, Earth, and Mars are aligned, what is the average distance from Venus to Mars?

Answer:

Convince Me!
In Example 1 and the Try it, why did you move the decimal point to get the final answer?

Try It!

There are 1 × 10¹⁴ good bacteria in the human body. There are 2.6 x 10¹⁸ good bacteria among the spectators in a soccer stadium. About how many spectators are in the stadium? Express your answer in scientific notation.

KEY CONCEPT
Operations with very large or very small numbers can be carried out more efficiently using scientific notation. The properties of exponents apply when carrying out operations.

Do You Understand?

Question 1.
? Essential Question
How does using scientific notation help when computing with very small or very large numbers?

Question 2.
Use Structure When multiplying and dividing two numbers in scientific notation, why do you sometimes have to rewrite one factor?
Answer:

Question 3.
Use Structure For the sum of (5.2 × 10⁴) and (6.95 × 10⁴) in scientific notation, why will the power of 10 be 10⁵?
Answer:

Do You Know How?

Question 4.
A bacteriologist estimates that there are 5.2 × 10⁴ bacteria growing in each of 20 petri dishes. About how many bacteria in total are growing in the petri dishes? Express your answer in scientific notation.
Answer:

Question 5.
The distance from Earth to the Moon is approximately 1.2 × 10⁹ feet. The Apollo 11 spacecraft was approximately 360 feet long. About how many spacecraft of that length would fit end to end from Earth to the Moon? Express your answer in scientific notation.

Answer:

Question 6.
The mass of Mars is 6.42 × 10²³ kilograms. The mass of Mercury is 3.3 × 10²³ kilograms.
a. What is the combined mass of Mars and Mercury expressed in scientific notation?
b. What is the difference in the mass of the two planets expressed in scientific notation?
Answer:

Practice & Problem Solving

Leveled Practice In 7 and 8, perform the operation and express your answer in scientific notation.

Question 7.
(7 × 10^-6)(7 × 10^-6)

Answer:

Question 8.
(3.76 × 10⁵) + (7.44 × 10⁵)

Answer:

Question 9.
What is the value of n in the equation
1.9 × 10⁷ = (1 × 10⁵)(1.9 × 10ⁿ)?
Answer:

Question 10.
Find (5.3 × 10³) – (8 × 10²).
Express your answer in scientific notation.
Answer:

Question 11.
What is the mass of 30,000 molecules? Express your answer in scientific notation.

Answer:

Question 12.
Critique Reasoning Your friend says that the product of 4.8 × 10⁸ and 2 × 10^-3 is 9.6 × 10^-5. Is this answer correct? Explain.
Answer:

Question 13.
Find \(\frac{7.2 \times 10^{-8}}{3 \times 10^{-2}}\). Write your answer in scientific white notation.
Answer:

Question 14.
A certain star is 4.3 × 10² light years from Earth. One light year is about 5.9 × 10¹² miles. How far from Earth (in miles) is the star? Express your answer in scientific notation.

Question 15.
The total consumption of fruit juice in a particular country in 2006 was about 2.28 × 10⁹ gallons. The population of that country that year was 3 × 10⁸. What was the average number of gallons consumed per person in the country in 2006?
Answer:

Question 16.
The greatest distance between the Sun and Jupiter is about 8.166 × 10⁸ kilometers. The greatest distance between the Sun and Saturn is about 1.515 × 10⁹ kilometers. What is the difference between these two distances?
Answer:

Question 17.
What was the approximate number of pounds of garbage produced per person in the country in one year? Express your answer in scientific notation.

Answer:

Question 18.
Higher Order Thinking
a. What is the value of n in the equation
1.5 × 10¹² = (5 × 10⁵)(3 × 10ⁿ)?
b. Explain why the exponent on the left side of the equation is not equal to the sum of the exponents on the right side.

Assessment Practice

Question 19.
Find (2.2 × 10⁵) ÷ (4.4 × 10^-3). When you regroup the factors, what do you notice about the quotient of the decimal factors? How does this affect the exponent of the quotient?
Answer:

Question 20.
Which expression has the least value?
A. (4.7 × 10⁴) + (8 × 10⁴)
B. (7.08 × 10³) + (2.21 × 10³)
C. (5.43 × 10⁸) – (2.33 × 10⁸)
D. (9.35 × 10⁶) – (6.7 × 10⁶)
Answer:

Topic 1 Review

? Topic Essential Question
What are real numbers? How are real numbers used to solve problems?

Vocabulary Review

Use Vocabulary in Writing
Use vocabulary words to explain how to find the length of each side of a square garden with an area of 196 square inches.

Concepts and Skills Review

LESSON 1.1 Rational Numbers as Decimals

Quick Review

You can write repeating decimals in fraction form by writing two equations. You multiply each side of one equation by a power of 10. Then you subtract the equations to eliminate the repeating decimal.

Practice
Write each number as a fraction or a mixed number.

Question 1.
0.\(\overline{7}\)
Answer:

Question 2.
0.0\(\overline{4}\)
Answer:

Question 3.
4.\(\overline{45}\)

Question 4.
2.191919….
Answer:

LESSON 1.2 Understand Irrational Numbers

Quick Review
An irrational number is a number that cannot be written in the form \(\frac{a}{b}\), where a and b are integers and b ≠ 0. Rational and irrational numbers together make up the real number system.

Practice

Question 1.
Determine which numbers are irrational. Select all that apply.
\(\sqrt{36}\)
\(\sqrt{23}\)
-4.232323….
0.151551555….
0.3\(\overline{5}\)
π

Question 2.
Classify -0.\(\overline{25}\) as rational or irrational. Explain.
Answer:

LESSON 1.3 Compare and Order Real Numbers

Quick Review
To compare and order real numbers, it helps to first write each number in decimal form.

Practice

Question 1.
Between which two whole numbers does \(\sqrt{89}\) lie?
\(\sqrt{89}\) is between and
Answer:

Question 2.
Compare and order the following numbers. Locate each number on a number line. 2.\(\overline{3}\), \(\sqrt{8}\), 2.5, 2\(\frac{1}{4}\)
Answer:

LESSON 1.4 Evaluate Square Roots and Cube Roots

Quick Review
Remember that a perfect square is the square of an integer. A square root of a number is a number that when multiplied by itself is equal to the original number. Similarly, a perfect cube is the cube of an integer. A cube root of a number is a number that when cubed is equal to the original number.

Practice
Classify each number as a perfect square, a perfect cube, both, or neither.

Question 1.
27
Answer:

Question 2.
100
Answer:

Question 3.
64
Answer:

Question 4.
24
Answer:

Question 5.
A gift box is a cube with a volume of 512 cubic inches. What is the length of each edge of the box?
Answer:

LESSON 1.5 Solve Equations Using Square Roots and Cube Roots

Quick Review

You can use square roots to solve equations involving squares. You can use cube roots to solve equations involving cubes. Equations with square roots often have two solutions. Look at the context to see whether both solutions are valid.

Practice Solve for x.

Question 1.
x³ = 64
Answer:

Question 2.
x² = 49
Answer:

Question 3.
x³ = 25
Answer:

Question 4.
x² = 125
Answer:

Question 5.
A container has a cube shape. It has a volume of 216 cubic inches. What are the dimensions of one face of the container?

LESSON 1.6 Use Properties of Integer Exponents

Quick Review
These properties can help you write equivalent expressions that contain exponents.
Product of Powers Property
a^m.aⁿ = a^m+n
Power of Powers Property
(a^m)ⁿ = a^mn
Power of Products Property
aⁿ ∙ bⁿ = (a ∙ b)ⁿ
Quotient of Powers Property
a^m ÷ aⁿ = a^m-n, when a ≠ 0

Practice
Use the properties of exponents to write an equivalent expression for each given expression.

Question 1.
6⁴ ∙ 6³
Answer:

Question 2.
(3⁶)^-2
Answer:

Question 3.
7³ ∙ 2³
Answer:

Question 4.
4¹⁰ ÷ 4⁴
Answer:

LESSON 1.7 More Properties of Integer Exponents

Quick Review
The Zero Exponent Property states that any nonzero number raised to the power of 0 is equal to 1. The Negative Exponent Property states that for any nonzero rational number a and integer n, a^-n = \(\frac{1}{a^{n}}\)

Practice
Write each expression using positive exponents.

Question 1.
9^-4
Answer:

Question 2.
\(\frac{1}{3^{-5}}\)
Answer:

Evaluate each expression for x = 2 and y = 5

Question 3.
-4x^-2 + 3y⁰
Answer:

Question 4.
2x⁰y^-2
Answer:

LESSON 1.8 Use Powers of 10 to Estimate Quantities

Quick Review
You can estimate very large and very small quantities by writing the number as a single digit times a power of 10.

Practice

Question 1.
In the year 2013 the population of California was about 38,332,521 people. Write the estimated population as a single digit times a power of 10.
Answer:

Question 2.
The wavelength of green light is about 0.00000051 meter. What is this estimated wavelength as a single digit times a power of 10?
Answer:

Question 3.
The land area of Connecticut is about 12,549,000,000 square meters. The land area of Rhode Island is about 2,707,000,000 square meters. How many times greater is the land area of Connecticut than the land area of Rhode Island?
Answer:

LESSON 1.9 Understand Scientific Notation

Quick Review
A number in scientific notation is written as a product of two factors, one greater than or equal to 1 and less than 10, and the other a power of 10.

Practice

Question 1.
Write 803,000,000 in scientific notation.
Answer:

Question 2.
Write 0.0000000068 in scientific notation.
Answer:

Question 3.
Write 1.359 × 10⁵ in standard form.
Answer:

Question 4.
The radius of a hydrogen atom is 0.000000000025 meter. How would you express this radius in scientific notation?

LESSON 1.10 Operations with Numbers in Scientific Notation

Quick Review
When multiplying and dividing numbers in scientific notation, multiply or divide the first factors. Then multiply or divide the powers of 10. When adding and subtracting numbers in scientific notation, first write the numbers with the same power of 10. Then add or subtract the first factors, and keep the same power of 10.
If the decimal part of the result is not greater than or equal to 1 and less than 10, move the decimal point and adjust the exponent.

Practice
Perform each operation. Express your answers in scientific notation.

Question 1.
(2.8 × 10⁴) × (4 × 10⁵)
Answer:

Question 2.
(6 × 10⁹) ÷ (2.4 × 10³)

Question 3.
(4.1 × 10⁴) + (5.6 × 10⁶)
Answer:

Question 4.
The population of Town A is 1.26 × 10⁵ people. The population of Town B is 2.8 × 10⁴ people. How many times greater is the population of Town A than the population of Town B?
Answer:

Topic 1 Fluency Practice

Crisscrossed

Solve each equation. Write your answers in the cross-number puzzle below. Each digit, negative sign, and decimal point of
your answer goes in its own box.

Across
A -377 = x – 1,000
B x³ = 1,000
C x³ = -8
D x + 7 = -209
F x + 19 = -9
J 14 + x = -9
L m – 2.02 = -0.58
M -3.09 + x = -0.7
N -2.49 = -5 + x
Q x – 3.5 = -3.1
T q – 0.63 = 1.16
V 8.3 + x = 12.1

Down
A y – 11 = 49
B x + 8 = 20
C z³ = -1,331
D 11 + x = 3
E x – 14 -7.96
F 14 + x = -19
G d + 200 = 95
H x² = 144
K -12 = t – 15.95
P 0.3 + x = 11
R x – 3 = -21
S – 7 = -70 + y

Practice with the help of enVision Math Common Core Grade 7 Answer Key Topic 6 Use Sampling to Draw Inferences About Populations regularly and improve your accuracy in solving questions.

enVision Math Common Core 7th Grade Answers Key Topic 6 Use Sampling To Draw Inferences About Populations

TOPIC 6 GET READY!

Review What You Know!

Vocabulary

Choose the best term from the box to complete each definition.

Question 1.
A ____ is how data values are arranged.
Answer:
Data Distribution,

Explanation:
A data distribution is how data values are arranged.

Question 2.
The part of a data set where the middle values are concentrated is called the ___ of the data
Answer:
Center,

Explanation:
The part of a data set where the middle values are concentrated is called the center of the data.

Question 3.
A ___ anticipates that there will be different answers when gathering information.
Answer:
Statistical Question,

Explanation:
A statistical question anticipates that there will be different answers when gathering information.

Question 4.
____ is a measure that describes the spread of values in a data set.
Answer:
Variability,

Explanation:
Variability is a measure that describes the spread of values in a data set.

Statistical Measures

Use the following data to determine each statistical measure.
9, 9, 14, 7, 12, 8, 11, 19, 15, 11

Question 5.
mean : Add all the given numbers, then divide by the amount of numbers
Answer:
Mean is 11.5,

Explanation :
Sum or total = 9+9+14+7+12+8+11+19+15+11=115 (total of given numbers)
MEAN = 115/10=11.5.

Question 6.
median :
7,8,9,9,11,11,12,14,15,19
Answer:
Median is Eleven (11),

Explanation:
Put the numbers from smallest to the largest, the number in the middle is Median,
if two numbers are in the middle, then add them and divide by 2.
Median = (11+11)/2 = 22/2 = 11.

Question 7.
range
Answer:
Range is Twelve (12),

Explanation:
The range is the difference between the lowest to the highest value in the
given numbers 7,8,9,9,11,11,12,14,15,19,
Lowest number is : 7
Highest number is : 19
Range = 19 – 7 = 12.

Question 8.
mode
Answer:
Mode is Two (2),

Explanation:
The value around which there is the greatest concentration is called mode.
Count how many of each value appears in the given numbers or values
Here the modes are 2, that is 9, 9 and 11,11
Mode =3(Median) – 2(Mean),
= 3(11) – 2(11.5),
= 33 – 23,
= 10.

Question 9.
inter quartile range (IQR)
Answer:
Inter quartile Range is Five (5),

Explanation:
Given statistical measures : 9, 9, 14, 7, 12, 8, 11, 19, 15, 11
Arranged in order 7,8,9,9,11,11,12,14,15,19
First half 7,8,9,9,11,
second half  11,12,14,15,19.

Quartiles : The observations which divide the whole set of observations into four equal parts.
lower quartile LQ : Mid number of or median of a given series
First half is lower quartile LQ = 9,

Upper quartile UQ : Mid number of or median of a given series
second half is lower quartile, UQ = 14
interquartile range(IQR) : The difference between the upper quartile and
the lower quartile is called the interquartile range(IQR)
IQR= UQ – LQ = 14 – 9 = 5.

Question 10.
mean absolute deviation (MAD)
Answer:
MAD is 2.8,

Explanation:
The mean absolute deviation of a data set is the average distance between
each data point and the mean. It gives us an idea about the variability in a data set

Step 1:
Calculate the mean.
sum or total = 9+9+14+7+12+8+11+19+15+11=115 (total of given numbers)
MEAN = 115/10=11.5,

Step 2:
 Calculate how far away each data point is from the mean using positive distances.
These are called absolute deviations.
Data Point    Distance from Mean
1)  9              I   9 – 11.5   I    = 2.5
2)  9              I   9 – 11.5   I    = 2.5
3)  14            I   14 – 11.5   I    = 2.5
4)  7              I    7 – 11.5   I    = 4.5
5)  12            I   12 – 11.5   I   =0.5
6)  8              I   8 – 11.5   I    = 3.5
7)  11            I 11 – 11.5 I    = 0.5
8)  19           I 19 – 11.5 I    = 7.5
9)  15           I 15 – 11.5 I    = 3.5
10)  11           I 11 – 11.5 I   = 0.5

Total measure is 2.5+2.5+2.5+4.5+0.5+3.5+0.5+7.5+3.5+0.5=28.0,
Divide the total measure by the number of observation = 28.0/10 =2.8,
MAD = 28/10 = 2.8.

Data Representations
Make each data display using the data from Problems 5-7.

Question 11.
Answer:
Displayed data is collected from problems 5-7,
Box Plot :
a simple way of representing statistical data on a plot in which a rectangle is drawn to
represent the second and third quartiles, usually with a vertical line inside to
indicate the median value. The lower and upper quartiles are shown as horizontal lines
either side of the rectangle.

Explanation:
Box plot:

Question 12.
Dot plot
Answer:
Dot Plot:
A dot plot, also known as a strip plot or dot chart, is a simple form of data visualization
that consists of data points plotted as dots on a graph with an x- and y-axis.
These types of charts are used to graphically depict certain data trends or groupings.

Explanation:

Statistical Questions

Question 13.
Which is NOT a statistical question that might be used to gather data from a certain group?
A. In what state were you born?
B. What is the capital of the United States?
C. How many pets do you have?
D. Do you like strawberry yogurt?

Answer :
B is NOT statistical questions
Explanation :
A statistical question anticipates that there will be different answers when gathering information, where as Capital of United States is the same answer, so the question What is the capital of the United States? might be used to gather data from a certain group.

Language Development

Fill in the graphic organizer. Write each definition in your own words. Illustrate or cite supporting examples.

Answer:

PICK A PROJECT

PROJECT 6A

What types of changes would you like to see in your community?
PROJECT: WRITE TO YOUR REPRESENTATIVE
Answer:

PROJECT 6B
How could you combine physical activity and a fundraiser?
PROJECT: ANALYZE AN ACTIVITY
Answer:
Physical Activity Marathon is one of the fundraiser.
Explanation:

PROJECT 6C
If you could study an animal population in depth, which animal would you choose, and why?
PROJECT: SIMULATE A POPULATION STUDY

Answer : Tiger
After a century of decline, overall wild tiger numbers are starting to tick upward. Based on the best available information, tiger populations are stable or increasing in India, Nepal, Bhutan, Russia and China.

PROJECT 6D
If you were to design a piece of art that moved, how would you make it move?
PROJECT: BUILD A MOBILE

Answer:

Explanation:

Mobiles are free-hanging sculptures that can move in the air. These sculptures are not only artistic, but they are also a great demonstration of balanced forces. If you look at a traditional mobile more closely you will usually notice that it is made of various horizontal rods.

Materials:
Heavy construction paper or cardstock (various colors work well)
Hole punch, Pen, Markers, Scissors, Tape, String, Straws, at least 6
Ceiling or doorframe you can hang the mobile from (and a chair or adult to help in hanging it)

Preparation:
Carefully cut out the different shapes with your scissors. If you like, you can decorate each of them.
Punch a hole into the top center of each of the cut-out shapes.Attach a piece of string to each of the shapes by threading it through the punched hole and tying a knot. Try to vary the length of string attached to each shape so that they are not all the same.

Procedure:
Start with one layer of your mobile. Attach a piece of string to the center of one of your straws. Hold the straw by the string so it is hanging freely in the air. Once the straw is balanced tie your first shape to one end of the straw.Tie a second shape to the other end of the straw then hold the straw up in the air again.Balance the straw by moving one of the shapes along the straw. Use a second straw and two more shapes to build another balanced structure.

Lesson 6.1 Populations and Samples

Solve & Discuss It!

The table shows the lunch items sold on one day at the middle school cafeteria. Use the given information to help the cafeteria manager complete his food supply order for next week.

Generalize
what conclusions can you draw from the lunch data?
Answer:
Highest Sold items are Hot Dog and least sold items are Veggie Burger.
Explanation:
In the middle school cafeteria Highest Sold items are Hot Dog and least sold items are Veggie Burger, this information help the cafeteria manager to complete his food supply order for next week.

Focus on math practices
Construct Arguments Why might it be helpful for the cafeteria manager to look at the items ordered on more than one day?

Sales of food items information to help the cafeteria manager complete his food supply order to avoid wastage of excess food items and the loos occur due to the less or non saleable items.

Essential Question
How can you determine a representative sample of a population?
Answer:
A subset of a population that seeks to accurately reflect the characteristics of the larger group.
Explanation:
In case of Morgan and her friends are sub set of the registered voted of Morgan town for construction of new stadium.

Try It!
Miguel thinks the science teachers in his school give more homework than the math teachers. He is researching the number of hours middle school students in his school spend doing math and science homework each night.
Answer:
The population includes all the students in Miguel’s middle school.
A possible sample is some students from each of the grades in the middle school.

Convince Me!
Why is it more efficient to study a sample rather than an entire population?
Answer:
To study the whole population is often very expensive and time consuming because of the number of people involved.
For example every 10th person of the population and reduce the time by 10 and still get a representative results.

Try It!
A produce manager is deciding whether there is customer demand for expanding the organic food section of her store. How could she obtain the information she needs?
Answer:
The manager can interview the customers random sample for customer demand for expanding the organic food section of her store. As a sample out of the population.

Try It!
Ravi is running against two other candidates for student council president. All of the 750 students in Ravi’s school will vote for student council president. How can Ravi generate a representative sample that will help him determine whether he will win the election?
Answer:
Ravi can ask one in every 10 people to reduce the number of people he needs to interview in order to make a sample out of 750 students, as we know that a random sample 75 students, Ravi generate a representative sample that will help him determine whether he will win the election.

Try It!
The table at the right shows the random sample that Jeremy generated from the same population as Morgan’s and Maddy’s samples. Compare Jeremy’s sample to Morgan’s and Maddy’s.

Answer :
Jeremy’s sample also contains 20samples. and the sample shares the value 36 with Morgan’s sample and one value 126 with Maddy’s sample, the distribution of values in Jeremy’s sample is different then that of Morgans and Maddy.

KEY CONCEPT
A population is an entire group of objects-people, animals, plants—from which data can be collected. A sample is a subset of the population. When you ask a statistical question about a population, it is often more efficient to gather data from a sample of the population.
A representative sample of a population has the same characteristics as the population. Generating a random sample is one reliable way to produce a representative sample of a population. You can generate multiple random samples that are different but that are each representative of the population.

Do You Understand?

Question 1.
Essential Question
How can you determine a representative sample of a population?
Answer:
A representative sample of a population has the same characteristics as the population but generating a random sample.

Question 2.
Construct Arguments Why does a sample need to be representative of a population?
Answer:
A sample is need for Reliability.
Explanation:
A random sample is one reliable way to produce a representative sample of a population.

Question 3.
Be Precise The quality control manager of a peanut butter manufacturing plant wants to ensure the quality of the peanut butter in the jars coming down the assembly line. Describe a representative sampling method she could use.

Answer:
The quality control manager of a peanut butter manufacturing plant must adopt repeat in line quality checking method to ensure of the quality of the peanut butter in the jar coming down the assembly line. the manager or representative must check every 4th peanut butter jars coming down.

Do You Know How?

Question 4.
A health club manager wants to determine whether the members would prefer a new sauna or a new steam room. The club surveys 50 of its 600 members. What is the population of this study?
Answer:
The population of this study is 600.

Question 5.
A journalism teacher wants to determine whether his students would prefer to attend a national writing convention or tour of a local newspaper press. The journalism teacher has a total of 120 students in 4 different classes. What would be a representative sample in this situation?
Answer:
Representative sample is Four(4)
Explanation:
The journalism teacher will prefer 4 representatives, one sample from each class. total 4 representative sample for collecting the student prefer for attending National writing convention or Local newspaper press.

Question 6.
Garret wants to find out which restaurant people think serves the best beef brisket in town.
a. What is the population from which Garret should find a sample?
b. What might be a sample that is not representative of the population?
Answer:
a. A population is an entire group of people in the town visits restaurant for the best beef brisket.

b. A representative sample of a population in a restaurant is who does not prefer the beef brisket.

Practice & Problem Solving

Leveled Practice In 7 and 8, complete each statement with the correct number.

Question 7.
Of a group of 200 workers, 15 are chosen to participate in a survey about the number of miles they drive to work each week.
Answer:
Sample consists of 15 workers out of 500.
Explanation:
In this situation, the sample consists of the 15 workers selected to participate in the survey are random samples.
The population consists of 200 workers.

Question 8.
The ticket manager for a minor league baseball team awarded prizes by drawing four numbers corresponding to the ticket stub numbers of four fans in attendance.
Answer:
In this situation, the sample consists of the 4 people selected to win a prize.
Explanation:
The population consists of 4 the spectators who purchased tickets to attend the game.

Question 9.
A supermarket conducts a survey to find the approximate number of its customers who like apple juice. What is the population of the survey?
Answer:
Representative Sample survey.
Explanation:
A representative sample of a population has the same characteristics as the population. All the population of from that town.

Question 10.
A national appliance store chain is reviewing the performances of its 400 sales associate trainees. How can the store choose a representative sample of the trainees?
Answer:
Random Sample.
Explanation:
 Random sampling is a part of the sampling in which each sample has an equal probability of being chosen. 

Of the 652 passengers on a cruise ship, 30 attended the magic show on board.
a. What is the sample?
b. What is the population?
Answer:
a. sample is 30
b. population is 652
Explanation:
Total number of passengers is equal to the population in ship i.e.,652
Out of which 30 attended magic show, so the sample became 30.

Question 12.
Make Sense and Persevere
The owner of a landscaping company is investigating whether his 120 employees would prefer a water cooler or bottled water. Determine the population and a representative sample for this situation.

Answer:
The Population is 120 employees.
Explanation:
A representative sample is a subset of a population that seeks to accurately reflect the characteristics of the larger group that is 120employees, 12 employees can be considered as representative sample for investigating whether his 120 employees would prefer a water cooler or bottled water.

Question 13.
Higher Order Thinking
A bag contains 6 yellow marbles and 18 red marbles. If a representative sample contains 2 yellow marbles, then how many red marbles would you expect it to contain? Explain.

Answer:
6 marbles are expected.
Explanation:

Question 14.
Chung wants to determine the favorite hobbies among the teachers at his school. How could he generate a representative sample? Why would it be helpful to generate multiple samples?
Answer:
A representative sample is a subset of a population that seeks to accurately reflect the characteristics of the larger group.
It would be helpful to determine the favorite hobbies among teachers.
Explanation:
For example,
A classroom of 30 teachers with 15 males and 15 females could generate a representative sample,
it can help to generate multiple samples, to determine favorite hobbies.

Question 15.
The table shows the results of a survey conducted to choose a new mascot. Yolanda said that the sample consists of all 237 students at Tichenor Middle School.
a. What was Yolanda’s error?
Answer:
Yolanda’s error is the total population is all students must be a round figure 240
Explanation:
Errors happen when you take a sample from the population rather than using the entire population. In other words, it’s the difference between the statistic you measure and the parameter you would find if you took a census of the entire population.
Then 24 out of 240 be a 10% sample
Instead of 24 of 237 is 10.12%
Main reason for sample size in the population is important.

b. What is the sample size? Explain.
Answer:
sample size is 40.
A sample size is a part of the population chosen for a survey or experiment.
Explanation:
For example, you might take a survey of car owner’s brand preferences. You won’t want to survey all the millions of Cars owners in the country, so you take a sample size. That may be several thousand owners. The sample size is a representation of all car owner’s brand preferences. If you choose your sample wisely, it will be a good representation.

Question 16.
Reasoning
To predict the outcome of the vote for the town budget, the town manager assigned random numbers and selected 125 registered voters. He then called these voters and asked how they planned to vote. Is the town manager’s sample representative of the population? Explain.
Answer:
YES, the town manager’s sample representative of the population.
Explanation:
A representative sample is where your sample matches some characteristic of your population, usually the characteristic you’re targeting with your research. the town manager selected 125 registered voters randomly to ask how they plan to vote.

Question 17.
David wants to determine the number of students in his school who like Brussels sprouts. What is the population of David’s study?
Answer:
The population of David’s Study is the the number of students in his school those who like brussels sprouts and dose not like.
Explanation:
A population is a whole, it’s every member of a group. A population is the opposite to a sample, which is a fraction or percentage of a group.

Question 18.
Researchers want to determine the percentage of Americans who have visited The Florida Everglades National Park in Florida. The diagram shows the population of this study, as well as the sample used by the researchers. After their study, the researchers conclude that nearly 75% of Americans have visited the park.
a. What error was likely made by the researchers?

Answer:
A sampling error is a statistical error that occurs when an analyst does not select a sample that represents the entire population of data. As a result, the results found in the sample do not represent the results that would be obtained from the entire population. Here the researchers conclude that nearly 75% of Americans have visited the park.

b. Give an example of steps researchers might take to improve their study.
Answer:
Sampling errors are easy to identify. Here are a few simple steps to reduce sampling error:

1. Increase sample size: A larger sample size results in a more accurate result because the study gets closer to the actual population size.
2. Divide the population into groups: Test groups according to their size in the population instead of a random sample. For example, if people of a specific demographic make up 20% of the population, make sure that your study is made up of this variable.
3. Know your population: Study your population and understand its demographic mix. Know what demographics use your product and service and ensure you only target the sample that matters.

Question 19.
An art teacher asks a sample of students if they would be interested in studying art next year. Of the 30 students he surveys, 81% are already enrolled in one of his art classes this year. Only 11% of the school’s students are studying art this year. Did the teacher survey a representative sample of the students in the school? Explain.

Answer:
The teacher surveys 30 students, total population of the school is not given to infer the survey results.

Question 20.
Make Sense and Persevere
A supermarket wants to conduct a survey of its customers to find whether they enjoy oatmeal for breakfast. Describe how the supermarket could generate a representative sample for the survey.
Answer:
The manager of the super market can interview people at random.
for example in the store every 10th customer use the random sample as a representative sample of the population, enjoy oatmeal for breakfast.

Question 21.
Critique Reasoning
Gwen is the manager of a clothing store. To measure customer satisfaction, she asks each shopper who makes big purchases for a rating of his or her overall shopping experience. Explain why Gwen’s sampling method may not generate a representative sample.
Answer: In general customer visit the store with positive attitude and satisfaction. Here the mistake made by Gwen was sampling big purchases customer for a rating of his or her overall shopping experience rather then considering all the customers of a clothing store.

Assessment Practice

Question 22.
Sheila wants to research the colors of houses on a highly populated street. Which of the following methods could Sheila use to generate a representative sample? Select all that apply.
Assign each house a number and use a random number generator to produce a list of houses for the sample.
Choose every house that has at least 3 trees in the front yard.
Choose only the houses of the people you know.
List the house numbers on slips of paper and draw at least 20% of the numbers out of a box.
Choose all of the houses on the street that have shutters.
Answer:
The statements that apply are
Assign each house a number and use a random number generator to produce a list of houses for the sample
List the house numbers on slips of paper and draw at least 20% of the numbers out of a box.

Question 23.
A national survey of middle school students asks how many hours a day they spend doing homework. Which sample best represents the population?
PART A
A. A group of 941 students in eighth grade in
B. A group of 886 students in sixth grade in a certain county
C. A group of 795 students in seventh grade in different states
D. A group of 739 students in different middle school grade levels from various states

Answer: D
D. A group of 739 students in different middle school grade levels from various states
is correct

PART B
Explain the reasoning for your answer in Part A.
Answer:
Option D is the only answer that covers multiple grades in different states of the country. That way we have the most representative sample among those four.

Lesson 6.2 Draw Inferences from Data

Solve & Discuss It!

The students in Ms. Miller’s class cast their votes in the school-wide vote for which color to paint the cafeteria walls. Based on the data, what might you conclude about how the rest of the school will vote?

Make Sense and Persevere
How many students are in Ms. Miller’s class? How many students voted for each color?
Answer:
30 students are in Ms. Miller’s class.
Explanation:
Number of students voted for each color.
Radical Rule RED = 7
Box plot BLUE = 12
Geometric mean GREEN = 4
y Plane YELLOW = 3
odd Number ORANGE = 4

Focus on math practices
Reasoning How can you determine whether a sample is representative of a population?
Answer :
Reliability
Explanation:
A sample is a subset of the population. A representative sample of a population has the same characteristics as the population. Generating a random sample is one reliable way to produce a representative sample of a population.

Essential Question
How can inferences be drawn about a population from data gathered from samples?
Answer:
By using sample statistics.
Explanation:
A samples are referred to as sample statistics while values concerning a population are referred to as population parameters. The process of using sample statistics to make conclusions about population parameters is known as inferential statistics.

Try It!
Dash collects data on the hair lengths of a random sample of seventh-grade boys in his school.
Answer:
The data are clustered between 1/2  and 2 inches and between 3½ and 4½ inches. Dash can infer from the data that seventh-grade boys in his school have both short and long hair.
Convince Me!
How does a dot plot help you make inferences from data?
Answer :
A Dot Plot is a type of simple histogram-like chart used in statistics for relatively small data sets where values fall into a number of discrete bins

Try It!
Alexis surveys three different samples of 20 students selected randomly from the population of 492 students in the seventh grade about their choice for class president. In each sample, Elijah receives the fewest votes. Alexis infers that Elijah will not win the election. Is her inference valid? Answer:
Yes, Her inference valid.
explanation:
More then 10% for population 492 is surveyed by Alexis by selecting 20 students of 3 groups is total 60 students, Elijah receives less votes.

Try It!
For his report, Derek also collects data from a random sample of eighth graders in his school, and finds that 18 out of 20 eighth graders have cell phones. If there are 310 eighth graders in his school, estimate the number of eighth graders who have cell phones.
Answer:
The number of eighth garden who have cell phones are 279.
Explanation:

KEY CONCEPT

You can analyze numerical data from a random sample to draw inferences about the population. Measures of center, like mean and median, and measures of variability, like range, can be used to analyze the data in a sample.

Do You Understand?

Question 1.
Essential Question
How can inferences be drawn about a population from data gathered from samples?
Answer:
Inferential statistics is a way of making inferences about populations based on samples.
Explanation:
The inferences about the population Hours of sleep per night is 9pm to 9:30pm has more population

Question 2.
Reasoning Why can you use a random sample to make an inference?
Answer:
A random sample is the subset of the population selected without bias in order to make inferences about the entire population.
Explanation:
Random samples are more likely to contain data that can be used to make predictions about a whole population. The size of a sample influences the strength of the inference about the population.

Question 3.
Critique Reasoning
Darrin surveyed a random sample of 10 students from his science class about their favorite types of TV shows. Five students like detective shows, 4 like comedy shows, and 1 likes game shows. Darrin concluded that the most popular type of TV show among students in his school is likely detective shows. Explain why Darrin’s inference is not valid.
Answer:
Darrin’s inference is not valid because he concluded on most of the students like detective shows.
Explanation:
Out of 10 sample students
5 like – detective shows
4 like – comedy shows
1 like – game shows
Darrin’s concluded on most of the students like detective shows.

Do You Know How?

Question 5.
In a carnival game, players get 5 chances to throw a basketball through a hoop. The dot plot shows the number of baskets made by 20 different players.

a. Make an inference by looking at the shape of the data.
Answer:
Total 20 players, each player get 5 chance, except 2 players, 18 players throw a basketball through a hoop successfully.
Explanation:
2 players – Zero out of five score
3 players – 5 out of five score
3 players – 1 out of five score
4 players – 2 out of five score
4 players – 3 out of five score
4 players – 4 out of five score

b. What is the median of the data? What is the mean? Do these measures of center support the inference you made in part (a)?
Answer:
Median =  10
Explanation:
Throw a basketball through a hoop if we arrange in ascending order 0,3,8,12,15,16 as the measures, and the average of 8 and 12 will be the median
(8+12)/2=20/2=10

Answer:
Mean = 9
Explanation:
If we add all the measures and divided by the number as shown below
(0+3+8+12+15+16)/6=54/6=9

Question 6.
In the dot plot above, 3 of 20 players made all 5 baskets. Based on this data, about how many players out of 300 players will make all 5 baskets?
Answer:
45 players will make all 5 baskets.
Explanation:
3 of 20 players made 5 baskets
X of 300 players will make 5 baskets
cross multiply as shown below

Question 7.
The manager of a box office gathered data from two different ticket windows where tickets to a music concert were being sold. Does the data shown in the box plots below support the inference that most of the tickets sold were about $40? Explain.

Answer:
NO, the box plot will not support the inference.
Explanation :
As per the Box Plot most of the tickets sold were about $50 to $60 as IQR or Q2 lies between the 50-60

Practice & Problem Solving

Leveled Practice In 8-10, use the sample data to answer the questions.

Alicia and Thea are in charge of determining the number of T-shirts to order to sell in the school store. Each student collected sample data from the population of 300 students. Alicia surveyed 50 students in the cafeteria. Thea surveyed the first 60 students who arrived at school one morning.

Question 8.
Use Alicia’s data to estimate the number of T-shirts they should order.

Answer:
180 T- shirts should be order.
Explanation:

They should order about 180 T-shirts.

Question 9.
Use Thea’s data to estimate the number of T-shirts they should order.

Answer:
255 T-shirts should be order.
Explanation:

They should order about 255 T-shirts

Question 10.
Construct Arguments Can Alicia or Thea make a valid inference? Explain.
Answer:
Thea : As per my survey 255 students like T- shirts of 300 students
Alicia: As per my survey 180 T-shirts to be ordered for sale of 300 students

Question 11.
Three of the five medical doctors surveyed by a biochemist prefer his newly approved Brand X as compared to the leading medicine. The biochemist used these results to write the TV advertisement shown. Is the inference valid? Explain your answer.

Answer:
Yes, it is valid
Explanation:
60% biochemist survey results approved Brand X as compared to the leading medicine.

Question 12.
Aaron conducted a survey of the type of shoes worn by a random sample of students in his school. The results of his survey are shown at the right.

a. Make a valid inference that compares the number of students who are likely to wear gym shoes and those likely to wear boots.
b. Make a valid inference that compares the number of students who are likely to wear boots and those likely to wear sandals.
Answer:
a) number of students who are likely to wear gym shoes three times more then those likely to wear boots.
b) number of students who are likely to wear boots are two times less then those likely to wear sandals.

Question 13.
Shantel and Syrus are researching the types of novels that people read. Shantel asks every ninth person at the entrance of a mall. She infers that about 26% of the population prefers fantasy novels. Syrus asks every person in only one store. He infers that about 47% of the population prefers fantasy novels.
a. Construct Arguments Whose inference is more likely to be valid? Explain.
b. What mistake might Syrus have made?
Answer:
a) Shantel asks every ninth person at the entrance of a mall is the correct sample survey for researching the type of novels the people read, which gives 26% of the population prefers fantasy novels.
b) Syrus askes in only one store can not give good results even he infers 47% of the population prefers the fantasy navels.

Question 14.
Higher Order Thinking A national TV news show conducted an online poll to find the nation’s favorite comedian. The website showed the pictures of 5 comedians and asked visitors of the site to vote. The news show inferred that the comedian with the most votes was the funniest comedian in the nation.

a. Is the inference valid? Explain.
b. How could you improve the poll? Explain.
Answer:
YES, its valid.
Explanation : conducting survey by a national TV news show by online poll to find the nation’s favorite comedian. the comedian #3 with the most votes was the funniest comedian in the nation.
Broadcasting the news in other channels can participate more population for more accurate results.
In 15 and 16, use the table of survey results from a random sample of people about the way they prefer to view movies.

Question 15.
Lindsay infers that out of 400 people, 300 would prefer to watch movies in a theater. Is her inference valid? Explain.

Answer:
NO, her inference is not valid.
Explanation :
As per survey results form a random sample of people preference is given to Streaming rather then Theater.

Question 16.
Which inferences are valid? Select all that apply.
Going to a theater is the most popular way to watch a movie.
About twice as many people would prefer to stream movies instead of watching in a theater.
About 3 times as many people would prefer to watch a movie on DVD instead of watching in a theater.
About 8 times as many people would prefer to watch a movie on DVD instead of streaming.
Most people would prefer streaming over any other method.
Answer:
Most people would prefer streaming over any other method

Question 17.
Monique collects data from a random sample of seventh graders in her school and finds that 10 out of 25 seventh graders participate in after-school activities. Write and solve a proportion to estimate the number of seventh graders, n, who participate in after-school activities if 190 seventh graders attend Monique’s school.
Answer:
76 students participated.
Explanation:

Question 18.
Each of the 65 participants at a basketball camp attempted 20 free throws. Mitchell collected data for the first 10 participants, most of whom were first-time campers. Lydia collected data for the next 10 participants, most of whom had attended the camp for at least one week.

a. Using only his own data, what inference might Mitchell make about the median number of free throws made by the 65 participants?
Answer:
Median is 9 as per the Mitchell data
Explanation:
IQR remains same even more entries raise.
b. Using only her own data, what inference might Lydia make about the median number of free throws made by the 65 participants?
Answer:
Median is 12 as per the Lydia’s Data
Explanation IQR remain same

c. Who made a valid inference? Explain.
Answer:
Both Mitchell and Lydia made a valid inference.
Explanation:
Mitchell collected data of first 10 participants, most of whom were first-time campers. Lydia collected data for the next 10 participants, most of whom had attended the camp for at least one week.

Assessment Practice

Question 19.
June wants to know how many times most people have their hair cut each year. She asks two of her friends from Redville and Greenburg, respectively, to conduct a random survey. The results of the surveys are shown below.
Redville surveyed on 50 people
Median number of haircuts: 7
Mean number of haircuts: 7.3

Greenburg: 60 people surveyed
Median number of haircuts: 6.5
Mean number of haircuts: 7.6
June infers that most people get 7 haircuts per year. Based on the survey results, is this a valid inference? Explain.
Answer:
YES, its valid inference.
Explanation :
The mean and median are the averages of the survey measures or data collected and number 7 lies in between the 6.5 to 7.6 its a valid

TOPIC 6 MID-TOPIC CHECKPOINT

Question 1.
Vocabulary Krista says that her chickens lay the most eggs of any chickens in the county. To prove her claim, she could survey chicken farms to see how many eggs each of their chickens laid that day. In this scenario, what is the population and what is a possible representative sample?
Answer:
A population is an entire group of objects-people, animals, plants—from which data can be collected.
A representative sample of a population has the same characteristics as the population.

Question 2.
Marcy wants to know which type of book is most commonly checked out by visitors of her local public library. She surveys people in the children’s reading room between 1:00 and 2:00 on Saturday afternoon. Select all the statements about Marcy’s survey that are true. Lesson 6-1
Marcy’s sample is not representative because not all of the library’s visitors go to the children’s reading room.
Marcy’s sample is a representative sample of the population.
Marcy will get a random sample by surveying as many people in the children’s reading room as possible.
The population of Marcy’s study consists of all visitors of the public library.
The results of Marcy’s survey include a mode, but neither a mean nor a median.

For Problems 3-5, use the data from the table.

Question 3.
Michael surveyed a random sample of students in his school about the number of sports they play. There are 300 students in Michael’s school. Use the results of the survey to estimate the number of students in Michael’s school who play exactly one sport. Explain your answer. Lesson 6-2

Answer:
45 students play exactly one sport
Explanation:

Question 4.
What inference can you draw about the number of students who play more than one sport? Lesson 6-2
Answer:
160 students play more then 1 sport.
Explanation:

Question 5.
Avi says that Michael’s sample was not random because he did not survey students from other schools. Is Avi’s statement correct?
Explain. Lesson 6-1
Answer:
No, Avi’s statement is not correct.
Explanation:
Michael’s survey is about his school students and their play, not about other school.
How well did you do on the mid-topic checkpoint? Fill in the stars.

Topic 6 MID-TOPIC PERFORMANCE TASK

Sunil is the ticket manager at a local soccer field. He wants to conduct a survey to determine how many games most spectators attend during the soccer season.

PART A
What is the population for Sunil’s survey? Give an example of a way that Sunil could collect a representative sample of this population.
Answer:
The population in Sunil’s survey is 150.
He could collect the sample from ticket manager at a local Soccer field.
Explanation:
According to part B population of sunil’s survey is determined.

PART B
Sunil conducts the survey and obtains the results shown in the table below. What can Sunil infer from the results of the survey?

Answer:
Only 1 or 2 games be conducted.
Explanation:
From the above survey he concluded.

PART C

Suppose 2,400 spectators attend at least one game this soccer season. Use the survey data to estimate the number of spectators who attended 5 or more games this season. Explain how you made your estimate.
Answer:
The survey data to estimate the number of spectators who attended 5 or more games this season is 2,400.

Lesson 6.3 Make Comparative Inferences About Populations

Explore It!

Ella surveys a random sample of 20 seventh graders about the number of siblings they have.

The table shows the results of her survey.

A. Model with Math Draw a model to show how Ella can best display her data.
Answer:

Explanation:
In the above data number of students siblings are displayed on DOT PLOT.
B. Explain why you chose that model.
Dot Plot is easy and effective to show the data.
Explanation:
Dot plot and Box plot are the types of math draw models, that data can be shown in chart format, here Dot plot taken as it is easy and effective way of showing data on charts,

Focus on math practices
Reasoning Using your data display, what can you infer about the number of siblings that most seventh graders have? Explain.
Answer:
Only 1 sibling.
Explanation:
As shown in the above dot plot of 7th grade students has numbers of siblings 1 are more

Essential Question
How can data displays be used to compare populations?
Answer :
Data can be displayed using Dot plot, Box plot or Histogram to compare the population for concluding valid reasons.

Try It!

Kono gathers the heights of a random sample of sixth graders and seventh graders and displays the data in box plots. What can he say about the two data sets?

Answer:
The median of the 7th grade sample is greater than the median of the 6th grade sample.
The 7th grade sample has greater variability.
Explanation:
By comparing both the box plots, he concluded that 7th grade has greater variability. Most 7th grade students have one sibling 8 out of 20 students

Convince Me!
How can you visually compare data from two samples that are displayed in box plots?
Answer:
Guidelines for comparing boxplots from two sam

Try It!

A local recreation center offers a drop-in exercise class in the morning and in the evening. The attendance data for each class over the first month is shown in the box plots at the right. What can you infer about the class attendance?

Answer:
The line for the median of evening attendance data set is to the right of the line for the median of morning attendance data set.
so, morning attendance data can say that the median of evening attendance data set is greater.
Explanation:
The box for evening attendance data set is longer then the morning data set.so, evening attendance data is more spread out or grater variability

KEY CONCEPT
You can use data displays, such as box plots, to make informal comparative inferences about two populations. You can compare the shapes of the data displays or the measures of center and variability.

Do You Understand?

Question 1.
Essential Question How can data displays be used to compare populations?
Answer:
The median and variability of measures are compared between the data set A and set B

Question 2.
Generalize What measures of variability are used when comparing box plots? What do these measures tell you?
Answer:
The median and variability of measures are compared between the data set A and set B.
Explanation:
Here the median is same 5, and the data set B, variability is 9 , that is greater compared to data set A  is 7.
So ,these measures tell us the greater or smaller comparison

Question 3.
Make Sense and Persevere Two data sets both have a median value of 12.5. Data Set A has an interquartile range of 4 and Data Set B has an interquartile range of 2. How do the box plots for the two data sets compare?
Answer:
BOX PLOT
Explanation:

Do You Know How?

The box plots describe the heights of flowers selected randomly from two gardens. Use the box plots to answer 4 and 5.

Question 4.
Find the median of each sample.
Garden Y median = ___ inches
Garden Z median = ___  inches
Answer:
Garden Y median = 6 inches
Garden Z median = 4 inches
Explanation :
In the above box plot the median or IQR is shown in the graph

Question 5.
Make a comparative inference about the flowers in the two gardens.
Answer:
Heights of the flowers in the Garden .
Y is greater then the Garden Z as the median of the Garden Y flowers is right to the Garden Z flowers.
Garden Y is more spread out or grater variability compare to the Garden Z.
Explanation:
Compare the gardens of Y and Z, displayed in the box plot.

Practice & Problem Solving

Leveled Practice For 6-8, complete each statement.

Question 6.
Water boils at lower temperatures as elevation increases. Rob and Ann live in different cities. They both boil the same amount of water in the same size pan and repeat the experiment the same number of times. Each records the water temperature just as the water starts to boil. They use box plots to display their data. Compare the medians of the box plots.
The median of Rob’s data is the median of Ann’s data.
This means Rob is at elevation than Ann.

Explanation:

Question 7.
Liz is analyzing two data sets that compare the amount of food two animals eat each day for one month.

a. The median of Animal 2’s data is than the median of Animal 1’s data
b. Liz can infer that there is variability in the data for Animal 1 than for Animal 2.
c. Liz can infer that Animal generally eats more food.
Answer:

Question 8.
The box plots show the heights of a sample of two types of trees.
The median height of Tree is greater.

Answer:
Tree 1 height is greater then Tree 2.
Explanation:
The median of the Tree 1 is right side of the Tree 2 median,
So the Height of the Tree 1 is greater.

Question 9.
Reasoning A family is comparing home prices in towns where they would like to live. The family learns that the median home price in Hometown is equal to the median home price in Plainfield and concludes that the homes in Hometown and Plainfield are similarly priced.
What is another statistical measure that the family might consider when deciding where to purchase a home?

Answer:
If median is same, then Mean can be a another statistical measure to check for the best option, and variability of the space of the house plot can be compared for the greater one in space.
Explanation:
Mean and median both try to measure the “central tendency” in a data set. The goal of each is to get an idea of a “typical” value in the data set. The mean is commonly used, but sometimes the median is preferred.

Question 10.
Higher Order Thinking The box plots show the daily average high temperatures of two cities from January to December. Which city should you live in if you want a greater variability in temperature? Explain.

Answer:
City X has greater variability.
Explanation :
City X and Y are of same temperature variability, but the median of City X is less then the City Y.

Assessment Practice

Question 11.
Paul compares the high temperatures in City 1 and City 2 for one week. In City 1, the range in temperature is 10°F and the IQR is 5°F. In City 2, the range in temperature is 20°F and the IQR is 5°F. What might you conclude about the weather pattern in each city based on the ranges and interquartile ranges?
A. The weather pattern in City 1 is more consistent than the weather pattern in City 2.
B. The weather patterns in City 1 and City 2 are equally consistent.
C. The weather pattern in City 2 is more consistent than the weather pattern in City 1.
D. The range and interquartile range do not provide enough information to make a conclusion.
Answer:
Option A
Explanation:
The IQR of both the Cities are same (IQR is 5°F) but the rage in temperature city 1 has less then the city 2.

Lesson 6.4 Make More Comparative Inferences About Populations

Explore It!

Jackson and his brother Levi watch Jewel Geyser erupt one afternoon. They record the time intervals between eruptions. The dot plot shows their data.
Jackson estimates that the average time between eruptions is 8 minutes. Levi estimates that the average time between eruptions is 8\(\frac{1}{2}\) minutes.

A. Construct Arguments Construct an argument to support Jackson’s position.
Jackson estimates the average time between eruptions is 8 minutes.
Levi estimates the average time between eruptions is 8\(\frac{1}{2}\) minutes. Jackson estimate is nearer to the median value of the time as shown in the above dot plot

B. Construct Arguments Construct an argument to support Levi’s position.
Jackson estimates the average time between eruptions is 8 minutes. Levi estimates the average time between eruptions is 8\(\frac{1}{2}\) minutes. Levi’s estimate is exactly value of the time  as shown in the above dot plot

Focus on math practices
Reasoning How can you determine the best measure of center to describe a set of data?

Answer:
we can determine the best measure of center to describe a set of data is by finding the Mean and Median of data.
Explanation:
The mean (average) of a data set is found by adding all numbers in the data set and then dividing by the number of values in the set.
The median is the middle value when a data set is ordered from least to greatest. The mode is the number that occurs most often in a data set.

Essential Question
How can dot plots and statistical measures be used to compare populations?
Answer:
By calculating the Mean and Median of data and variability and range.
Explanation:
The mean (average) of a data set is found by adding all numbers in the data set and then dividing by the number of values in the set.
The median is the middle value when a data set is ordered from least to greatest. The mode is the number that occurs most often in a data set.
 Variability is also referred to as spread, scatter or dispersion.
Range: the difference between the highest and lowest values.

Try It!

Quinn also collects data about push-ups. Does it appear that students generally did more push-ups last year or this year? Explain your reasoning.

Answer:
No, it does not give any inference.
Explanation:
The students do less push-ups last year then this year.

Convince Me!
How does the range of these data sets affect the shape of the dot plots?

Answer:
In a dot plot, range is the difference between the values represented by the farthest. left and farthest right dots.
Explanation :
Range is 12 – 3 = 9

Try It!

Peter surveyed a random sample of adults and a random sample of teenagers about the number of hours that they exercise in a typical week. He recorded the data in the table below. What comparative inference can Peter make from the data sets?

Answer:
The mean is 4.4 of adults is less then the Teenagers 7.9 the average the number of hours that they exercise in a typical week is more for teenagers.
Explanation:
Drawing comparative inferences may involve analyzing the data using mean, median, mean absolute deviation, interquartile range, range, and/or mode. In this lesson students will analyze data in different forms and draw informal comparative inferences about the populations involved.

KEY CONCEPT

You can use dot plots to make informal comparative inferences about two populations. You can compare the shapes of the data displays or the measures of center and variability.

The modes of Data Set B are greater than the modes of Data Set A.
The mean of Data Set B is greater than the mean of Data Set A.
You can infer that data points are generally greater in Data Set B.
The ranges and the MADs of the data sets are similar. You can infer that the variabilities of the two data sets are about the same.

Do You Understand?

Question 1.
Essential Question How can dot plots and statistical measures be used to compare populations?

Answer:
The statistical measures of the data of set A and B are compared with reference to the Median, Mode of the sets and the variability of the measures are compared for the population.

Question 2.
Reasoning How can you make predictions using data from samples from two populations?
Answer:
we call this making predictions using random sampling from two population. We basically take data from a random sample of two population and make predictions about the whole population based on that data. Random sample – A sample of a population that is random and the every elements of the population is equally likely to be chosen for the sample.

Question 3.
Construct Arguments Two data sets have the same mean but one set has a much larger MAD than the other. Explain why you may want to use the median to compare the data sets rather than the mean.
Answer:
Both set A and B have the same mean.

Do You Know How?

For 4 and 5, use the information below.

Coach Fiske recorded the number of shots on goal his first-line hockey players made during two weeks of hockey scrimmage.

Question 4.
Find the mean number of shots on goal for each week.
Answer:
Mean of week 1 is 5 and week 2 is 7
Explanation:
Mean  in week 1 is 5
(5+4+6+8+2+3+7)/5=35/7=5

Mean in week 2 is 7
(8+7+9+5+5+7+8)/7=49/7=7

Question 5.
a. Based on the mean for each week, in which week did his first line take more shots on goal?
b. Based on the comparison of the mean and the range for Week 1 and Week 2, what could the coach infer?
Answer:
a. week 2
b. the range in week 1 is more then the week 2 and the mean in week 1 is less then the week 2,
week 2 is better then the week 1 as Coach Fiske recorded the number of shots on goals made during two weeks of hockey.

Practice & Problem Solving

Question 6.
A study is done to compare the fuel efficiency of cars. Cars in Group 1 generally get about 23 miles per gallon. Cars in Group 2 generally get about 44 miles per gallon. Compare the groups by their means. Then make an inference and give a reason the inference might be true.

The mean for Group is less than the mean for Group .
The cars in Group generally are more fuel-efficient.
The cars in Group may be smaller.
Answer:

Question 7.
The dot plot shows a random sample of vertical leap heights of basketball players in two different basketball camps. Compare the mean values of the dot plots. Round to the nearest tenth.

The mean values tell you that participants in Camp jump higher in general.
Answer:
camp 1 mean = 28 in
camp 2 mean = 24 in

Explanation:

Question 8.
A researcher divides some marbles into two data sets. In Data Set 1, the mean mass of the marbles is 13.6 grams. In Data Set 2, the mean mass of the marbles is 14 grams. The MAD of both data sets is 2. What can you infer about the two sets of marbles?
Answer:
The mass of marbles in set 2 is higher then the marbles in set 1 as comparing the mean mass of the marbles of both sets.

Question 9.
Generalize Brianna asks 8 classmates how many pencils and erasers they carry in their bags. The mean number of pencils is 11. The mean number of erasers is 4. The MAD of both data sets is 2. What inference could Brianna make using this data?
Answer:
Total 88 pencils and 32 erasers they carry in their bags.
Explanation:
Pencils =(P)/8 = 11
P= 11×8=88
Erasers = (E)/8=4
E=8×4 = 32

Question 10.
Higher Order Thinking Two machines in a factory are supposed to work at the same speed to pass inspection. The number of items built by each machine on five different days is recorded in the table. The inspector believes that the machines should not pass inspection because the mean speed of Machine X is much faster than the mean speed of Machine Y.

a. Which measures of center and variability should be used to compare the performances of each machine? Explain.
b. Is the inspector correct? Explain.
Answer:
a. Median and IQR are used.
Explanation:

b. YES, the inspector is correct.
Explanation:
The mean speed of Machine X is faster then the Machine Y, they should run same speed but they are varying with an average speed of 2.2.

Assessment Practice

Question 11.
The dot plots show the weights of a random sample of fish from two lakes. Which comparative inference about the fish in the two lakes is most likely correct?

A. There is about the same variation in weight between small and large fish in both lakes.
Answer:
No, there is variation.
Explanation:
Variation of weights in Round Lake is from 15 to 21 ounces, difference is 6 ounce. In South lake weights vary from 11 to 21 ounces, difference is 10 ounce
B. There is less variation in weight between small and large fish in South Lake than between small and large fish in Round Lake.
Answer:
No, there is variation.
Explanation:
Variation in south lake is higher then the Round lake, they differ with 10 ounce and 6 ounce respectively.

C. There is less variation in weight between small and large fish in Round Lake than between small and large fish in South Lake.
Answer:
YES, 6 ounce in Round lake.
Explanation:
Variation in south lake is higher then the Round lake, they differ with 10 ounce and 6 ounce respectively
D. There is greater variability in the weights of fish in Round Lake.
Answer:
No, there is variation.
Explanation:
Less variability in the weights of the fish in the round lake.

3-Act Mathematical Modeling: Raising Money

ACT 1

Question 1.
After watching the video, what is the first question that comes to mind?
Answer:

Question 2.
Write the Main Question you will answer.
Answer:

Question 3.
Make a prediction to answer this Main Question.
Answer:

Question 4.
Construct Arguments Explain how you arrived at your prediction.

Answer:

ACT 2

Question 5.
What information in this situation would be helpful to know? How would you use that information?

Answer:

Question 6.
Use Appropriate Tools What tools can you use to solve the problem? Explain how you would use them strategically.
Answer:

Question 7.
Model with Math Represent the situation using mathematics. Use your representation to answer the Main Question.
Answer:

Question 8.
What is your answer to the Main Question? Does it differ from your prediction? Explain.

Answer:

ACT 3

Question 9.
Write the answer you saw in the video.
Answer:

Question 10.
Reasoning Does your answer match the answer in the video? If not, what are some reasons that would explain the difference?

Answer:

Question 11.
Make Sense and Persevere Would you change your model now that you know the answer? Explain.

Answer:

ACT 3

Extension

Reflect

Question 12.
Model with Math Explain how you used a mathematical model to represent the situation. How did the model help you answer the Main Question?
Answer:

Question 13.
Critique Reasoning Explain why you agree or disagree with each of the arguments in Act 2.

SEQUEL

Question 14.
Use Appropriate Tools You and your friends are starting a new school club. Design a sampling method that is easy to use to help you estimate how many people will join your club. What tools will you use?

Answer:

TOPIC 6 REVIEW

Topic Essential Question

How can sampling be used to draw inferences about one or more populations?

Vocabulary Review

Complete each definition, and then provide an example of each vocabulary word used.

1. A population is a entire group of objects from which data can be collected.

2. Making a conclusion by interpreting data is called making an inference

3. A valid inference is one that is true about a population based on a representative sample.

4 A(n) representative sample accurately reflects the characteristics of an entire population.

Use Vocabulary in Writing

Do adults or teenagers brush their teeth more? Nelson surveys two groups: 50 seventh grade students from his school and 50 students at a nearby college of dentistry. Use vocabulary words to explain whether Nelson can draw valid conclusions.

Concepts and Skills Review

LESSON 6-1 Populations and Samples

Quick Review
A population is an entire group of people, items, or events. Most populations must be reduced to a smaller group, or sample, before surveying. A representative sample accurately reflects the characteristics of the population. In a random sample, each member of the population has an equal chance of being included.

Practice

Question 1.
Anthony opened a new store and wants to conduct a survey to determine the best store hours. Which is the best representative sample?
A. A group of randomly selected people who come to the store in one week
B. A group of randomly selected people who visit his website on one night
C. Every person he meets at his health club one night
D. The first 20 people who walk into his store one day
Answer:
Option A

Question 2.
Becky wants to know if she should sell cranberry muffins at her bakery. She asks every customer who buys blueberry muffins if they would buy cranberry muffins. Is this a representative sample? Explain.
Answer:
A representative sample accurately reflects the characteristics of the population. Those like blueberry muffins may not like cranberry muffins
up to their choice.

Question 3.
Simon wants to find out which shop has the best frozen fruit drink in town. How could Simon conduct a survey with a sample that is representative of the population?
Answer:
A representative sample accurately reflects the characteristics of the population. that means Simon select some representative samples of his friends as a random sample, to find out shop has the best frozen fruit drink in town.

LESSON 6-2 Draw Inferences from Data

Quick Review
An inference is a conclusion about a population based on data from a sample or samples. Patterns or trends in data from representative samples can be used to make valid inferences. Estimates can be made about the population based on the sample data.

Practice

Question 1.
Refer to the example. Polly surveys two more samples. Do the results from these samples support the inference made from the example?

Answer:
In all three samples Polly collected there is the least number of students that do crafts for a hobby
that means that Polly made the correct inference

LESSONS 6-3 AND 6-4
Make Comparative Inferences About Populations | Make More Comparative Inferences About Populations

Quick Review
Box plots and dot plots are common ways to display data gathered from samples of populations. Using these data displays makes it easier to visually compare sets of data and make inferences. Statistical measures such as mean, median, mode, MAD, interquartile range (IQR), and range can also be used to draw inferences when comparing data from samples of two populations.

Practice

Question 1.
The two data sets show the number of days that team members trained before a 5K race.

a. What inference can you draw by comparing the medians?
Answer :
Team B median is higher then the A median, The range of Team A is higher then the Team B of the number of days that team members trained before a 5K race

b. What inference can you draw by comparing the interquartile ranges?
Answer:
Team B IQR is right side of the Team A IQR  in the above box plot of days that team members trained before a 5K race
IQR of Team A is 20 and IQR of Team B is 24

Question 2.
The dot plots show how long it took students in Mr. Chauncey’s two science classes to finish their science homework last night. Find the means to make an inference about the data.

Answer:
i) mean of first period is 38.75 Minutes
ii) mean of second period is 35 Minutes

Explanation:
Mean of first period=(20×2+25×2+30×4+35×3+45×2+50×3+55×4)/20=775/20=38.75

Mean of Second Period=(15×1+20×2+25×4+30×4+35×2+45×3+50×1+55×2+60×1)/20=700/20=35

Second period home work taken less time then first period science home work

TOPIC 6 Fluency Practice

Riddle Rearranging

Find each percent change or percent error. Round to the nearest whole percent as needed. Then arrange the answers in order from least to greatest. The letters will spell out the answer to the riddle below.

V
A young tree is 16 inches tall. One year later, it is 20 inches tall. What is the percent increase in height?

Answer : 25%
Explanation:
The Tree grew for 4 inches, we need to find the % of the starting height that equals

A
A ship weighs 7 tons with no cargo. With cargo, it weighs 10.5 tons. What is the percent increase in the weight?
Answer: 50%
Explanation:
The ship’s weight changed for 3.5 tons.
Divide 3.5 by the weight of the ship with no cargo to calculate the percentage increase in the ships weight.

R
The balance of an account is $500 in April. In May it is $440. What is the percent decrease in the balance?
Answer: 12%
Explanation:
The change in the balance 60 out of 500.
divide the two values to calculate the percent decrease in the balance.

B
Ben thought an assignment would take 20 minutes to complete. It took 35 minutes. What is the percent error in his estimate of the time?
Answer : 42.86%
Explanation:
Divide the absolute value of the error (15) by the actual time it took to him to complete the assignment (35).

N
Natalie has $250 in savings. At the end of 6 months, she has $450 in savings. What is the percent increase in the amount of her savings?
Answer: 80%
Explanation:
The absolute increase of the money in Natalie’s bank account is $200 and we know that she started the period with $250 in her account.
Divide the two values to calculate the percent increase of the money in her account.

I
The water level of a lake is 22 feet. It falls to 18 feet during one month. What is the percent decrease in the water level?
Answer: 18%
Explanation :
The water lowered for 4 feet in the fall .
Divide that by the water level before the decrease the to calculate the percent decrease of the water level.

R
Shamar has 215 photos on his cell phone. He deletes some so that only 129 photos remain. What is the percent decrease in the number of photos?

Answer : 40%
Explanation:
Divide the number of pictures Shamar deleted(215-129=86) by the number of pictures Shamar had on his phone before he started deleting them(215).

K
Lita estimates she will read 24 books during the summer. She actually reads 9 books. What is the percent error of her estimate?

Answer: 167%
Explanation:
Divide the value of the absolute error.
Lita made(24-9=15) by the actual number of books she red (9), to calculate the percent error she made.

E
Camden estimates his backpack weighs 9 pounds with his books. It actually weighs 12 pounds. What is the percent error of his estimate?

Answer: 25%
Explanation:
The absolute error Camden made is 3
Divide that by the actual weight of the backpack(12) to calculate the percent error Camden made.

Answer: RIVER BANK

Explanation :
Arrange the values from smallest to largest
12<18<25=25<40<42.86<50<80<167
The letters put together give the solution to the Riddle.

Practice with the help of enVision Math Common Core Grade 7 Answer Key Topic 4 Generate Equivalent Expressions regularly and improve your accuracy in solving questions.

enVision Math Common Core 7th Grade Answers Key Topic 4 Generate Equivalent Expressions

Topic 4 Essential Question
How can properties of operations help to generate equivalent expressions that can be used in solving problems?

3-ACT MATH

Topic 4 enVision STEM Project

Did You Know?
In 2013, just over 30% of American consumers knew about activity trackers. By 2015, about 82% recognized them.

Continued research and development leads to technological advances and breakthroughs, such as the use of biosensing apparel to track activity.

Your Task: Analyze Activity Tracker Data
The ways that data are communicated and presented to the user are just as important as the types of data collected. You and your classmates will continue your exploration of activity trackers and use data to develop models based on individual fitness goals.

Topic 4 Get Ready

Review What You Know

Vocabulary

Choose the best term from the box to complete each definition.
evaluate
expression
factor
order of operations
substitute
term

Question 1.
When you __________ an expression, you replace each variable with a given value.

Answer:
When you evaluate an expression, you replace each variable with a given value.

Explanation:
In the above-given question,
given that,
when we evaluate an expression, we replace each variable with a given value.
for example:
Evaluate 3a-2b.
for a = 6 and b = 4.
3(6) – 2(4).
18 – 8.
10.

Question 2.
To evaluate a + 3 when a = 7, you can _________ 7 for a in the expression.

Answer:
To evaluate a + 3 when a = 7, you can substitute 7 for a in the expression.

Explanation:
In the above-given question,
given that,
if we evaluate a + 3 when a = 7.
we will substitute 7 for a in the expression.
a + 3.
7 + 3.
10.

Question 3
The set of rules used to determine the order in which operations are performed is called the _________

Answer:
The set of rules used to determine the order in which operations are performed is called the order of operations.

Explanation:
In the above-given question,
given that,
The set of rules used to determine the order in which operations are performed is called the order of operations.
for example:
3 + [6(11 + 1 – 4)]/8 x 2.
3+[6(8)]/8 x 2.
3 + 48 / 8 x 2.
3 + 6 x 2.
3 + 12.
15.

Question 4.
Each part of an expression that is separated by a plus or minus sign is a(n) __________.

Answer:
Each part of an expression that is separated by a plus or minus sign is the term.

Explanation:
In the above-given question,
given that,
Each part of an expression that is separated by a plus or minus sign is the term.
for example:
2x + 4y – 9.
where x and y are variables.
9 is the constant.
2 and 4 are coefficients.
terms are 2x, 4y, and 9.

Question 5.
A(n) __________ is a mathematical phrase that can contain numbers, variables, and operation symbols.

Answer:
An expression is a mathematical phrase that can contain numbers, variables, and operation symbols.

Explanation:
In the above-given question,
given that,
An expression is a mathematical phrase that can contain numbers, variables, and operation symbols.
for example:
n + 7 = 10.
x – 5 = 3.
3p = 15.
y/2 = 5.

Question 6.
When two numbers are multiplied to get a product, each number is called a(n) _________.

Answer:
When two numbers are multiplied to get a product, each number is called a factor.

Explanation:
In the above-given question,
given that,
When two numbers are multiplied to get a product, each number is called a factor.
for example:
3 x 5 = 15.
3 and 5 are the factors.
15 is the product.

Order of Operations

Evaluate each expression using the order of operations.
Question 7.
3(18 – 7) + 2

Answer:
3(18 – 7) + 2 = 35.

Explanation:
In the above-given question,
given that,
3(18 – 7) + 2.
3(11) + 2.
33 + 2.
35.
3(18 – 7) + 2 = 35.

Question 8.
(13 + 2) ÷ (9 – 4)

Answer:
(13 + 2) ÷ (9 – 4) = 3.

Explanation:
In the above-given question,
given that,
(13 + 2) ÷ (9 – 4).
(13 + 2) ÷ (9 – 4).
15 / 5.
3.
(13 + 2) ÷ (9 – 4) = 3.

Question 9.
24 ÷ 4 • 2 – 2

Answer:
24 ÷ 4 • 2 – 2 = 10.

Explanation:
In the above-given question,
given that,
24 ÷ 4 • 2 – 2.
6 . 2 – 2.
12 – 2.
10.
24 ÷ 4 • 2 – 2 = 10.

Equivalent Expressions

Evaluate each expression when a = -4 and b = 3.
Question 10.
ab

Answer:
ab = -12.

Explanation:
In the above-given question,
given that,
a = -4 and b = 3.
– 4 x 3.
-12.
ab = -12.

Question 11.
2a + 3b

Answer:
2a + 3b = 1.

Explanation:
In the above-given question,
given that,
a = -4 and b = 3.
2(-4) + 3(3).
-8 + 9.
1.
2a + 3b = 1.

Question 12.
2(a – b)

Answer:
2(a – b) = -14.

Explanation:
In the above-given question,
given that,
a = -4 and b = 3.
2(-4 – 3).
2(-7).
-14.
2(a – b) = -14.

Question 13.
Explain the difference between evaluating 3 • 7 – 4 ÷ 2 and evaluating 3(7 – 4) ÷ 2.

Answer:
The two expressions are different.

Explanation:
In the above-given question,
given that,
3 . 7 – 4 ÷ 2.
3 . 7 – 2.
21 – 2.
19.
3(7 – 4) ÷ 2.
3(3) / 2.
9 / 2.

Language Development

Complete each math statement using the word bank.

To evaluate an algebraic expression, substitute a __________ for the variable in the expression.

Answer:
To evaluate an algebraic expression, substitute properties of operations for the variable in the expression.

Explanation:
In the above-given question,
given that,
To evaluate an algebraic expression, substitute properties of operations for the variable in the expression.
for example:
n + 7 = 10.
x – 5 = 3.
3p = 15.
y/2 = 5.

In the algebraic expression 3(x – 2), 3 and x – 2 are ___________

Answer:
In the algebraic expression 3(x – 2), 3, and x – 2 are coefficients.

Explanation:
In the above-given question,
given that,
In the algebraic expression 3(x – 2), 3 and x – 2 are coefficients.
for example:
3(x – 2).
3 and x-2 are coefficients.

To generate equivalent expressions, you can use the __________

Answer:
To generate equivalent expressions, you can use the order of operations.

Explanation:
In the above-given question,
given that,
To generate equivalent expressions, you can use the order of operations.
for example:
5(x – 1) + 7.
5(x) + 5(-7) + 7.
5x – 5 + 7.
5x + 2.

In the expression 4x + 2x – 6y, you first need to __________

Answer:
In the expression 4x + 2x – 6y, you first need to add.

Explanation:
In the above-given question,
given that,
In the expression 4x + 2x – 6y, you first need to add.
4x + 2y – 6y.
4x – 4y.

You can use the Distributive Property to __________ the algebraic expression 5(x – 7).

Answer:
You can use the Distributive property to find the algebraic expression.

Explanation:
In the above-given question,
given that,
we can use the distributive property to find the algebraic expression.
for example:
5(x – 7).
5x – 35.

In the algebraic expression, 6x + 10, x is the ________ , 6 is the ________ and 10 is the ___________

Answer:
In the algebraic expression 6x + 10, x is the coefficient, 6 is the variable, and 10 is the constant.

Explanation:
In the above-given question,
given that,
In the algebraic expression 6x + 10, x is the coefficient, 6 is the variable, and 10 is the constant.
for example:
6x + 10.
where 6 is the variable.
x is coefficient.
10 is constant.

Four words that describe operations that can be used with expressions are _________, and ________, _________ and __________.

Answer:
The words that describe the operations are constants, terms, variables, and coefficients.

Explanation:
In the above-given question,
given that,
The words that describe the operations are constants, terms, variables, and coefficients.
for example:
2x + 4y – 9.
where x and y are variables.
9 is the constant.
2 and 4 are coefficients.
terms are 2x, 4y, and 9.

In the algebraic expression 5x + 4 + 6x – 3, you use the Commutative Property to _________ like terms next to each other and the Associative Property to _________ like terms together.

Answer:

Pick A Project

PROJECT 4A
Which emojis would you use to tell the story of your day so far?
PROJECT: WRITE AND ILLUSTRATE A CHILDREN’S BOOK

PROJECT 4B
How many different ways can you represent a dollar?
PROJECT: GENERATE EQUIVALENCE

PROJECT 4C
If you wrote a song, what would it sound like?
PROJECT: COMPOSE A SONG

PROJECT 4D
What was your favorite structure at a playground when you were younger?
PROJECT: BUILD A MODEL PLAYGROUND

Lesson 4.1 Write and Evaluate Algebraic Expressions

Solve & Discuss It!
Mr. Ramirez’s class was playing a game in which students need to match sticky notes that have equivalent expressions. How can you sort the expressions into groups?
I can… write and evaluate algebraic expressions.

Focus on math practices
Reasoning is there more than one way to group the expressions? Give an example.

Answer:
Yes, there are more than one way to group the expressions.

Explanation:
In the above-given question,
given that,
A numerical expression in mathematics can be a combination of numbers, integers combined using.
for example:
16 is an numerical expression.

Essential Question
How can algebraic expressions be used to represent and solve problems?

Answer:
We can use algebra to solve mathematical problems.

Explanation:
In the above-given question,
given that,
we can also interpret the solution in the context of the original problem.
for example:
2x + 5 = 43.
where 43 is the constant.
always has an equal symbol.
2x + 5 = algebraic expression.

Try It!

Misumi started with $217 in her bank account. She deposits $25.50 each week and never withdraws any money. What expression can Misumi use to determine her account balance after w weeks?

Answer:
The expression can Misumi use to determine her account balance after w weeks = 8.5 weeks.

Explanation:
In the above-given question,
given that,
Misumi started with $217 in her bank account.
She deposits $25.50 each week and never withdraws any money.
$217 / 25.50 = 8.5.
so the expression can Misumi use to determine her account balance after w weeks = 8.5.

Convince Me! How did you determine which value to use for the constant and which value to use for the coefficient?

Answer:
x is coefficient and 3 is constant.

Explanation:
In the above-given question,
given that,
2x + 3 is the expression.
x is the coefficient.
3 is the constant.

Try It!

The cost to rent a scooter is $15.50 per hour and the cost to rent a watercraft is $22.80 per hour. Use the expression 15.5s + 22.8w to determine how much it would cost to rent a scooter for 3\(\frac{1}{2}\) hours and a watercraft for 1\(\frac{3}{4}\) hours.

Answer:
The cost would cost to rent a scooter for 3(1/2) hours and watercraft for 1(3/4) hours = $54.25 and $40.

Explanation:
In the above-given question,
given that,
The cost to rent a scooter is $15.50 per hour and the cost to rent a watercraft is $22.80 per hour.
Use the expression 15.5s + 22.8w to determine how much.
15.5s + 22.8w.
3. 1/2 = 7/2.
1. 3/4 = 7/4.
15.5(7/2) + 22.8(7/4).
108.5/2 + 159.6/4.
54.25 + 39.9.

Try It!

Emelia earns $8.74 per hour plus a gas allowance of $3.50 per day at her job. How much does Emelia’s job pay in a day when she works 5\(\frac{1}{2}\) hours? Write an expression and evaluate for 5\(\frac{1}{2}\) hours.

Answer:
Emelia’s job pay in a day when she works 5(1/2) hours = $67.32.

Explanation:
In the above-given question,
given that,
Emelia earns $8.74 per hour plus a gas allowance of $3.50 per day at her job.
$8.74 + $3.50.
$12.24.
5(1/2) = 5.5.
5.5 x $12.24 = $67.32.
so Emelia’s job pay in a day when she works 5(1/2) hours = $67.32.

KEY CONCEPT
Algebraic expressions can be used to represent problems with unknown or variable values. Values can be substituted for variables to evaluate the expression.

Do You Understand?
Question 1.
Essential Question How are algebraic expressions used to represent and solve problems?

Answer:
Algebraic expressions are used to represent problems with unknowns or variable values.

Explanation:
In the above-given question,
given that,
Algebraic expressions are used to represent problems with unknowns or variable values.
Values can be substituted for variables to evaluate the expression.
for example:
2x + 3y = a.
where x = 2 and y = 3.
2 x 2 + 3 x 3 = a.
4 + 9 = a.
13 = a.

Question 2.
Use Structure How is a constant term different than a variable term for an expression that represents a real-world situation?

Answer:
a = 13.

Explanation:
In the above-given question,
given that,
2x + 3y = a.
where x = 2 and y = 3.
2 x 2 + 3 x 3 = a.
4 + 9 = a.
13 = a.

Question 3.
Look for Relationships Explain why you can have different values when evaluating an algebraic expression.

Answer:
To evaluate an algebraic expression we have to substitute a number for each variable and perform the arithmetic operations.

Explanation:
In the above-given question,
given that,
To evaluate an algebraic expression we have to substitute a number for each variable and perform the arithmetic operations.
for example:
x + 6.
where x = 6.
6 + 6 = 12.
if we know the value of our variables, we can replace the variables with their values and then evaluate the expression.

Do You Know How?
Question 4.
A tank containing 35 gallons of water is leaking at a rate of \(\frac{1}{4}\) gallon per minute. Write an expression to determine the number of gallons left in the tank after m minutes.

Answer:
The number of gallons left in the tank after m minutes = 8.75 gallons.

Explanation:
In the above-given question,
given that,
A tank containing 35 gallons of water is leaking at a rate of \(\frac{1}{4}\) gallon per minute.
35 x 1/4 = 35/4.
8.75.
so the number of gallons left in the tank after m minutes = 8.75 gallons.

Question 5.
Write an algebraic expression that Marshall can use to determine the total cost of buying a watermelon that weighs w pounds and some tomatoes that weigh t pounds. How much will it cost to buy a watermelon that weighs 18\(\frac{1}{2}\) pounds and 5 pounds of tomatoes?

Answer:
The much will it cost to buy a watermelon that weighs 18(1/2) pounds and 5 pounds of tomatoes = $29.25 and $3.4.

Explanation:
In the above-given question,
given that,
the cost of tomatoes is $3.25 per lb.
the cost of watermelons is $0.68 per lb.
18/2 = 9.
$3.25 x 9 = $29.25.
0.68 x 5 = $3.4.
so the much will it cost to buy a watermelon that weighs 18(1/2) pounds and 5 pounds of tomatoes = $29.25 and $3.4.

Question 6.
What is the value of \(\frac{3}{8}\)x – 4.5 when x = 0.4?

Answer:
(3/8)x – 4.5 = 4.35.

Explanation:
In the above-given question,
given that,
(3/8)x – 4.5.
where x = 0.4.
(3/8)0.4 – 4.5.
0.375 x 0.4 – 4.5.
0.15 – 4.5.
4.35.
(3/8)x – 4.5 = 4.35.

Question 7.
What is the value of 8.4n – 3.2p when n = 2 and p = 4?

Answer:
8.4n – 3.2p = 4.

Explanation:
In the above-given question,
given that,
8.4n – 3.2p.
8.4 (2) – 3.2 (4).
16.8 – 12.8.
4.
8.4n – 3.2p = 4.

Practice & Problem Solving

Leveled Practice For 8-10, fill in the boxes to complete the problems.
Question 8.
Evaluate 10.2x + 9.4y when x = 2 and y = 3.
10.2 (_______) + 9.4 (_______)
= _______ + 28.2
= _______

Answer:
10.2 x + 9.4 y = 48.6.

Explanation:
In the above-given question,
given that,
10.2 x + 9.4 y.
where x = 2 and y = 3.
10.2 x 2 + 9.4 x 3.
20.4 + 28.2.
48.6.
10.2 x + 9.4 y = 48.6.

Question 9.
Evaluate \(\frac{1}{2}\)t + \(\frac{3}{8}\) when t = \(\frac{1}{4}\)
\(\frac{1}{2}\)(________) + \(\frac{3}{8}\)
= ______ + \(\frac{3}{8}\)
= ______

Answer:
1/2 x 1/4 + 3/8 = 0.5.

Explanation:
In the above-given question,
given that,
1/2 = 0.5.
1/4 = 0.25.
0.5 x 0.25 + 3/8.
0.125 + 0.375.
0.5.
1/2 x 1/4 + 3/8 = 0.5.

Question 10.
Write an expression that represents the height of a tree that began at 6 feet and increases by 2 feet per year. Let y represent the number of years.
_____ + ______ y

Answer:
6x + 2y.

Explanation:
In the above-given question,
given that,
the height of a tree that began at 6 feet and increases by 2 feet per year.
6x + 2y.
where y represents the number of years.
so the expression is 6x + 2y.

For 11-14, evaluate each expression for the given value of the variable(s).
Question 11.
3d – 4
d = 1.2

Answer:
3d – 4 = 0.4.

Explanation:
In the above-given question,
given that,
3d – 4.
where d = 1.2.
3(1.2) – 4.
3.6 – 4.
0.4.
3d – 4 = 0.4.

Question 12.
0.5f – 2.39
f = 12, 9 = 2

Answer:
0.5f – 2.39 = 3.68.

Explanation:
In the above-given question,
given that,
0.5f – 2.39.
where f = 12 and 9 = 2.
0.5 x 12 – 2.32.
6 – 2.32.
3.68.
0.5f – 2.39 = 3.68.

Question 13.
p + 3
p = \(\frac{3}{5}\)

Answer:
p + 3 = 3.6.

Explanation:
In the above-given question,
given that,
p + 3.
where p = 3/5.
3/5 = 0.6.
0.6 + 3.
3.6.
p + 3 = 3.6.

Question 14.
34 + \(\frac{4}{9}\)w
w = –\(\frac{1}{2}\)

Answer:
34 + 4/9x w = 33.9.

Explanation:
In the above-given question,
given that,
34 + 4/9x w.
where w = -1/2.
34 + 4/9(-1/2).
34 + 0.4(-0.5).
34 – 0.1.
33.9.

Question 15.
Model with Math What expression can be used to determine the total cost of buying g pounds of granola for $3.25 per pound and f pounds of flour for $0.74 per pound?

Answer:
$3.25g and $0.74f.

Explanation:
In the above-given question,
given that,
the total cost of buying g pounds of granola for $3.25 per pound.
f pounds of flour for $0.74 per pound.
$3.25g + $0.74f

Question 16.
Model with Math Which expression can be used to determine the total weight of a box that by itself weighs 0.2 kilogram and contains p plaques that weigh 1.3 kilograms each?

A. 1.3p +0.2
B. 0.2p + 1.3
C. 0.2 – 1.3p
D. 1.2p

Answer:
Option A is correct.

Explanation:
In the above-given question,
given that,
total weight of a box that by itself weighs 0.2 kilogram.
and contains p plaques that weigh 1.3 kilograms each.
0.2 + 1.3p.
so option A is correct.

Question 17.
The expression -120 + 13n represents a submarine that began at a depth of 120 feet below sea level and ascended at a rate of 13 feet per minute. What was the depth of the submarine after 6 minutes?

Answer:
The depth of the submarine after 6 minutes = – 42 feet.

Explanation:
In the above-given question,
given that,
The expression -120 + 13n represents a submarine that began at a depth of 120 feet below sea level.
ascended at a rate of 13 feet per minute.
-120 + 13(6).
-120 + 78.
-42.
so the depth of the submarine after 6 minutes = -42 feet.

Question 18.
Be Precise A full grain silo empties at a constant rate. Write an expression to determine the amount of grain left after s seconds.

Answer:
The amount of grain left after 5 seconds = 2982.5 cubic feet.

Explanation:
In the above-given question,
given that,
A full grain silo empties at a constant rate.
the capacity of food grain is 3000 cubic feet.
3000 – 3.5/s.
3000 – 3.5(5).
3000 – 17.5.
2982.5.
so the amount of grain left after 5 seconds = 2982.5 cubic feet.

Question 19.
Higher Order Thinking For the expression 5 – 5x to have a negative value, what must be true about the value of x?

Answer:
The value of x = 4.

Explanation:
In the above-given question,
given that,
the expression is 5 – 5x.
where x = 4.
5 – 5(4).
5 – 20.
-15.

Assessment Practice

Question 20.
Joe bought g gallons of gasoline for $2.85 per gallon and c cans of oil for $3.15 per can.
PART A
What expression can be used to determine the total amount Joe spent on gasoline and oil?

Answer:
The total amount joe spend on gasoline and oil = $2.85g + $3.15c.

Explanation:
In the above-given question,
given that,
Joe bought g gallons of gasoline for $2.85 per gallon.
c cans of oil for $3.15 per can.
$2.85g + $3.15c.
so the total amount joe spends on gasoline and oil = $2.85g + $3.15c.

PART B
Joe spent $15. He bought 2 cans of oil. About how many gallons of gasoline did he buy?
A. 2.5
B. 3
C. 3.5
D. 4

Answer:
The gallons of gasoline did he buy = 3 gallons.

Explanation:
In the above-given question,
given that,
Joe spent $15. He bought 2 cans of oil.
1.5 x 2.
3.
the gallons of gasoline did he buy = 3.
so option B is correct.

Question 21.
The outside temperature was 73°F at 1 P.M. and decreases at a rate of 1.5°F each hour. What expression can be used to determine the temperature h hours after 1 P.M.?

Answer:
The expression can be used to determine the temperature h hours after 1 P.M = 71.5°F.

Explanation:
In the above-given question,
given that,
The outside temperature was 73°F at 1 P.M. and decreases at a rate of 1.5°F each hour.
73 – 1.5.
71.5°F.
so the expression can be used to determine the temperature h hours after 1 P.M = 71.5°F.

Lesson 4.2 Generate Equivalent Expressions

Explore It!
A shipment of eggs contains some cartons with a dozen eggs and some cartons with a half-dozen eggs.
I can… write equivalent expressions for given expressions.

A. How can you represent the total number of eggs in the shipment using diagrams or images? Explain your diagram.

Answer:
1 dozen + 1/2 dozen eggs.

Explanation:
In the above-given question,
given that,
A shipment of eggs contains some cartons with a dozen eggs and some cartons with a half-dozen eggs.
1 dozen = 12.
1/2 doxen = 12/2.
12/2 = 6.
12 + 1/2 eggs.

B. How can you represent the total number of eggs in the shipment using expressions? What variables do you use? What do they represent?

Answer:
1 dozen + 1/2 dozen eggs.

Explanation:
In the above-given question,
given that,
A shipment of eggs contains some cartons with a dozen eggs and some cartons with a half-dozen eggs.
1 dozen = 12.
1/2 doxen = 12/2.
12/2 = 6.
12 + 1/2 eggs.

Focus on math practices
Construct Arguments How do the two representations compare? How are they different?

Essential Question
What are equivalent expressions?

Try It!

Nancy wrote the expression 3x – 12 to represent the relationship in a table of values. Use properties of operations to write two equivalent expressions.
3(x – _____)
_____ + 3x

Answer:
The two equivalent expressions are 36 + 3x.

Explanation:
In the above-given question,
given that,
3x – 12.
3(x – 12).
3x – 36.
36 + 3x.

Convince Me! What property can you use to write an equivalent expression for -5(x – 2)? Explain.

Answer:
-5(x – 2) = -5x – 10.

Explanation:
In the above-given question,
given that,
-5(x – 2).
-5x – 10.
we can use the distributive property.
-5x -10.

Try It!

Use properties of operations to write two expressions that are equivalent to \(\frac{3}{4}\)n + (8 + \(\frac{1}{3}\)z).

Answer:
3/4n + (8 + {1/3}) = 0.75n + 8.3z.

Explanation:
In the above-given question,
given that,
3/4n + (8 + {1/3})z.
3/4n + 8 + 0.3z.
0.75n + 8.3z.
3/4n + (8 + {1/3}) = 0.75n + 8.3z.

Try It!

Write two expressions that are equivalent to –\(\frac{5}{4}\)x – \(\frac{3}{4}\)

Answer:
-5/4 – 3/4 = 2.

Explanation:
In the above-given question,
given that,
-5/4 – 3/4.
-5/4 = 1.25.
3/4 = 0.75.
-1.25 – 0.75.
2.

KEY CONCEPT
You can use properties of operations to write equivalent expressions.

Do You Understand?
What are equivalent
Question 1.
Essential Question expressions?

Answer:
-1/2(x + 8), -1/2x + (-4) and -4 +(-1/2x) are equivalent.

Explanation:
In the above-given question,
given that,
-1/2(x + 8).
-1/2x + (-1/2) . 8.
-1/2x + (-4).
-4 + (-1/2x).
the three expressions are true.

Question 2.
Make Sense and Persevere For which operations is the Commutative Property true?

Answer:
(-4) + -1/2x.

Explanation:
In the above-given question,
given that,
(-4) + -1/2x.
we can use the commutative property for the expression.
-1/2x + (-4).

Question 3.
How can the Associative Property be applied when writing equivalent expressions with variables?

Answer:

Do You Know How?
Question 4.
Write an expression equivalent to -3 + \(\frac{2}{3}\)y – 4 – \(\frac{1}{3}\)y.

Answer:
-3 + (2/3)y – 4 – (1/3)y.

Explanation:
In the above-given question,
given that,
-3 + \(\frac{2}{3}\)y – 4 – \(\frac{1}{3}\)y.
-3 + (2/3)y – 4 – (1/3)y.
-3 – 4 + (2/3)y – (1/3)y.
-7 + 1/3y.
-3 + (2/3)y – 4 – (1/3)y = -7 + 1/3y.

Question 5.
Complete the tables to determine if the expressions are equivalent. If the expressions are equivalent, name the property or properties that make them equivalent.

Answer:
3(x – 5) = 3x – 15.

Explanation:
In the above-given question,
given that,
3(x – 5).
3x – 15.
x = 1.
3 – 15 = -12.
x = 2.
6 – 15 = -9.
x = 3.
9 – 15 = -6.

Question 6.
Use the properties of operations to write an expression equivalent to 4x + \(\frac{1}{2}\) + 2x – 3.

Answer:
4x +[{1/2}] + 2x – 3 = 3x -3.

Explanation:
In the above-given question,
given that,
4x + (1/2) + 2x – 3.
4x + 2x + (1/2) – 3.
6x + (1/2) -3.
3x – 3.
4x +[{1/2}] + 2x – 3 = 3x -3.

Practice & Problem Solving

For 7-9, write an equivalent expression.
Question 7.
-3(7 + 5g)

Answer:
-36g.

Explanation:
In the above-given question,
given that,
-3(7 + 5g).
-3 x 7 = -21.
-3 x 5 = 15.
-21 + (-15g).
-36g.

Question 8.
(x + 7) + 3y

Answer:
24xy.

Explanation:
In the above-given question,
given that,
(x + 7) + 3y.
3y x X + 3y x 7.
3xy + 21y.
24xy.

Question 9.
\(\frac{2}{9}\) – \(\frac{1}{5}\) • x

Answer:
2/9 – 1/5 . X =

Explanation:
In the above-given question,
given that,
2/9 x X – (1/5)x.
2/9 x – 1/5 x.

Question 10.
Which expression is equivalent to t + 4 + 3 – 2t?
A. t + 7
B. -t + 7
C. 6t
D. 10t

Answer:
t + 4 + 3 – 2t = -t + 7.

Explanation:
In the above-given question,
given that,
t + 4 + 3 – 2t.
t + 7 – 2t.
t – 2t + 7.
-t + 7.
t + 4 + 3 – 2t = -t + 7.

Question 11.
The distance in feet that Karina swims in a race is represented by 4d – 4, where d is the distance for each lap. What is an expression equivalent to 4d – 4?

Answer:
The expression equal to 4d – 4 = 4(d – 4).

Explanation:
In the above-given question,
given that,
4d – 4.
4(d – 4).
4d – 16.
4d – 4 = 4d – 16.

Question 12.
Use the Associative Property to write an expression equivalent to (w + 9) + 3.

Answer:
The expression is (3 + w) + 9.

Explanation:
In the above-given question,
given that,
the expression is (w + 9) + 3.
(3 + w) + 3.
6 + w.

Question 13.
Nigel is planning his training schedule for a marathon over a 4-day period. He is uncertain how many miles he will run on two d

Answer:
The number of miles he will run on two days = 14.5 miles.

Explanation:
In the above-given question,
given that,
Nigel is planning his training schedule for a marathon over a 4-day period.
on day 1 he will run 12 miles.
on day 2 he will run 14.5 miles.
on day 3 he will run 17 miles.
12 + 17 = 29.
29/2 = 14.5.
on day 2 he will run 14.5 miles.

Question 14.
Maria said the expression -4n+ 3 + 9n – 4 is equivalent to 4n. What error did Maria likely make?

Answer:
-4n + 3 + 9n – 4 = 5n – 1.

Explanation:
In the above-given question,
given that,
-4n + 3 + 9n – 4.
-4n + 9n = 5n.
5n + 3 – 4.
5n – 1.
-4n + 3 + 9n – 4 = 5n – 1.

Question 15.
Write an expression equivalent to x – 3y + 4.

Answer:
x – 3y + 4 = 4 + x – 3y.

Explanation:
In the above-given question,
given that,
x – 3y + 4.
4 + x – 3y.
x + 4 – 3y.

Question 16.
Andre wrote the expression -2 + 4x = 3 to represent the relationship shown in the table.

Write two other expressions that also represent the relationship shown in the table.

Answer:
-2 + 4x = 3.

Explanation:
In the above-given question,
given that,
-2 + 4x.
x = 0.
-2 + 0.
-2.
x = 6.
-2 + 4(6).
-2 + 24.
22.
x = 12.
-2 + 4(12).
-2 + 48.
46.

Question 17.
Higher Order Thinking to rent a car for a trip, four friends are combining their money. The group chat shows the amount of money that each puts in. One expression for their total amount of money is 189 plus p plus 224 plus q.

a. Use the Commutative Property to write two equivalent expressions.

Answer:
The expressions are 189 + p + 224q.

Explanation:
In the above-given question,
given that,
189 + p + 224q.
p + 189 + 224q.
224q + p + 189.

b. If they need $500 to rent a car, find at least two different pairs of numbers that p and q could be.

Answer:
$500 + p + 224q.

Explanation:
In the above-given question,
given that,
500 – 224 = 276.
276 + p + 224q.
p + 276 + 224q.
224q + 276 + p.

Assessment Practice

Question 18.
Select all expressions equivalent to \(\frac{3}{5}\)x + 3.

Answer:
The expressions equivalent to (3/5)x + 3 = 1 + 2/5x + 3 and 4/5x – 1/5x + 3.

Explanation:
In the above-given question,
given that,
(3/5)x + 3.
1 + 2/5x + 3.
4/5x – 1/5x + 3.
3/5x + 3.
so the expressions equivalent to (3/5)x + 3 = 1 + 2/5x + 3 and 4/5x – 1/5x + 3.

Lesson 4.3 Simplify Expressions

Solve & Discuss It!
How can the tiles below be sorted?
I can… use properties of operations to simplify expressions.

Focus on math practices
Reasoning Would sorting the tiles with positive coefficients together and tiles with negative coefficients together help to simplify an expression that involves all the tiles? Explain.

Answer:
The positive coefficients are 4.25, 3/5y, 3/8, 1/5, 2.1x, and 1/2x.
The negative coefficients are -0.5, -0.5x, -2.1y, and -2.

Explanation:
In the above-given question,
given that,
the coefficients are 4.25, 3/5y, 3/8, 1/5, 2.1x, -0.5, -0.5x, -2.1y, -2, and 1/2x.
the positive coefficients are 4.25, 3/5y, 3/8, 1/5, 2.1x, and 1/2x.
the negative coefficients are -0.5, -0.5x, -2.1y, and -2.

Essential Question
How are properties of operations used to simplify expressions?

Try It!

Simplify the expression – 6 – 6f + 7 – 3f – 9.
______ – 3f – _____ + 7 – ______
_____ – ______

Answer:
-6 – 6f + 7 – 3f – 9 = -9f – 23.

Explanation:
In the above-given question,
given that,
-6 – 6f + 7 – 3f – 9.
-6f  – 3f – 6 + 7 – 9.
-9f – 13 – 9.
-9f – 23.
-6 – 6f + 7 – 3f – 9 = -9f – 23.

Convince Me! How do you decide in what way to reorder the terms of an expression when simplifying it?

Try It!

Simplify each expression.
a. 59.95m – 30 + 7.95m + 45 + 9.49m

Answer:
59.95m – 30 + 7.95m + 45 + 9.49m = 52m – 5.51.

Explanation:
In the above-given question,
given that,
59.95m – 30 + 7.95m + 45 + 9.49.
59.95m + 7.95m – 30 + 45 + 9.49.
52m – 15 + 9.49.
52m – 5.51.

b. -0.5p + \(\frac{1}{2}\)p – 2.75 + \(\frac{2}{3}\)p

Answer:
-0.5p + (1/2)p – 2.75 + (2/3)p = 0.6p – 2.75.

Explanation:
In the above-given question,
given that,
-0.5p + (1/2)p – 2.75 + (2/3)p.
-0.5p + 0.5p – 2.75 + 0.6p.
0.6p – 2.75.
-0.5p + (1/2)p – 2.75 + (2/3)p = 0.6p – 2.75.

Try It!

Simplify the expression -3.7 +59 + 4k + 11.1 – 10g.
(______ – 10g) + 4k + (______ + 11.1)
= _______ + 4k + ______
The simplified expression is ________.

Answer:
-3.7 + 59 + 4k + 11.1 – 10g = 55.3 – 10g.

Explanation:
In the above-given question,
given that,
-3.7 + 59 + 4k + 11.1 – 10g.
-3.7 + 11.1 – 11.1 + 59 – 10g.
-3.7 + 59 – 10g.
55.3 – 10g.

KEY CONCEPT
When simplifying algebraic expressions, use properties of operations to combine like terms.
To simplify the expression below, group like terms.
\(\frac{3}{10}\)y – 3.5x – \(\frac{3}{8}\) +0.53x + 5.25 – 2.75y – 12
(-3.5x + 0.53x) + (\(\frac{3}{10}\)y – 2.75y) + (-\(\frac{3}{8}\) + 5.25 – 12)
Then combine like terms.
-2.97x – 2.45y – 7.125

Do You Understand?
Question 1.
Essential Question How are properties of operations used to simplify expressions?

Answer:
The properties of operations are used to combine like terms.

Explanation:
In the above-given question,
given that,
the properties of operations used to combine like terms.
for example:
\(\frac{3}{10}\)y – 3.5x – \(\frac{3}{8}\) +0.53x + 5.25 – 2.75y – 12
(-3.5x + 0.53x) + (\(\frac{3}{10}\)y – 2.75y) + (-\(\frac{3}{8}\) + 5.25 – 12)
Then combine like terms.
-2.97x – 2.45y – 7.125

Question 2.
Make Sense and Persevere Explain why constant terms expressed as different rational number types can be combined.

Answer:
The constant terms remain the same.

Explanation:
In the above-given question,
given that,
constant terms expressed as different rational number types can be combined.
for example:
\(\frac{3}{10}\)y – 3.5x – \(\frac{3}{8}\) +0.53x + 5.25 – 2.75y – 12
(-3.5x + 0.53x) + (\(\frac{3}{10}\)y – 2.75y) + (-\(\frac{3}{8}\) + 5.25 – 12)
Then combine like terms.
-2.97x – 2.45y – 7.125

Question 3.
Reasoning How do you know when an expression is in its simplest form?

Answer:
The expression is in its simplest form when it has only limited expressions.

Explanation:
In the above-given question,
given that,
for example:
-3.7 + 59 + 4k + 11.1 – 10g.
-3.7 + 11.1 – 11.1 + 59 – 10g.
-3.7 + 59 – 10g.
55.3 – 10g.

Do You Know How?
Question 4.
Simplify -4b + (-9k) – 6 – 3b + 12.

Answer:
-7b – 9k + 6.

Explanation:
In the above-given question,
given that,
-4b + (-9k) – 6 – 3b + 12.
-4b – 3b -9k – 6 + 12.
-7b – 9k + 6.

Question 5.
Simplify -2 + 6.45z – 6+ (-3.25z).

Answer:
-8 + 3.2z.

Explanation:
In the above-given question,
given that,
-2 + 6.45z – 6+ (-3.25z).
-2 + 6.45z – 6 – 3.25z.
-8 + 3.2z.

Question 6.
Simplify –9 + (-\(\frac{1}{3}\)y) +6 – \(\frac{4}{3}\)y.

Answer:
-3 – 5/3y.

Explanation:
In the above-given question,
given that,
–9 + (-\(\frac{1}{3}\)y) +6 – \(\frac{4}{3}\)y.
-9 – 1/3y + 6 – 4/3y.
-9 + 6 – 5/3y.
-3 – 5/3y.

Practice & Problem Solving

In 7-10, simplify each expression.
Question 7.
–2.8f +0.96 – 12 – 4

Answer:
-2.8f + 0.96 – 12 – 4 = -2.8f – 15.04.

Explanation:
In the above-given question,
given that,
-2.8f + 0.96 – 12 – 4.
-2.8f + 0.96 – 16.
-2.8f – 15.04.
-2.8f + 0.96 – 12 – 4 = -2.8f – 15.04.

Question 8.
3.2 – 5.1n – 3n + 5

Answer:
3.2 – 5.1n – 3n + 5 = 8.2 – 8.1n.

Explanation:
In the above-given question,
given that,
3.2 – 5.1n – 3n + 5.
3.2 – 8.1n + 5.
8.2 – 8.1n.
3.2 – 5.1n – 3n + 5 = 8.2 – 8.1n.

Question 9.
2n + 5.5 – 0.9n – 8 + 4.5p

Answer:
2n + 5.5 – 0.9n – 8 + 4.5p = 4.5p – 2.5 + 1.1n.

Explanation:
In the above-given question,
given that,
2n + 5.5 – 0.9n – 8 + 4.5p.
2n – 0.9n + 5.5 – 8 + 4.5p.
1.1n – 2.5 + 4.5p.
4.5p – 2.5 + 1.1n.
2n + 5.5 – 0.9n – 8 + 4.5p = 4.5p – 2.5 + 1.1n.

Question 10.
12 + (-4) – \(\frac{2}{5}\)j – \(\frac{4}{5}\)j + 5

Answer:
12 + (-4) – 2/5j – 4/5j + 5 = 13 – 6/5j.

Explanation:
In the above-given question,
given that,
12 + (-4) – 2/5j – 4/5j + 5.
12 -4 – 6/5j + 5.
8 – 6/5j + 5.
13 – 6/5j.
12 + (-4) – 2/5j – 4/5j + 5 = 13 – 6/5j.

Question 11.
Which expression is equivalent to -5v + (-2) + 1 + (-2v)?
A. -9v
B. -4v
C. -7v – 1
D. -7V + 3

Answer:
Option C is correct.

Explanation:
In the above-given question,
given that,
-5v + (-2) + 1 + (-2v).
-5v -2 + 1 -2v.
-7v -1.
so option C is correct.

Question 12.
Which expression is equivalent to \(\frac{2}{3}\)x + (-3) + (-2) – \(\frac{1}{3}\)x?
A. x + 5
B. –\(\frac{1}{3}\)x + 5
C. \(\frac{1}{3}\)x – 1
D. \(\frac{1}{3}\)x – 5

Answer:
Option D is correct.

Explanation:
In the above-given question,
given that,
\(\frac{2}{3}\)x + (-3) + (-2) – \(\frac{1}{3}\)x.
2/3x -3 -2 -1/3x.
1/3x -5
so option D is correct.

Question 13.
The dimensions of a garden are shown. Write an expression to find the perimeter.

Answer:
The perimeter of the garden = x – 7.

Explanation:
In the above-given question,
given that,
the length of the garden = x.
the width of the garden = 1/2x – 7.
area of the garden = l x b.
x + 1/2x – 7.
2/2x – 7.
x – 7.
so the perimeter of the garden = x – 7.

Question 14.
Simplify the expression 8h + (-7.3d) – 14 + 5d – 3.2h.

Answer:
8h + (-7.3d) – 14 + 5d – 3.2h = 5.2h – 2.3d – 14.

Explanation:
In the above-given question,
given that,
8h + (-7.3d) – 14 + 5d – 3.2h.
8h – 7.3d – 14 + 5d – 3.2h.
5.2h – 2.3d – 14.
8h + (-7.3d) – 14 + 5d – 3.2h = 5.2h – 2.3d – 14.

Question 15.
Simply 4 – 2y + (-8y) + 6.2.

Answer:
4 – 2y + (-8y) + 6.2 = 10.2 – 10y.

Explanation:
In the above-given question,
given that,
4 – 2y + (-8y) + 6.2.
4 – 2y – 8y + 6.2.
4 – 10y + 6.2.
10.2 – 10y.
4 – 2y + (-8y) + 6.2 = 10.2 – 10y.

Question 16.
Simplify \(\frac{4}{9}\)z – \(\frac{3}{9}\)z + 5 – \(\frac{5}{9}\)z – 8.

Answer:
4/9z – 3/9z + 5 – 5/9z – 8 = -4/9z – 3.

Explanation:
In the above-given question,
given that,
4/9z – 3/9z + 5 – 5/9z – 8.
1/9z + 5 – 5/9z – 8.
-4/9z + 5 – 8.
-4/9z – 3.
4/9z – 3/9z + 5 – 5/9z – 8 = -4/9z – 3.

Question 17.
Construct Arguments Explain whether 11t – 4t is equivalent to 4t – 11t. Support your answer by evaluating the expression for t = 2.

Answer:
The values are the same but 11t – 4t is positive and 4t – 11t is negative.

Explanation:
In the above-given question,
given that,
11t – 4t is equivalent to 4t – 11t.
t = 2.
11(2) – 4(2).
22 – 8.
14.
4t – 11t.
4(2) – 11(2).
8 – 22.
-14.

Question 18.
The signs show the costs of different games at a math festival. How much would it cost n people to play Decimal Decisions and Ratio Rage?

Answer:
The cost would take to n people to play Decimal Decisions and Ratio Rage = 6.6n/4 – 3.

Explanation:
In the above-given question,
given that,
the cost of 1 Game is 5.5n – 3.
the cost of 1 Game is n/4.
5.5n – 3 + n/4.
6.6n/4 – 3.
the cost would take to n people to play Decimal Decisions and Ratio Rage = 6.6n/4 – 3.

Question 19.
Higher Order Thinking in the expression ax + bx, a is a decimal and b is a fraction. How do you decide whether to write a as a fraction or b as a decimal?

Answer:
Yes, we can write an as a fraction and b as a decimal.

Explanation:
In the above-given question,
given that,
in the expression ax + bx, a is a decimal and b is a fraction.
for example:
a = 1.1.
b = 1/2.
ax + bx.
1.1x + 1/2x.
so 1.1 is a decimal and 1/2 is a fraction.

Assessment Practice

Question 20.
Select all expressions equivalent to -6z + (-5.5) + 3.5z + 5y – 2.5.
☐ -8 + 5y + 2.52
☐ -2.5z + 5y – 8
☐ -8 + 5y +(-2.5z)
☐ 2.5y + (-2.5z) – 5.5
☐ 5y – 8 – 2.5z

Answer:
Option B and C are correct.

Explanation:
In the above-given question,
given that,
-6z + (-5.5) + 3.5z + 5y – 2.5.
-6z + 3.5z – 5.5 – 2.5 + 5y.
-2.5z -8 + 5y.
so options B and C are correct.

Lesson 4.4 Expand Expressions

Solve & Discuss It!
The school is planning to add a weight room to the gym. If the total area of the gym and weight room should stay under 5,500 square feet, what is one possible length for the new weight room? Show your work. Are there other lengths that would work? Why or why not? -90 ft
I can… expand expressions using the Distributive Property.

Look for Relationships
What is the relationship between the areas of the gym and weight room?

Answer:
The relationship between the areas of the gym and weight room = 550 ft.

Explanation:
In the above-given question,
given that,
The school is planning to add a weight room to the gym.
If the total area of the gym and weight room should stay under 5,500 square feet.
the area of the school is l x b.
where l = 90 ft and b = 55 ft.
area = l x b.
area = 90 x 55.
area = 4950.
5500 – 4950 = 550.
so the relationship between the areas of the gym and weight room = 550 ft.

Focus on math practices
Model with Math What is an expression using x that represents the total area of the gym and the weight room?

Answer:
The relationship between the areas of the gym and weight room = 550 ft.

Explanation:
In the above-given question,
given that,
The school is planning to add a weight room to the gym.
If the total area of the gym and weight room should stay under 5,500 square feet.
the area of the school is l x b.
where l = 90 ft and b = 55 ft.
area = l x b.
area = 90 x 55.
area = 4950.
5500 – 4950 = 550.
so the relationship between the areas of the gym and weight room = 550 ft.

Essential Question
How does the value of an expression change when it is expanded?

Try It!

What is the expanded form of the expression 3.6(t + 5)?

3.6(t + 5)
= ________t + _______ • 5
= _______ + _______
The expanded expression is _______.

Answer:
3.6t + 18.

Explanation:
In the above-given question,
given that,
3.6(t + 5).
3.6 x t = 3.6t.
3.6t + 3.6 x 5.
3.6t + 18.

Convince Me! If you know the value of t, would the evaluated expression be different if you added the known value of t and 5 and then multiplied by 3.6? Explain.

Try It!

Expand the expression t(-1.2w + 3).

Answer:
The expression is -1.2tw + 3t.

Explanation:
In the above-given question,
given that,
the expression is t(-1.2w + 3).
-1.2tw + 3t.
so the expanded expression is -1.2tw + 3t.

Try It!

Simplify the expression –\(\frac{2}{5}\)(10 + 15m – 20n).

Answer:
The expression is -4 -6m – 8n.

Explanation:
In the above-given question,
given that,
–\(\frac{2}{5}\)(10 + 15m – 20n).
-2/5 (10 + 15m – 20n).
-2/5(10 + 15m – 20n).
-20/5 – 30/5m – 40/5n.
-4 – 6m – 8n.

KEY CONCEPT
You can expand an expression using the Distributive Property.
Multiply, or distribute, the factor outside the parentheses with each term inside the parentheses.
-7(3y – 1)
= (-7)(3y) + (-7)(-1)
= -21y + 7
The sign of each term is included in all calculations.

Do You Understand?
Question 1.
Essential Question How does the value of an expression change when it is expanded?

Answer:
The value of an expression change when it is expanded.

Explanation:
In the above-given question,
given that,
the value of an expression change when it is expanded.
for example:
-8(2y – 2).
-8(-2y) + (-8) (-2).
-16y + -16.

Question 2.
Use Structure How does the subtraction part of the expression change when a(b – c) is expanded?

Answer:
The subtraction part of the expression change.

Explanation:
In the above-given question,
given that,
a(b – c).
ax b – a x c.
ab – ac.
the product of in terms is multiplied with outterms.

Question 3.
Make Sense and Persevere When does expanding and simplifying a(b + c) result in a positive value for ac?

Answer:
ab + ac.

Explanation:
In the above-given question,
given that,
the expression is a(b + c).
a x b + a x c.
ab + ac.
the sign is positive.
so the value for ac is also positive.

Do You Know How?
Question 4.
Shoes and hats are on sale. The expression \(\frac{1}{4}\)(s + 24.80) can be used to determine the discount when you buy shoes with a retail price of s dollars and a hat with a retail price of $24.80. Write another expression that can be used to determine the discount.

Answer:
Another expression is $1.55.

Explanation:
In the above-given question,
given that,
Shoes and hats are on sale.
The expression \(\frac{1}{4}\)(s + 24.80).
when you buy shoes with a retail price of s dollars and a hat with a retail price of $24.80.
1/4 (s + 24.80).
s/4 + 24.80/4.
s/4 + 6.2.
s/4 = – 6.2.
s = -6.2/4.
s = 1.55.
so the retail price of the shoes = $1.55.

Question 5.
Expand x(4 – 3.4y).

Answer:
The expression is 4x – 3.4xy.

Explanation:
In the above-given question,
given that,
x(4 – 3.4y).
4x X – 3.4 x X x Y.
4x – 3.4xy.
so the expanded expression is 4x – 3.4xy.

Question 6.
Expand –\(\frac{2}{10}\)(1 – 2x + 2).

Answer:
The expanded expression is -3/5 – 2/5x.

Explanation:
In the above-given question,
given that,
–\(\frac{2}{10}\)(1 – 2x + 2).
-2/10 (1 – 2x + 2).
-1/5 (1 – 2x + 2).
-1/5 – 2/5x – 2/5.
-3/5 – 2/5x.

Practice & Problem Solving

Leveled Practice For 7-8, fill in the boxes to expand each expression.
Question 7.
3(n + 7)
= (3) (_____) + (3) (_____)
= ____ + _____

Answer:
3n + 21.

Explanation:
In the above-given question,
given that,
3(n + 7).
3 x n + 3 x 7.
3n + 21.

Question 8.
4(x – 3)
= ______ x – ______ (3)
= ______ – ______

Answer:
4x – 12.

Explanation:
In the above-given question,
given that,
4(x – 3).
4 x X – 4 x 3.
4x – 12.

For 9-14, write the expanded form of the expression.
Question 9.
y(0.5 + 8)

Answer:
y(0.5 + 8) = 8.5y.

Explanation:
In the above-given question,
given that,
y(0.5 + 8).
0.5y + 8y.
8.5y.
y(0.5 + 8) = 8.5y.

Question 10.
4(3 + 4x – 2)

Answer:
4(3 + 4x – 2) = 4 + 16x.

Explanation:
In the above-given question,
given that,
4(3 + 4x – 2).
4 x 3 + 4x x 4 – 2 x 4.
12 + 16x – 8.
4 + 16x.
4(3 + 4x – 2) = 4 + 16x.