Vijaya Sree

If you are looking for Chapter 7 Go Math Grade 5 Answer Key then you have reached the right destination.  We have mentioned Go Math Grade 5 Answer Key Chapter 7 Multiply Fractions with straight forward explanation for all the Problems. Parents who wanted to give their kids enough practice on concepts of Multiplying Fractions can give them the Go Math 5th Grade Answer Key Chapter 7 Multiply Fractions. Enhance your Math Skills by practicing from the HMH Go Math Grade 5 Answer Key Ch 7 Multiply Fractions.

Go Math Grade 5 Answer Key Chapter 7 Multiply Fractions

Do you feel the concept of Multiplying Fractions difficult? Not anymore with our Go Math Answer Key Grade 5 Chapter 7 Multiply Fractions as they have concepts explained in detail. Start learning the basics involved right from the basic level with our 5th Grade Go Math Answer Key Ch 7 Multiply Fractions. Test your preparation level by answering the questions at the end of the Go Math Grade 5 Answer Key for Multiply Fractions. By doing so, you can have an estimation of your preparation standards.

Chapter 7 – Lesson 1: Find Part of a Group

• Share and Show – Page No. 293
• Problem Solving – Page No. 294

Chapter 7 – Lesson 2: Investigate • Multiply Fractions and Whole Numbers

• Share and Show – Page No. 297
• Problem Solving – Page No. 298

Chapter 7 – Lesson 3: Fraction and Whole Number Multiplication

• Share and Show – Page No. 301
• UNLOCK the Problem – Page No. 302

Chapter 7 – Lesson 4: Investigate • Multiply Fractions

• Share and Show – Page No. 304
• Share and Show – Page No. 305
• Problem Solving – Page No. 306

Chapter 7 – Lesson 5: Compare Fraction Factors and Products

• Share and Show – Page No. 309
• Problem Solving – Page No. 310

Chapter 7 – Lesson 6: Fraction Multiplication

• Share and Show – Page No. 313
• Problem Solving – Page No. 314

Chapter 7 – Mid-Chapter Checkpoint

• Mid-Chapter Checkpoint – Page No. 315
• Mid-Chapter Checkpoint – Page No. 316

Chapter 7 – Lesson 7: Investigate • Area and Mixed Numbers

• Share and Show – Page No. 319
• Problem Solving – Page No. 320

Chapter 7 – Lesson 8: Compare Mixed Number Factors and Products

• Share and Show – Page No. 323
• Problem Solving – Page No. 324

Chapter 7 – Lesson 9: Multiply Mixed Numbers

• Share and Show – Page No. 327
• Share and Show Connect to health – Page No. 328

Chapter 7 – Lesson 10: Problem Solving • Find Unknown Lengths

• Share and Show – Page No. 331
• On Your Own – Page No. 332

Chapter 7 – Chapter 7 Review/Test

• Chapter Review/Test – Page No. 333
• Chapter Review/Test – Page No. 334
• Chapter Review/Test – Page No. 335
• Chapter Review/Test – Page No. 336

Share and Show – Page No. 293

Question 1.
Complete the model to solve.

\(\frac{7}{8}\) of 16, or \(\frac{7}{8}\) × 16
How many rows of counters are there?
_____ rows

Answer: 8
By seeing the above figure we can say that the number of counters is 8 rows.

Question 1.
How many counters are in each row?
_____ counters

Answer: 2
There are 2 counters in each row.

Question 1.
Circle ____ rows to solve the problem.
_____ rows

Answer: 7

• • • • • • •
• • • • • • •
• • • • • • •
• • • • • • •
• • • • • • •
• • • • • • •
• • • • • • •
• • • • • • •

Question 1.
How many counters are circled?
\(\frac{7}{8}\) of 16=
or \(\frac{7}{8}\) × 16 =
_____ counters

Answer: 14
\(\frac{7}{8}\) × 16
8 divides 16 two times.
So, \(\frac{7}{8}\) × 16 = 7 × 2 = 14
Therefore 14 counters are circled.

Use a model to solve.

Question 2.
\(\frac{2}{3}\) × 18 = _____

Answer: 12

Explanation:
\(\frac{2}{3}\) × 18
3 divides 18 six times.
2 × 6 = 12

On Your Own

Use a model to solve.

Question 5.
\(\frac{5}{8}\) × 24 = _____

Answer: 15

Explanation:
\(\frac{5}{8}\) × 24
8 divides 24 three times.
5 × 3 = 15
\(\frac{5}{8}\) × 24 = 15

Question 6.
\(\frac{3}{4}\) × 24 = _____

Answer: 18

Explanation:
\(\frac{3}{4}\) × 24
4 divides 24 six times.
\(\frac{3}{4}\) × 24 = 3 × 6 = 18
So, \(\frac{3}{4}\) × 24 = 18

Question 7.
\(\frac{4}{7}\) × 21 = _____

Answer: 12

Explanation:
\(\frac{4}{7}\) × 21
7 divides 21 three times.
4 × 3 = 12
\(\frac{4}{7}\) × 21 = 12

Question 8.
\(\frac{2}{9}\) × 27 = _____

Answer: 6

Explanation:
\(\frac{2}{9}\) × 27
9 divides 27 three times.
2 × 3 = 6
\(\frac{2}{9}\) × 27 = 6

Problem Solving – Page No. 294

Use the table for 11-12.

Question 11.
Four-fifths of Zack’s stamps have pictures of animals. How many stamps with pictures of animals does Zack have? Use a model to solve.
_____ stamps

Answer: 24 stamps

Explanation:
Given that, Four-fifths of Zack’s stamps have pictures of animals.
Number of stamps that Zack collected is 30
30 × \(\frac{4}{5}\)
5 divides 30 six times.
6 × 4 = 24
Zack has 24 stamps with pictures of animals.

Question 12.
Zack, Teri, and Paco combined the foreign stamps from their collections for a stamp show. Out of their collections, \(\frac{3}{10}\) of Zack’s stamps, \(\frac{5}{6}\) of Teri’s stamps, and \(\frac{3}{8}\) of Paco’s stamps were from foreign countries. How many stamps were in their display? Explain how you solved the problem.
_____ stamps

Answer: 33 stamps

Explanation:
Zack, Teri, and Paco combined the foreign stamps from their collections for a stamp show.
Out of their collections, \(\frac{3}{10}\) of Zack’s stamps, \(\frac{5}{6}\) of Teri’s stamps, and \(\frac{3}{8}\) of Paco’s stamps were from foreign countries.
Number of stamps Zack collected = 30
Number of stamps Teri collected = 18
Number of stamps Paco collected = 24
\(\frac{3}{10}\) of 30
\(\frac{3}{10}\) × 30 = 3 × 3 = 9
\(\frac{5}{6}\) × 18 = 5 × 3 = 15
\(\frac{3}{8}\) × 24 = 3 × 3 = 9
Now add all the stamps = 9 + 9 + 15 = 33

Question 14.
Test Prep Barry bought 21 stamps from a hobby shop. He gave \(\frac{3}{7}\) of them to his sister. How many stamps did he have left?

Options:
a. 3 stamps
b. 6 stamps
c. 9 stamps
d. 12 stamps

Answer: 9 stamps

Explanation:
Test Prep Barry bought 21 stamps from a hobby shop. He gave \(\frac{3}{7}\) of them to his sister.
\(\frac{3}{7}\) × 21
7 divides 21 three times.
3 × 3 = 9 stamps.
Thus the correct answer is option C.

Share and Show – Page No. 297

Use the model to find the product.

Question 1.
\(\frac{5}{6}\) × 3

______ \(\frac{□}{□}\)

Answer: 2 \(\frac{1}{2}\)

Explanation:
Place three whole fractions strips side by side.
Find six fraction strips all with the same denominator that fits exactly under the three whole numbers.
Circle \(\frac{5}{6}\) of 3 on the model you drew.
Complete the number sentence. \(\frac{5}{6}\) of 3
\(\frac{5}{6}\) × 3 = \(\frac{5}{2}\)
2 \(\frac{1}{2}\)

Question 2.
2 × \(\frac{5}{6}\)

______ \(\frac{□}{□}\)

Answer: 1 \(\frac{2}{3}\)

Explanation:
Place two whole fractions strips side by side.
Find six fraction strips all with the same denominator that fits exactly under the two whole numbers.
2 of \(\frac{5}{6}\) = \(\frac{5}{6}\) × 2
\(\frac{5}{3}\)
The mixed fraction of \(\frac{5}{3}\) is 1 \(\frac{2}{3}\)

Find the product.

Question 4.
9 × \(\frac{1}{3}\) = ______

Answer: 3

Explanation:
9 × \(\frac{1}{3}\)
Place nine whole fractions strips side by side.
Find three fraction strips all with the same denominator that fits exactly under the two whole numbers.
9 × \(\frac{1}{3}\)
3 divides 9 three times.
9 × \(\frac{1}{3}\) = 3
Thus 9 × \(\frac{1}{3}\) = 3

Question 5.
\(\frac{7}{8}\) × 4 = ______ \(\frac{□}{□}\)

Answer: 3 \(\frac{1}{2}\)

Explanation:
\(\frac{7}{8}\) × 4
Place four whole fractions strips side by side.
\(\frac{7}{8}\) × 4
4 divides 8 two times.
\(\frac{7}{8}\) × 4 = \(\frac{7}{2}\)
The mixed fraction of  \(\frac{7}{2}\) is 3 \(\frac{1}{2}\)
\(\frac{7}{8}\) × 4 = 3 \(\frac{1}{2}\)

Question 6.
4 × \(\frac{3}{5}\) = ______ \(\frac{□}{□}\)

Answer: 2 \(\frac{2}{5}\)

Explanation:
4 × \(\frac{3}{5}\)
Place four whole fractions strips side by side.
Place three \(\frac{1}{5}\) fraction strips all with the same denominator that fits exactly under the two whole numbers.
4 of \(\frac{3}{5}\)
4 × \(\frac{3}{5}\) = \(\frac{12}{5}\)
The mixed fraction of \(\frac{12}{5}\) is 2 \(\frac{2}{5}\)

Question 7.
\(\frac{7}{8}\) × 2 = ______ \(\frac{□}{□}\)

Answer: 1 \(\frac{3}{4}\)

Explanation:
\(\frac{7}{8}\) × 2
Place two whole fractions strips side by side.
Place seven \(\frac{1}{8}\) fraction strips all with the same denominator that fits exactly under the two whole numbers.
\(\frac{7}{8}\) of 2
\(\frac{7}{8}\) × 2 = \(\frac{7}{4}\)
The mixed fraction of \(\frac{7}{4}\) is 1 \(\frac{3}{4}\)

Question 8.
7 × \(\frac{2}{5}\) = ______ \(\frac{□}{□}\)

Answer: 2 \(\frac{4}{5}\)

Explanation:
7 × \(\frac{2}{5}\)
Place seven whole fractions strips side by side.
Place two \(\frac{1}{5}\) fraction strips all with the same denominator that fits exactly under the two whole numbers.
7 × \(\frac{2}{5}\) = \(\frac{14}{5}\)
The mixed fraction of \(\frac{14}{5}\) = 2 \(\frac{4}{5}\)

Question 9.
\(\frac{3}{8}\) × 4 = ______

Answer: \(\frac{3}{2}\)

Explanation:
\(\frac{3}{8}\) × 4
Place four whole fractions strips side by side.
Place three \(\frac{1}{8}\) fraction strips all with the same denominator that fits exactly under the two whole numbers.

Question 11.
\(\frac{4}{15}\) × 5 = ______ \(\frac{□}{□}\)

Answer: 5 \(\frac{1}{3}\)

Explanation:
\(\frac{4}{15}\) × 5
Place five whole fractions strips side by side.
Place four \(\frac{1}{15}\) fraction strips all with the same denominator that fits exactly under the two whole numbers.
\(\frac{4}{15}\) of 5
\(\frac{4}{15}\) × 5 = \(\frac{4}{3}\)
Convert the improper fraction to the mixed fraction.
\(\frac{4}{3}\) = 5 \(\frac{1}{3}\)

Question 12.
Matt has a 5-pound bag of apples. To make a pie, he needs to use \(\frac{3}{5}\) of the bag. How many pounds of apples will he use for the pie? Explain what a model for this problem might look like.
______ pound(s)

Answer: 3 pounds

Explanation:
Given, Matt has a 5-pound bag of apples.
To make a pie, he needs to use \(\frac{3}{5}\) of the bag.
\(\frac{3}{5}\) × 5 = 3
Therefore Matt used 3 pounds of apples to make a pie.

Problem Solving – Page No. 298

Pose a Problem

Question 13.
Tarique drew the model below for a problem. Write 2 problems that can be solved using this model. One of your problems should involve multiplying a whole number by a fraction and the other problem should involve multiplying a fraction by a whole number.

Pose problems.                                       Solve your problems.
How could you change the model to give you an answer of 4 \(\frac{4}{5}\)?
Explain and write a new equation.
Type below:
_________

Answer:
The five children in the Smith family each spend 2/5 of an hour doing household chores on Saturday. How much time did the spend altogether on their chores?
Multiply the numerator with the whole number.
5 × \(\frac{2}{5}\) = \(\frac{10}{5}\) = 2

Share and Show – Page No. 301

Find the product. Write the product in simplest form.

Question 1.
3 × \(\frac{2}{5}\) =
• Multiply the numerator by the whole number. Write the product over the denominator.

• Write the answer as a mixed number in simplest form.

______ \(\frac{□}{□}\)

Answer: 1 \(\frac{1}{5}\)

Explanation:
Multiply the whole number with the numerator.
3 \(\frac{2}{5}\) = \(\frac{6}{5}\)
Now write the improper fraction in the form of the mixed fraction.
\(\frac{6}{5}\) = 1 \(\frac{1}{5}\)

Question 2.
\(\frac{2}{3}\) × 5 = ______ \(\frac{□}{□}\)

Answer: 3 \(\frac{1}{3}\)

Explanation:
Multiply the whole number with the numerator.
\(\frac{2}{3}\) × 5 = \(\frac{10}{3}\)
Now write the improper fraction in the form of the mixed fraction.
\(\frac{10}{3}\) = 3 \(\frac{1}{3}\)

On Your Own

Find the product. Write the product in simplest form.

Question 5.
5 × \(\frac{2}{3}\) = ______ \(\frac{□}{□}\)

Answer: 3 \(\frac{1}{3}\)

Explanation:
5 × \(\frac{2}{3}\)
Multiply the whole number with the numerator.
5 × \(\frac{2}{3}\) = \(\frac{10}{3}\)
Now write the improper fraction in the form of the mixed fraction.
3 \(\frac{1}{3}\)
5 × \(\frac{2}{3}\) = 3 \(\frac{1}{3}\)

Question 6.
\(\frac{1}{4}\) × 3 = ______ \(\frac{□}{□}\)

Answer: \(\frac{3}{4}\)

Explanation:
\(\frac{1}{4}\) × 3
Multiply the whole number with the numerator.
\(\frac{1}{4}\) × 3 = \(\frac{3}{4}\)

Question 7.
7 × \(\frac{7}{8}\) = ______ \(\frac{□}{□}\)

Answer: 6 \(\frac{1}{8}\)

Explanation:
7 × \(\frac{7}{8}\)
Multiply the whole number with the numerator.
\(\frac{49}{8}\)
Now write the improper fraction in the form of the mixed fraction.
\(\frac{49}{8}\) = 6 \(\frac{1}{8}\)
Thus, 7 × \(\frac{7}{8}\) = 6 \(\frac{1}{8}\)

Question 8.
2 × \(\frac{4}{5}\) = ______ \(\frac{□}{□}\)

Answer: 1 \(\frac{3}{5}\)

Explanation:
2 × \(\frac{4}{5}\)
Multiply the whole number with the numerator.
2 × \(\frac{4}{5}\) = \(\frac{8}{5}\)
Now write the improper fraction in the form of the mixed fraction.
\(\frac{8}{5}\) = 1 \(\frac{3}{5}\)

Question 9.
4 × \(\frac{3}{4}\) = ______

Answer: 3

Explanation:
Multiply the whole number with the numerator.
4 × \(\frac{3}{4}\) = \(\frac{12}{4}\)
4 divides 12 three times.
So, \(\frac{12}{4}\) = 3
4 × \(\frac{3}{4}\) = 3

Question 10.
\(\frac{7}{9}\) × 2 = ______ \(\frac{□}{□}\)

Answer: 1 \(\frac{5}{9}\)

Explanation:
\(\frac{7}{9}\) × 2
Multiply the whole number with the numerator.
\(\frac{7}{9}\) × 2 = \(\frac{14}{9}\)
Now write the improper fraction in the form of the mixed fraction.
\(\frac{14}{9}\) = 1 \(\frac{5}{9}\)

Practice: Copy and Solve. Find the product. Write the product in simplest form.

Question 13.
\(\frac{5}{8}\) × 3 = ______ \(\frac{□}{□}\)

Answer: 1 \(\frac{7}{8}\)

Explanation:
\(\frac{5}{8}\) × 3
Multiply the whole number with the numerator.
\(\frac{5}{8}\) × 3 = \(\frac{15}{8}\)
Now write the improper fraction in the form of the mixed fraction.
\(\frac{15}{8}\) = 1 \(\frac{7}{8}\)

Algebra Find the unknown digit.

Question 14.
\(\frac{■}{2}\) × 8 = 4
■ = ______

Answer: 1

Explanation:
\(\frac{■}{2}\) × 8 = 4
\(\frac{■}{2}\) = 4/8
■ = 4 × 2/8 = 1
■ = 1

Question 15.
■ × \(\frac{5}{6}\) = \(\frac{20}{6}\) or 3 \(\frac{1}{3}\)
■ = ______

Answer: 4

Explanation:
■ × \(\frac{5}{6}\) = \(\frac{20}{6}\)
■ = 20/6 × 6/5
■ = 20/5 = 4
■ = 4

Question 16.
\(\frac{1}{■}\) × 18 = 3
■ = ______

Answer: 6

Explanation:
\(\frac{1}{■}\) × 18 = 3
\(\frac{1}{3}\) × 18 = ■
■ = 18/3 = 6
■ = 6

UNLOCK the Problem – Page No. 302

Question 17.
The caterer wants to have enough turkey to feed 24 people. If he wants to provide \(\frac{3}{4}\) of a pound of turkey for each person, how much turkey does he need?
a. What do you need to find?
Type below:
__________

Answer: I need to find How much turkey the caterer needs to provide for each person.

Question 17.
b. What operation will you use?
Type below:
__________

Answer: I will use the multiplication operation to solve the problem.

Question 17.
c. What information are you given?
Type below:
__________

I am given the information about the number of people to feed and the fraction of pounds of turkey each person gets.

Question 17.
d. Solve the problem.
Type below:
__________

Answer:
The caterer wants to serve 24 people
\(\frac{3}{4}\) × 24
4 divides 24 six times.
3 × 6 = 18
Thus the caterer needs 18 pounds of Turkey.

Question 17.
e. Complete the sentences.
The caterer wants to serve 24 people _____ of a pound of turkey each.
He will need ____ × ____ , or ______ pounds of turkey.
Type below:
__________

Answer: \(\frac{3}{4}\) × 24

Question 17.
f. Fill in the bubble for the correct answer choice.
Options:
a. 72 pounds
b. 24 pounds
c. 18 pounds
d. 6 pounds

Answer: 18 pounds

Explanation:
The caterer wants to serve 24 people
\(\frac{3}{4}\) × 24
4 divides 24 six times.
3 × 6 = 18
The correct answer is option C.

Question 18.
Patty wants to run \(\frac{5}{6}\) of a mile every day for 5 days. How far will she run in that time?
Options:
a. 25 miles
b. 5 miles
c. 4 \(\frac{1}{6}\) miles
d. 1 \(\frac{2}{3}\) miles

Answer: 4 \(\frac{1}{6}\) miles

Explanation:
Patty wants to run \(\frac{5}{6}\) of a mile every day for 5 days.
\(\frac{5}{6}\) × 5 = \(\frac{25}{6}\)
Convert the improper fraction to the mixed fraction.
\(\frac{25}{6}\) = 4 \(\frac{1}{6}\) miles
Thus the correct answer is option C.

Question 19.
Doug has 33 feet of rope. He wants to use \(\frac{2}{3}\) of it for his canoe. How many feet of rope will he use for his canoe?
Options:
a. 11 feet
b. 22 feet
c. 33 feet
d. 66 feet

Answer: 22 feet

Explanation:
Doug has 33 feet of rope. He wants to use \(\frac{2}{3}\) of it for his canoe.
\(\frac{2}{3}\) × 33 feet
3 divides 33 eleven times.
2 × 11 = 22 feet
The correct answer is option B.

Share and Show – Page No. 304

Use the model to find the product.

Question 1.

\(\frac{3}{5} \times \frac{1}{3}=\)
\(\frac{□}{□}\)

Answer: \(\frac{1}{5}\)

Explanation:
The fraction \(\frac{3}{5}\) represents the rows and columns.
The fraction \(\frac{1}{3}\) indicates the shaded part of the figure.
\(\frac{3}{5}\) × \(\frac{1}{3}\) = \(\frac{1}{5}\)

Question 2.

\(\frac{2}{3} \times \frac{3}{5}=\)
\(\frac{□}{□}\)

Answer: \(\frac{2}{5}\)

Explanation:

The above figure shows that the circle is divided into 5 parts in which 2 parts are non shaded and 3 parts are shaded.
So, the fraction of the circle is \(\frac{2}{3}\)
The fraction for the shaded part of the circle is \(\frac{3}{5}\)
\(\frac{2}{3}\) × \(\frac{3}{5}\) = \(\frac{2}{5}\)

Share and Show – Page No. 305

Find the product. Draw a model.

Question 5.
\(\frac{3}{5} \times \frac{1}{3}=\) \(\frac{□}{□}\)

Answer: \(\frac{1}{5}\)

Explanation:
\(\frac{3}{5}\) × \(\frac{1}{3}\)
Multiply the denominators and the numerators of the fractions.
\(\frac{3}{5}\) × \(\frac{1}{3}\) = \(\frac{3}{15}\)
\(\frac{3}{15}\) = \(\frac{1}{5}\)
\(\frac{3}{5} \times \frac{1}{3}=\) \(\frac{1}{5}\)

Question 6.
\(\frac{3}{4} \times \frac{1}{6}=\) \(\frac{□}{□}\)

Answer: \(\frac{1}{8}\)

Explanation:
\(\frac{3}{4}\) × \(\frac{1}{6}\)
Multiply the denominators and the numerators of the fractions.
\(\frac{3}{4}\) × \(\frac{1}{6}\) = \(\frac{3}{24}\)
3 divides 24 eight times.
So, \(\frac{3}{24}\) = \(\frac{1}{8}\)
Thus, \(\frac{3}{4} \times \frac{1}{6}=\) \(\frac{1}{8}\)

Question 7.
\(\frac{2}{5} \times \frac{5}{6}=\) \(\frac{□}{□}\)

Answer: \(\frac{1}{3}\)

Explanation:
\(\frac{2}{5}\) × \(\frac{5}{6}\)
Multiply the denominators and the numerators of the fractions.
\(\frac{2}{5}\) × \(\frac{5}{6}\) = \(\frac{10}{30}\)
10 divides 30 three times.
\(\frac{10}{30}\) = \(\frac{1}{3}\)
\(\frac{2}{5} \times \frac{5}{6}=\) \(\frac{1}{3}\)

Question 8.
\(\frac{5}{6} \times \frac{3}{5}=\) \(\frac{□}{□}\)

Answer: \(\frac{1}{2}\)

Explanation:
\(\frac{5}{6}\) × \(\frac{3}{5}\)
Multiply the denominators and the numerators of the fractions.
\(\frac{5}{6}\) × \(\frac{3}{5}\) = \(\frac{15}{30}\)
\(\frac{5}{6}\) × \(\frac{3}{5}\) = \(\frac{15}{30}\) = \(\frac{1}{2}\)
\(\frac{5}{6} \times \frac{3}{5}=\) \(\frac{1}{2}\)

Problem Solving – Page No. 306

What’s the Error?

Question 9.
Cheryl and Marcus are going to make a two-tiered cake. The smaller tier is \(\frac{2}{3}\) the size of the larger tier. The recipe for the bottom tier calls for \(\frac{3}{5}\) cup of water. How much water will they need to make the smaller tier?

They made a model to represent the problem. Cheryl says they need \(\frac{6}{9}\) cup of water. Marcus says they need \(\frac{2}{5}\) cup water. Who is correct? Explain.

Cheryl’s answer               Marcus’ answer
Type below:
_________

Answer: Marcus’ answer is correct.

Explanation:
Cheryl and Marcus are going to make a two-tiered cake.
The smaller tier is \(\frac{2}{3}\) the size of the larger tier.
The recipe for the bottom tier calls for \(\frac{3}{5}\) cup of water.
\(\frac{3}{5}\) × \(\frac{2}{3}\) = \(\frac{2}{5}\)

Share and Show – Page No. 309

Complete the statement with equal to, greater than, or less than.

Question 1.
4 × \(\frac{7}{8}\) will be ___________ \(\frac{7}{8}\)

_________

Answer: Greater than

Explanation:
4 × \(\frac{7}{8}\) = \(\frac{7}{2}\)
The denominator with a greater number will be the smallest number.
So, \(\frac{7}{2}\) is greater than \(\frac{7}{8}\)

Question 2.
\(\frac{3}{5} \times \frac{2}{7}\) will be ___________ \(\frac{3}{5}\)

Answer: Less than

Explanation:
\(\frac{3}{5}\) × \(\frac{2}{7}\) = \(\frac{6}{35}\)
The denominator with the greatest number will be the smallest fraction.
So, \(\frac{6}{35}\) is less than \(\frac{3}{5}\)

Question 5.
\(8 \times \frac{7}{8}\) will be ___________ 8

Answer: Less than

Explanation:
8 × \(\frac{7}{8}\)= 7
7 is less than 8.
\(8 \times \frac{7}{8}\) will be less than 8.

On Your Own

Complete the statement with equal to, greater than, or less than.

Question 6.
\(\frac{4}{9} \times \frac{3}{8}\) will be ___________ \(\frac{3}{8}\)

Answer: Less than

Explanation:
\(\frac{4}{9}\) × \(\frac{3}{8}\) = \(\frac{12}{72}\)
= \(\frac{1}{6}\)
\(\frac{1}{6}\) is less than \(\frac{3}{8}\)
\(\frac{4}{9} \times \frac{3}{8}\) will be less than \(\frac{3}{8}\)

Question 7.
\(7 \times \frac{9}{10}\) will be ___________ \(\frac{9}{10}\)

Answer: Greater than

Explanation:
7 × \(\frac{9}{10}\) = \(\frac{63}{10}\)
Denominators are same so compare the numerators.
\(\frac{63}{10}\) is greater than \(\frac{9}{10}\)

Question 8.
\(5 \times \frac{1}{3}\) will be ___________ \(\frac{1}{3}\)

Answer: Greater than

Explanation:
5 × \(\frac{1}{3}\) = \(\frac{5}{3}\)
Denominators are same so compare the numerators.
\(\frac{5}{3}\) is greater than \(\frac{1}{3}\)

Question 9.
\(\frac{6}{11} \times 1\) will be ___________ \(\frac{6}{11}\)

Answer: Equal to

Explanation:
\(\frac{6}{11}\) × 1 = \(\frac{6}{11}\)
\(\frac{6}{11}\) is equal to \(\frac{6}{11}\).

Problem Solving – Page No. 310

Question 12.
Lola is making cookies. She plans to multiply the recipe by 3 so she can make enough cookies for the whole class. If the recipe calls for \(\frac{2}{3}\) cup of sugar, will she need more than \(\frac{2}{3}\) or less than \(\frac{2}{3}\) cup of sugar to make all the cookies?
_________ \(\frac{2}{3}\) cup of sugar

Answer: More than

Explanation:
ola is making cookies. She plans to multiply the recipe by 3 so she can make enough cookies for the whole class.
3 × \(\frac{2}{3}\) = 2
So, Lola needs more than \(\frac{2}{3}\) cup of sugar.

Question 14.
Test Prep Rochelle saves \(\frac{1}{4}\) of her allowance. If she decides to start saving \(\frac{1}{2}\) as much, which statement below is true?
Options:
a. She will be saving the same amount.
b. She will be saving more.
c. She will be saving less.
d. She will be saving twice as much.

Answer: She will be saving more

Explanation:
Test Prep Rochelle saves \(\frac{1}{4}\) of her allowance.
\(\frac{1}{4}\) is greater than \(\frac{1}{2}\)
So, the answer is option B.

Connect to Art

A scale model is a representation of an object with the same shape as the real object. Models can be larger or smaller than the actual object but are often smaller.

Architects often make scale models of the buildings or structures they plan to build. Models can give them an idea of how the structure will look when finished. Each measurement of the building is scaled up or down by the same factor.

Bob is building a scale model of his bike. He wants his model to be \(\frac{1}{5}\) as long as his bike.

Question 15.
If Bob’s bike is 60 inches long, how long will his model be?
_____ in.

Answer: 12 inches

Explanation:
Given that, Bob is building a scale model of his bike. He wants his model to be \(\frac{1}{5}\) as long as his bike.
If Bob’s bike is 60 inches long then multiply with the fraction \(\frac{1}{5}\)
\(\frac{1}{5}\) × 60 = 12 inches
The model will be 12 inches long.

Question 16.
If one wheel on Bob’s model is 4 inches across, how many inches across is the actual wheel on his bike? Explain.
\(\frac{□}{□}\) in.

Answer: \(\frac{4}{5}\) in.

Explanation:
Given that, one wheel on Bob’s model is 4 inches across.
4 × \(\frac{1}{5}\) = \(\frac{4}{5}\) in.

Share and Show – Page No. 313

Find the product. Write the product in simplest form.

Question 1.
\(6 \times \frac{3}{8}\)
\(\frac{6}{1} \times \frac{3}{8}\) = \(\frac{■}{■}\)
______ \(\frac{□}{□}\)

Answer: 2 \(\frac{1}{4}\)

Explanation:
\(\frac{6}{1} \times \frac{3}{8}\) = \(\frac{■}{■}\)
6 × \(\frac{3}{8}\) = \(\frac{18}{8}\) = \(\frac{9}{4}\)
\(\frac{9}{4}\) = 2 \(\frac{1}{4}\)
2 \(\frac{1}{4}\) = \(\frac{■}{■}\)
\(\frac{■}{■}\) = 2 \(\frac{1}{4}\)

Question 2.
\(\frac{3}{8} \times \frac{8}{9}\) = \(\frac{□}{□}\)

Answer: \(\frac{1}{3}\)

Explanation:
\(\frac{3}{8} \times \frac{8}{9}\) = \(\frac{□}{□}\)
\(\frac{3}{8}\) × \(\frac{8}{9}\) = \(\frac{1}{3}\)
Thus, \(\frac{3}{8} \times \frac{8}{9}\) = \(\frac{1}{3}\)

Question 3.
\(\frac{2}{3} \times 27\) = ______

Answer: 18

Explanation:
27 × \(\frac{2}{3}\)
3 divides 27 nine times.
Thus, 27 × \(\frac{2}{3}\) = 18

Question 6.
\(\frac{2}{3} \times \frac{4}{5}\) = \(\frac{□}{□}\)

Answer: \(\frac{8}{15}\)

Explanation:
\(\frac{2}{3}\) × \(\frac{4}{5}\)
Multiply the numerators and the denominators.
\(\frac{2}{3}\) × \(\frac{4}{5}\) = \(\frac{8}{15}\)

Question 7.
\(\frac{1}{3} \times \frac{5}{8}\) = \(\frac{□}{□}\)

Answer: \(\frac{5}{24}\)

Explanation:
\(\frac{1}{3}\) × \(\frac{5}{8}\)
Multiply the numerators and the denominators.
\(\frac{1}{3} \times \frac{5}{8}\) = \(\frac{5}{24}\)

Question 8.
\(4 \times \frac{1}{5}\) = \(\frac{□}{□}\)

Answer: \(\frac{4}{5}\)

Explanation:
Multiply the numerator with the whole number.
4 × \(\frac{1}{5}\) = \(\frac{4}{5}\)
\(4 \times \frac{1}{5}\) = \(\frac{4}{5}\)

On Your Own

Find the product. Write the product in simplest form.

Question 9.
\(2 \times \frac{1}{8}\) = \(\frac{□}{□}\)

Answer: \(\frac{1}{4}\)

Explanation:
Multiply the whole number with the numerator.
2 × \(\frac{1}{8}\)
2 divides 8 four times.
2 × \(\frac{1}{8}\) = \(\frac{1}{4}\)
\(2 \times \frac{1}{8}\) = \(\frac{1}{4}\)

Question 10.
\(\frac{4}{9} \times \frac{4}{5}\) = \(\frac{□}{□}\)

Answer: \(\frac{16}{45}\)

Explanation:
\(\frac{4}{9}\) × \(\frac{4}{5}\)
Multiply the numerators and the denominators.
\(\frac{4}{9}\) × \(\frac{4}{5}\) = \(\frac{16}{45}\)
\(\frac{4}{9} \times \frac{4}{5}\) = \(\frac{16}{45}\)

Question 11.
\(\frac{1}{12} \times \frac{2}{3}\) = \(\frac{□}{□}\)

Answer: \(\frac{1}{18}\)

Explanation:
\(\frac{1}{12}\) × \(\frac{2}{3}\)
Multiply the numerators and the denominators.
\(\frac{1}{12}\) × \(\frac{2}{3}\) = \(\frac{2}{36}\)
\(\frac{2}{36}\) = \(\frac{1}{18}\)
\(\frac{1}{12} \times \frac{2}{3}\) = \(\frac{1}{18}\)

Question 12.
\(\frac{1}{7} \times 30\) = _____ \(\frac{□}{□}\)

Answer: 4 \(\frac{2}{7}\)

Explanation:
30 × \(\frac{1}{7}\) = \(\frac{30}{7}\)
Convert improper fraction to the mixed fraction.
\(\frac{30}{7}\) = 4 \(\frac{2}{7}\)
\(\frac{1}{7} \times 30\) = 4 \(\frac{2}{7}\)

Algebra Evaluate for the given value.

Question 15.
\(\frac{2}{5}\) × c for c = \(\frac{4}{7}\)
\(\frac{□}{□}\)

Answer: \(\frac{8}{35}\)

Explanation:
\(\frac{2}{5}\) × c = \(\frac{4}{7}\)
c = \(\frac{4}{7}\) × \(\frac{2}{5}\)
c = \(\frac{8}{35}\)

Question 16.
m × \(\frac{4}{5}\) for m = \(\frac{7}{8}\)
\(\frac{□}{□}\)

Answer: \(\frac{7}{10}\)

Explanation:
m = \(\frac{4}{5}\) × \(\frac{7}{8}\)
Multiply the numerators and denominators.
\(\frac{4}{5}\) × \(\frac{7}{8}\) = \(\frac{7}{10}\)

Question 17.
\(\frac{2}{3}\) × t for t = \(\frac{1}{8}\)
\(\frac{□}{□}\)

Answer: \(\frac{1}{12}\)

Explanation:
\(\frac{2}{3}\) × t for t = \(\frac{1}{8}\)
t = \(\frac{1}{8}\) × \(\frac{2}{3}\)
t = \(\frac{1}{12}\)

Question 18.
y × \(\frac{2}{3}\) for y = 5
_______

Answer: 4

Explanation:
y × \(\frac{2}{3}\) for y = 5
6 × \(\frac{2}{3}\) = 4

Problem Solving – Page No. 314

Speedskating is a popular sport in the Winter Olympics. Many young athletes in the U.S. participate in speedskating clubs and camps.

Question 19.
At a camp in Green Bay, Wisconsin, \(\frac{7}{9}\) of the participants were from Wisconsin. Of that group, \(\frac{3}{5}\) were 12 years old. What fraction of the group was from Wisconsin and 12 years old?
\(\frac{□}{□}\)

Answer: \(\frac{7}{15}\)

Explanation:
Given that,
At a camp in Green Bay, Wisconsin, \(\frac{7}{9}\) of the participants were from Wisconsin.
Of that group, \(\frac{3}{5}\) were 12 years old.
To find the fraction of the group was from Wisconsin and 12 years old
We have to multiply the fraction \(\frac{7}{9}\) and \(\frac{3}{5}\)
\(\frac{7}{9}\) × \(\frac{3}{5}\) = \(\frac{21}{45}\)
\(\frac{21}{45}\) = \(\frac{7}{15}\)
Thus the fraction of the group was from Wisconsin and 12 years old is \(\frac{7}{15}\).

Question 20.
Maribel wants to skate 1 \(\frac{1}{2}\) miles on Monday. If she skates \(\frac{9}{10}\) mile Monday morning and \(\frac{2}{3}\) of that distance Monday afternoon, will she reach her goal? Explain.
_____

Answer: Yes

Explanation:
Maribel wants to skate 1 \(\frac{1}{2}\) miles on Monday.
To find whether Maribel reached her goal we have to multiply the fractions \(\frac{9}{10}\) and \(\frac{2}{3}\)
\(\frac{9}{10}\) × \(\frac{2}{3}\) = \(\frac{3}{5}\)
By this we can say that Maribel reaches her goal.
So, the answer is yes.

Question 21.
On the first day of camp, \(\frac{5}{6}\) of the skaters were beginners. Of the beginners, \(\frac{1}{3}\) were girls. What fraction of the skaters were girls and beginners? Explain why your answer is reasonable.
\(\frac{□}{□}\)

Answer: \(\frac{5}{18}\)

Explanation:
On the first day of camp, \(\frac{5}{6}\) of the skaters were beginners. Of the beginners, \(\frac{1}{3}\) were girls.
Multiply the fraction of the skaters were beginning and the fraction of skaters were girls.
\(\frac{5}{6}\) × latex]\frac{1}{3}[/latex] = latex]\frac{5}{18}[/latex]
The fraction of the skaters were girls and beginners are latex]\frac{5}{18}[/latex]

Question 22.
Test Prep On Wednesday, Danielle skated \(\frac{2}{3}\) of the way around the track in 2 minutes. Her younger brother skated \(\frac{3}{4}\) of Danielle’s distance in 2 minutes. What fraction of the track did Danielle’s brother finish in 2 minutes?
Options:
a. \(\frac{1}{3}\)
b. \(\frac{1}{2}\)
c. \(\frac{5}{7}\)
d. \(\frac{3}{4}\)

Answer: \(\frac{1}{2}\)

Explanation:
Test Prep On Wednesday, Danielle skated \(\frac{2}{3}\) of the way around the track in 2 minutes.
Her younger brother skated \(\frac{3}{4}\) of Danielle’s distance in 2 minutes.
Multiply the fraction of Danielle skated and her younger brother skated.
\(\frac{2}{3}\) × \(\frac{3}{4}\) = \(\frac{1}{2}\)
Thus the correct answer is option B.

Mid-Chapter Checkpoint – Page No. 315

Concept and Skills

Question 1.
Explain how you would model 5 × \(\frac{2}{3}\)
Type below:
__________

Answer: \(\frac{10}{3}\)

Question 2.
When you multiply \(\frac{2}{3}\) by a fraction less than one, how does the product compare to the factors?
Type below:
__________

Answer: \(\frac{2}{3}\) × \(\frac{1}{2}\)
= \(\frac{1}{3}\)

Find the product. Write the product in simplest form.

Question 3.
\(\frac{2}{3} \times 6\)
______

Answer: 4

Explanation:
6 × \(\frac{2}{3}\)
Multiply the numerator with the whole numbers.
\(\frac{1}{3}\)

Question 6.
\(\frac{7}{8} \times \frac{3}{8}\)
\(\frac{□}{□}\)

Answer: \(\frac{21}{64}\)

Explanation:
\(\frac{7}{8}\) × \(\frac{3}{8}\)
Multiply the numerators and denominators of the fractions.
\(\frac{7}{8}\) × \(\frac{3}{8}\) = \(\frac{21}{64}\)

Question 7.
\(\frac{1}{2} \times \frac{3}{4}\)
\(\frac{□}{□}\)

Answer: \(\frac{3}{8}\)

Explanation:
Multiply the numerators and denominators of the fractions.
\(\frac{1}{2} \times \frac{3}{4}\)
\(\frac{1}{2}\) × \(\frac{3}{4}\) = \(\frac{3}{8}\)
\(\frac{1}{2} \times \frac{3}{4}\) = \(\frac{3}{8}\)

Question 8.
\(\frac{7}{8} \times \frac{4}{7}\)
\(\frac{□}{□}\)

Answer: \(\frac{1}{2}\)

Explanation:
Multiply the numerators and denominators of the fractions.
7 in the numerator and 7 in the denominator will be canceled.
4 divides 8 two times.
Thus the fraction is \(\frac{1}{2}\)
\(\frac{7}{8} \times \frac{4}{7}\) = \(\frac{1}{2}\)

Question 9.
\(2 \times \frac{3}{11}\)
\(\frac{□}{□}\)

Answer: \(\frac{6}{11}\)

Explanation:
Multiply the numerator with the whole numbers.
2 × \(\frac{3}{11}\)
2 × 3 = 6
2 × \(\frac{3}{11}\) = \(\frac{6}{11}\)
Thus, \(2 \times \frac{3}{11}\) = \(\frac{6}{11}\)

Question 10.
\(\frac{5}{8} \times \frac{2}{3}\)
\(\frac{□}{□}\)

Answer: \(\frac{5}{12}\)

Explanation:
\(\frac{5}{8} \times \frac{2}{3}\)
Multiply the numerators and denominators of the fractions.
\(\frac{5}{8}\) × \(\frac{2}{3}\) = \(\frac{10}{24}\)
\(\frac{10}{24}\) = \(\frac{5}{12}\)
\(\frac{5}{8} \times \frac{2}{3}\) = \(\frac{5}{12}\)

Question 11.
\(\frac{7}{12} \times 8\)
______ \(\frac{□}{□}\)

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

Explanation:
8 × \(\frac{7}{12}\)
Multiply the numerator with the whole numbers.
8 × \(\frac{7}{12}\) = \(\frac{56}{12}\) = \(\frac{14}{3}\)
Convert the improper fraction to the mixed fraction.
\(\frac{14}{3}\) = 4 \(\frac{2}{3}\)

Complete the statement with equal to, greater than, or less than.

Mid-Chapter Checkpoint – Page No. 316

Question 14.
There is \(\frac{5}{6}\) of an apple pie left from dinner. Tomorrow, Victor plans to eat \(\frac{1}{6}\) of the pie that was left. How much of the whole pie will be left after he eats tomorrow?
\(\frac{□}{□}\) of the whole pie

Answer: \(\frac{25}{36}\) of the whole pie

Explanation:
Gīven that,
An apple pie left from the dinner is \(\frac{5}{6}\)
Victor plans to eat pie which was left is \(\frac{1}{6}\)
The whole pie will be left after Victor eats tomorrow =?
Pie left from dinner = \(\frac{5}{6}\)
Victor plans to eat pie which was left  = \(\frac{1}{6}\)
\(\frac{5}{6}\) × \(\frac{1}{6}\) = \(\frac{5}{36}\)
To find the whole pie will be left after he eats tomorrow:
\(\frac{5}{6}\) – \(\frac{5}{36}\)
LCD = 36
\(\frac{5}{6}\) × \(\frac{6}{6}\) – \(\frac{5}{36}\)
\(\frac{30}{36}\) – \(\frac{5}{36}\) = \(\frac{25}{36}\)
Therefore, whole pie left after Victor eats tomorrow is \(\frac{25}{36}\)

Question 15.
Everett and Marie are going to make fruit bars for their family reunion. They want to make 4 times the amount the recipe makes. If the recipe calls for \(\frac{2}{3}\) cup of oil, how much oil will they need?
______ \(\frac{□}{□}\) cup of oil

Answer: 2 \(\frac{2}{3}\)

Explanation:
Everett and Marie are going to make fruit bars for their family reunion.
They want to make 4 times the amount the recipe makes.
4 × \(\frac{2}{3}\) = \(\frac{8}{3}\)
The mixed fraction of \(\frac{8}{3}\) is 2 \(\frac{2}{3}\)
Thus Everett and Marie need Everett and Marie of oil.

Question 16.
Matt made the model below to help him solve his math problem. Write an expression that matches Matt’s model.

Type below:
__________

Answer: \(\frac{3}{4}\) × \(\frac{1}{3}\)

Explanation:
By seeing the above figure we can say that the fraction for the Matt’s model is \(\frac{3}{4}\) and \(\frac{2}{3}\).
Multiply the fractions \(\frac{3}{4}\) × \(\frac{2}{3}\) = \(\frac{1}{4}\)

Share and Show – Page No. 319

Use the grid to find the area. Let each square represent \(\frac{1}{3}\) meter by \(\frac{1}{3}\) meter.

Question 1.
1 \(\frac{2}{3}\) × 1 \(\frac{1}{3}\)
• Draw a diagram to represent the dimensions.

• How many squares cover the diagram?
• What is the area of each square?
• What is the area of the diagram?
______ \(\frac{□}{□}\)

Answer: 2 \(\frac{2}{9}\)

Explanation:
20 squares cover the diagram.
Each square represents \(\frac{1}{9}\) square meter
20 × \(\frac{1}{9}\) = \(\frac{20}{9}\)
Convert the fraction into the mixed fraction.
\(\frac{20}{9}\) = 2 \(\frac{2}{9}\)
Thus the area of the diagram is 2 \(\frac{2}{9}\)

Use the grid to find the area. Let each square represent \(\frac{1}{4}\) meter by \(\frac{1}{4}\) meter.

Question 2.
1 \(\frac{3}{4}\) × 1 \(\frac{2}{4}\)

______ \(\frac{□}{□}\)

Answer: 2 \(\frac{5}{8}\)

Explanation:
42 squares cover the diagram.
Each square represents \(\frac{1}{16}\) square meters.
42 × \(\frac{1}{16}\) = \(\frac{21}{8}\)
Convert the fraction into the mixed fraction.
\(\frac{21}{8}\) = 2 \(\frac{5}{8}\)
The area of the diagram is 2 \(\frac{5}{8}\) square meter.

Question 3.
1 \(\frac{1}{4}\) × 1 \(\frac{1}{2}\)

______ \(\frac{□}{□}\)

Answer: 1 \(\frac{7}{8}\)

Explanation:
30 squares cover the diagram.
Each square represents \(\frac{1}{16}\) square meters.
30 × \(\frac{1}{16}\) = \(\frac{15}{8}\)
Convert the fraction into the mixed fraction.
\(\frac{15}{8}\) = 1 \(\frac{7}{8}\)

Use an area model to solve.

Question 6.
1 \(\frac{1}{9}\) × 1 \(\frac{2}{3}\)
______ \(\frac{□}{□}\)

Answer: 1 \(\frac{23}{27}\)

Explanation:
130 squares the diagram.
Each square represents \(\frac{1}{16}\) square meters.
1 \(\frac{1}{9}\) × 1 \(\frac{2}{3}\)
\(\frac{10}{9}\) × \(\frac{5}{3}\) = \(\frac{50}{27}\)
Convert the fraction into the mixed fraction.
\(\frac{50}{27}\) = 1 \(\frac{23}{27}\)

Question 7.
Explain how finding the area of a rectangle with whole-number side lengths compares to finding the area of a rectangle with fractional side lengths.
Type below:
__________

Answer:

15 squares cover the diagram.
Each square is \(\frac{1}{16}\) square unit.
The area of the diagram is \(\frac{15}{16}\) square units.

Problem Solving – Page No. 320

Pose a Problem

Question 8.
Terrance is designing a garden. He drew the following diagram of his garden. Pose a problem using mixed numbers that can be solved using his diagram.


Pose a Problem.                 Solve your problem.
Describe how you decided on the dimensions of Terrance’s garden.
Type below:
__________

Answer:
how finding the area of a rectangle with mixed fractions side compares to finding the area of the rectangle with the fractional side lengths.
6 × 1 \(\frac{1}{8}\) = \(\frac{□}{□}\)
Let each square represent \(\frac{1}{2}\) meter by \(\frac{1}{2}\)
From the above figure, we can say that the number of squares is 27.
So, 27 squares cover the diagram.
Each square is \(\frac{1}{4}\) square unit.
27 × \(\frac{1}{4}\)  = \(\frac{27}{4}\)
Convert the fraction into the mixed fraction.
\(\frac{27}{4}\) = 6 \(\frac{3}{4}\)

Share and Show – Page No. 323

Complete the statement with equal to, greater than, or less than.

Question 1.
\(\frac{5}{6}\) × 2 \(\frac{1}{5}\) will be __________ 2 \(\frac{1}{5}\)
Shade the model to
show \(\frac{5}{6}\) × 2 \(\frac{1}{5}\) .
__________

Answer: Less than

Explanation:
\(\frac{5}{6}\) × 2 \(\frac{1}{5}\)
Convert the mixed fraction to the improper fraction.
2 \(\frac{1}{5}\) = \(\frac{11}{5}\)
\(\frac{5}{6}\) × \(\frac{11}{5}\) = \(\frac{55}{30}\)
1 \(\frac{25}{30}\) = 1 \(\frac{5}{6}\)
Thus 1 \(\frac{5}{6}\) is less than 2 \(\frac{1}{5}\)

On Your Own

Complete the statement with equal to, greater than, or less than.

Question 4.
\(\frac{2}{2}\) × 1 \(\frac{1}{2}\) will be __________ 1 \(\frac{1}{2}\)

Answer: Equal to

Explanation:
\(\frac{2}{2}\) × \(\frac{3}{2}\) = \(\frac{6}{4}\)
\(\frac{6}{4}\) = 1 \(\frac{1}{2}\)
1 \(\frac{1}{2}\) is equal to 1 \(\frac{1}{2}\)
\(\frac{2}{2}\) × 1 \(\frac{1}{2}\) will be equal to 1 \(\frac{1}{2}\)

Question 5.
\(\frac{2}{3}\) × 3 \(\frac{1}{6}\) will be __________ 3 \(\frac{1}{6}\)

Answer: Less than

Explanation:
\(\frac{2}{3}\) × 3 \(\frac{1}{6}\)
First, Convert the mixed fraction to the improper fraction.
\(\frac{2}{3}\) × \(\frac{19}{6}\) = \(\frac{38}{18}\)
\(\frac{38}{18}\) = 2 \(\frac{2}{18}\)
2 \(\frac{2}{18}\) is less than 3 \(\frac{1}{6}\)
\(\frac{2}{3}\) × 3 \(\frac{1}{6}\) will be less than 3 \(\frac{1}{6}\)

Question 6.
2 × 2 \(\frac{1}{4}\) will be __________ 2 \(\frac{1}{4}\)

Answer: Greater than

Explanation:
2 × 2 \(\frac{1}{4}\)
First, Convert the mixed fraction to the improper fraction.
2 × \(\frac{9}{4}\) = \(\frac{18}{4}\)
\(\frac{18}{4}\) = 4 \(\frac{2}{4}\)
4 \(\frac{1}{2}\) is greater than 2 \(\frac{1}{4}\)

Question 7.
4 × 1 \(\frac{3}{7}\) will be __________ 1 \(\frac{3}{7}\)

Answer: Greater than

Explanation:
4 × 1 \(\frac{3}{7}\)
First, Convert the mixed fraction to the improper fraction.
4 × \(\frac{10}{7}\) = \(\frac{40}{7}\)
4 × 1 \(\frac{3}{7}\) = 5 \(\frac{5}{7}\)
5 \(\frac{5}{7}\) is greater than 1 \(\frac{3}{7}\)

Algebra Tell whether the unknown factor is less than 1 or greater than 1.

Question 8.
■ × 1 \(\frac{2}{3}\) = \(\frac{5}{6}\)
The unknown factor is __________ 1.

Answer: Less than

Explanation:
■ × 1 \(\frac{2}{3}\) = \(\frac{5}{6}\)
■ × \(\frac{5}{3}\) = \(\frac{5}{6}\)
■ = \(\frac{1}{2}\)
Thus the unknown factor is \(\frac{1}{2}\)
\(\frac{1}{2}\) is less than 1.

Question 9.
■ × 1 \(\frac{1}{4}\) = 2 \(\frac{1}{2}\)
The unknown factor is __________ 1.

Answer: Greater than

Explanation:
■ × 1 \(\frac{1}{4}\) = 2 \(\frac{1}{2}\)
■ = 2 \(\frac{1}{2}\) ÷ 1 \(\frac{1}{4}\)
■ = 2 × 2 = 4
■ = 4
Thus the unknown factor is 4
4 is greater than 1.

Problem Solving – Page No. 324

Question 10.
Kyle is making a scale drawing of his math book. The dimensions of his drawing will be \(\frac{1}{3}\) the dimensions of his book. If the width of his book is 8 \(\frac{1}{2}\) inches, will the width of his drawing be equal to, greater than, or less than 8 \(\frac{1}{2}\) inches?
__________

Answer: Less than

Explanation:
Given that,
Kyle is making a scale drawing of his math book.
The dimensions of his drawing will be \(\frac{1}{3}\) the dimensions of his book.
\(\frac{1}{3}\) × 8 \(\frac{1}{2}\)
First, Convert the mixed fraction to the improper fraction.
\(\frac{1}{3}\) × \(\frac{17}{2}\) = \(\frac{17}{6}\)
Convert the fraction into the mixed fraction.
\(\frac{17}{6}\) = 2 \(\frac{5}{6}\)
2 \(\frac{5}{6}\) is less than 8 \(\frac{1}{2}\) inches.

Question 11.
Sense or Nonsense?
Penny wants to make a model of a beetle that is larger than life-size. Penny says she is going to use a scaling factor of \(\frac{7}{12}\). Does this make sense or is it nonsense? Explain.
Type below:
__________

Answer: It is nonsense because Penny wants to make beetle Larger than life size. So, the scaling factor \(\frac{7}{12}\) is not corresponding, because when we multiply any value with the number less than 1 we get a smaller number.

Question 12.
Shannon, Mary, and John earn a weekly allowance. Shannon earns an amount that is \(\frac{2}{3}\) of what John earns. Mary earns an amount that is 1 \(\frac{2}{3}\) of what John earns. John earns $20 a week. Who earns the greatest allowance? Who earns the least?
__________ earns the greatest allowance.
__________ earns the least allowance

Answer:
Mary earns the greatest allowance.
Shannon earns the least allowance.

Explanation:

Shannon, Mary, and John earn a weekly allowance.
Shannon earns an amount that is \(\frac{2}{3}\) of what John earns.
Mary earns an amount that is 1 \(\frac{2}{3}\) of what John earns.
John earns $20 a week.
\(\frac{2}{3}\) ________ 1 \(\frac{2}{3}\)
Convert the mixed fraction into the improper fraction.
1 \(\frac{2}{3}\) = \(\frac{5}{3}\)
\(\frac{2}{3}\) is less than 1 \(\frac{2}{3}\)
Thus Shannon earns the least allowance and Mary earns the greatest allowance.

Question 13.
Test Prep Addie’s puppy weighs 1 \(\frac{2}{3}\) times what it weighed when it was born. It weighed 1 \(\frac{1}{3}\) pounds at birth. Which statement below is true?
Options:
a. The puppy weighs the same as it did at birth.
b. The puppy weighs less than it did at birth.
c. The puppy weighs more than it did at birth.
d. The puppy weighs twice what it did at birth.

Answer: The puppy weighs more than it did at birth.

Explanation:
Test Prep Addie’s puppy weighs 1 \(\frac{2}{3}\) times what it weighed when it was born.
It weighed 1 \(\frac{1}{3}\) pounds at birth.
1 \(\frac{2}{3}\) is greater than 1 \(\frac{1}{3}\).
So, the puppy weighs more than it did at birth.
Thus the correct answer is option C.

Share and Show – Page No. 327

Find the product. Write the product in simplest form.

Question 1.
1 \(\frac{2}{3}\) × 3 \(\frac{4}{5}\) = \(\frac{■}{3}\) × \(\frac{■}{5}\)
= \(\frac{■}{■}\)
=?
_____ \(\frac{□}{□}\)

Answer: 6 \(\frac{1}{3}\)

Explanation:

1 \(\frac{2}{3}\) × 3 \(\frac{4}{5}\)
\(\frac{5}{3}\) × \(\frac{19}{5}\)
\(\frac{■}{3}\) × \(\frac{■}{5}\) = \(\frac{5}{3}\) × \(\frac{19}{5}\)
\(\frac{5}{3}\) × \(\frac{19}{5}\) = 6 \(\frac{1}{3}\)

Question 2.
\(\frac{1}{2}\) × 1 \(\frac{1}{3}\)

Shade the model to find the product.
\(\frac{□}{□}\)

Answer: \(\frac{2}{3}\)

Explanation:
\(\frac{1}{2}\) × 1 \(\frac{1}{3}\)
1 \(\frac{1}{3}\) = \(\frac{4}{3}\)
\(\frac{1}{2}\) × \(\frac{4}{3}\) = \(\frac{4}{6}\)

Question 5.
\(1 \frac{2}{7} \times 1 \frac{3}{4}\) = ______ \(\frac{□}{□}\)

Answer: 2 \(\frac{1}{4}\)

Explanation:
1 \(\frac{2}{7}\) × 1 \(\frac{3}{4}\)
Multiply the numerators and the denominators.
Convert the mixed fraction to the improper fraction.
\(\frac{9}{7}\) × \(\frac{7}{4}\) = \(\frac{63}{28}\)
\(\frac{63}{28}\) = 2 \(\frac{1}{4}\)
\(1 \frac{2}{7} \times 1 \frac{3}{4}\) = 2 \(\frac{1}{4}\)

Question 6.
\(\frac{3}{4} \times 1 \frac{1}{4}\) = ______ \(\frac{□}{□}\)

Answer: \(\frac{15}{16}\)

Explanation:
\(\frac{3}{4}\) × 1 \(\frac{1}{4}\)
\(\frac{3}{4}\) × \(\frac{5}{4}\) = \(\frac{15}{16}\)
\(\frac{3}{4} \times 1 \frac{1}{4}\) = \(\frac{15}{16}\)

Use the Distributive Property to find the product.

Question 7.
\(16 \times 2 \frac{1}{2}\) = ______

Answer: 40

Explanation:
\(16 \times 2 \frac{1}{2}\)
(16 × 2) + (16 × \(\frac{1}{2}\))
32 + 8 = 40
\(16 \times 2 \frac{1}{2}\) = 40

Question 8.
\(1 \frac{4}{5} \times 15\) = ______

Answer: 27

Explanation:
\(1 \frac{4}{5} \times 15\)
15 × 1 \(\frac{4}{5}\)
(1 × 15) + (15 × \(\frac{4}{5}\))
15 + \(\frac{60}{5}\)
15 + 12 = 27
Thus \(1 \frac{4}{5} \times 15\) = 27

On Your Own

Find the product. Write the product in simplest form.

Question 9.
\(\frac{3}{4} \times 1 \frac{1}{2}\) = ______ \(\frac{□}{□}\)

Answer: 1 \(\frac{1}{8}\)

Explanation:
\(\frac{3}{4}\) × 1 \(\frac{1}{2}\)
\(\frac{3}{4}\) × \(\frac{3}{2}\) = \(\frac{9}{8}\)
Now convert the improper fraction to the mixed fraction.
\(\frac{9}{8}\) = 1 \(\frac{1}{8}\)

Question 12.
\(2 \frac{1}{2} \times 5 \frac{1}{5}\) = ______

Answer: 13

Explanation:
2 \(\frac{1}{2}\) × 5 \(\frac{1}{5}\)
\(\frac{5}{2}\) × \(\frac{26}{5}\) = \(\frac{130}{10}\)
10 divides 130 thirteen times.
\(\frac{130}{10}\) = 13
\(2 \frac{1}{2} \times 5 \frac{1}{5}\) = 13

Question 13.
\(12 \frac{3}{4} \times 2 \frac{2}{3}\) = ______

Answer: 34

Explanation:
12 \(\frac{3}{4}\) × 2 \(\frac{2}{3}\)
\(\frac{51}{4}\) × \(\frac{6}{3}\)
3 divides 51 seventeen times.
17 × 2 = 34

Question 14.
\(3 \times 4 \frac{1}{2}\) = ______ \(\frac{□}{□}\)

Answer: 13 \(\frac{1}{2}\)

Explanation:
3 × 4 \(\frac{1}{2}\)
3 × \(\frac{9}{2}\) = \(\frac{27}{2}\)
Convert the fraction to the mixed fraction
\(\frac{27}{2}\) = 13 \(\frac{1}{2}\)

Question 15.
\(2 \frac{3}{8} \times \frac{4}{9}\) = ______ \(\frac{□}{□}\)

Answer: 1 \(\frac{1}{18}\)

Explanation:
2 \(\frac{3}{8}\) × \(\frac{4}{9}\)
\(\frac{19}{8}\) × \(\frac{4}{9}\) = \(\frac{76}{72}\)
\(\frac{76}{72}\) = 1 \(\frac{1}{18}\)
\(2 \frac{3}{8} \times \frac{4}{9}\) = 1 \(\frac{1}{18}\)

Question 16.
\(1 \frac{1}{3} \times 1 \frac{1}{4} \times 1 \frac{1}{5}\) = ______

Answer: 2

Explanation:

1 \(\frac{1}{3}\) × 1 \(\frac{1}{4}\) × 1 \(\frac{1}{5}\)
\(\frac{4}{3}\) × \(\frac{5}{4}\) × \(\frac{6}{5}\) = 2
\(1 \frac{1}{3} \times 1 \frac{1}{4} \times 1 \frac{1}{5}\) = 2

Use the Distributive Property to find the product.

Share and Show Connect to health – Page No. 328

Changing Recipes

You can make a lot of recipes more healthful by reducing the amounts of fat, sugar, and salt.

Kelly has a muffin recipe that calls for 1 \(\frac{1}{2}\) cups of sugar. She wants to use \(\frac{1}{2}\) that amount of sugar and more cinnamon and vanilla. How much sugar will she use?
Multiply 1 \(\frac{1}{2}\) by \(\frac{1}{2}\) to find what part of the original amount of sugar to use.
Write the mixed number as a fraction greater than 1. Then, multiply.
\(\frac{1}{2} \times 1 \frac{1}{2}=\frac{1}{2} \times \frac{3}{2}\)
= \(\frac{3}{4}\)
So, Kelly will use \(\frac{3}{4}\) cup of sugar.

Question 19.
Michelle has a recipe that calls for 2 \(\frac{1}{2}\) cups of vegetable oil. She wants to use \(\frac{2}{3}\) that amount of oil and use applesauce to replace the rest. How much vegetable oil will she use?
______ \(\frac{□}{□}\) cups

Answer: 1 \(\frac{2}{3}\)

Explanation:
Michelle has a recipe that calls for 2 \(\frac{1}{2}\) cups of vegetable oil.
She wants to use \(\frac{2}{3}\) that amount of oil and use applesauce to replace the rest
Multiply 2 \(\frac{1}{2}\) by \(\frac{2}{3}\) to find how much vegetable oil she will use.
2 \(\frac{1}{2}\) × \(\frac{2}{3}\)
Convert the mixed fractions into the fractions.
\(\frac{5}{2}\) × \(\frac{2}{3}\) = \(\frac{10}{6}\)
\(\frac{10}{6}\) = \(\frac{5}{3}\) = 1 \(\frac{2}{3}\)
She will use 1 \(\frac{2}{3}\) cups of vegetable oil.

Question 20.
Tony’s recipe for soup calls for 1 \(\frac{1}{4}\) teaspoons of salt. He wants to use \(\frac{1}{2}\) that amount. How much salt will he use?
\(\frac{□}{□}\) teaspoon

Answer: \(\frac{5}{8}\)

Explanation:
Tony’s recipe for soup calls for 1 \(\frac{1}{4}\) teaspoons of salt.
He wants to use \(\frac{1}{2}\) that amount.
Multiply the fractions to find how much salt he will use in the recipe for soup.
1 \(\frac{1}{4}\) × \(\frac{1}{2}\)
Convert the mixed fractions to the improper fractions.
\(\frac{5}{4}\) × \(\frac{1}{2}\) = \(\frac{5}{8}\)
Thus Tony use \(\frac{5}{8}\) teaspoon of salt for soup.

Question 21.
Jeffrey’s recipe for oatmeal muffins calls for 2 \(\frac{1}{4}\) cups of oatmeal and makes one dozen muffins. If he makes 1 \(\frac{1}{2}\) dozen muffins for a club meeting, how much oatmeal will he use?
_____ \(\frac{□}{□}\) cups

Answer: 3 \(\frac{3}{8}\)

Explanation:
Jeffrey’s recipe for oatmeal muffins calls for 2 \(\frac{1}{4}\) cups of oatmeal and makes one dozen muffins.
To find how much oatmeal he will use we need to multiply the fractions.
2 \(\frac{1}{4}\) × 1 \(\frac{1}{2}\)
Convert the mixed fractions to the improper fractions.
\(\frac{9}{4}\) × \(\frac{3}{2}\)
\(\frac{27}{8}\) = 3 \(\frac{3}{8}\)
Thus he will use 3 \(\frac{3}{8}\) cups of oatmeal to make oatmeal muffins.

Question 22.
Cara’s muffin recipe calls for 1 \(\frac{1}{2}\) cups of flour for the muffins and \(\frac{1}{4}\) cup of flour for the topping. If she makes \(\frac{1}{2}\) of the original recipe, how much flour will she use?
\(\frac{□}{□}\) cup of flour

Answer: \(\frac{7}{8}\)

Explanation:
Convert mixed fractions into improper fractions.
1 \(\frac{1}{2}\) = \(\frac{3}{2}\)
\(\frac{3}{2}\) + \(\frac{1}{4}\) = \(\frac{7}{4}\)
Now we can find how much flour she will use to make \(\frac{1}{2}\) of the original recipe, when multiply
\(\frac{7}{4}\) by \(\frac{1}{2}\)
\(\frac{7}{4}\) × \(\frac{1}{2}\) = \(\frac{7}{8}\)

Share and Show – Page No. 331

Question 1.
When Pascal built a dog house, he knew he wanted the floor of the house to have an area of 24 square feet. He also wanted the width to be \(\frac{2}{3}\) the length. What are the dimensions of the dog house?
First, choose two numbers that have a product of 24.
Guess: ____ feet and ____ feet
Then, check those numbers. Is the greater number \(\frac{2}{3}\) of the other number?
Check: \(\frac{2}{3}\) × _____ = _____
My guess is ______.
Finally, if the guess is not correct, revise it and check again. Continue until you find the correct answer.
_____ feet by _____ feet

Answer: 4 feet by 6 feet

Explanation:
When Pascal built a dog house, he knew he wanted the floor of the house to have an area of 24 square feet.
He also wanted the width to be \(\frac{2}{3}\) the length.
My guess for 24 square feet is 4 feet and 6 feet.
Now let us check the numbers.
6 × \(\frac{2}{3}\) = 4
So my guess is correct.
Thus the dimensions are 4 feet by 6 feet

Question 3.
Leo wants to paint a mural that covers a wall with an area of 1,440 square feet. The height of the wall is \(\frac{2}{5}\) of its length. What is the length and the height of the wall?
_____ feet by _____ feet

Answer: 24 feet by 60 feet

Explanation:
Leo wants to paint a mural that covers a wall with an area of 1,440 square feet. The height of the wall is \(\frac{2}{5}\) of its length.
Guess: 1,440 square feet = 24 feet × 60 feet
\(\frac{2}{5}\) × 60 = 24
So, our guess is correct.
.Thus the dimensions of the wall are 24 feet by 60 feet.

On Your Own – Page No. 332

Question 4.
Barry wants to make a drawing that is \(\frac{1}{4}\) the size of the original. If a tree in the original drawing is 14 inches tall, how tall will the tree in Barry’s drawing be?
_____ \(\frac{□}{□}\) inches

Answer: 3 \(\frac{1}{2}\) inches

Explanation:
Given:
Barry wants to make a drawing that is \(\frac{1}{4}\) the size of the original.
The tree is 14 inches tall in the drawing.
14 × \(\frac{1}{4}\) = \(\frac{14}{4}\) = \(\frac{7}{2}\)
Convert the fraction to the mixed fraction.
\(\frac{7}{2}\) = 3 \(\frac{1}{2}\) inches

Question 5.
A blueprint is a scale drawing of a building. The dimensions of the blueprint for Penny’s doll house are \(\frac{1}{4}\) of the measurements of the actual doll house. The floor of the doll house has an area of 864 square inches. If the width of the doll house is \(\frac{2}{3}\) the length, what are the dimensions of the floor on the blueprint of the doll house?
_____ inches by _____ inches

Answer: 9 inches by 6 inches

Explanation:
A blueprint is a scale drawing of a building.
The dimensions of the blueprint for Penny’s dollhouse are \(\frac{1}{4}\) of the measurements of the actual dollhouse.
The floor of the dollhouse has an area of 864 square inches.
The area of the dollhouse is 54 square inches.
My guess is 9 inches by 6 inches
Let us check the numbers
9 × \(\frac{2}{3}\) = 6
My guess is correct.
Therefore the dimensions of the floor on the blueprint of the dollhouse is 9 inches by 6 inches

Question 6.
Pose a Problem Look back at Exercise 4. Write a similar problem using a different measurement and a different fraction. Then solve your problem.
Type below:
__________

Answer:
Kyle is making reusable grocery bags and lunch bags. She used a 3/4 yard of cloth to make the grocery bag. A lunch bag requires 2/3 of the amount of cloth of a grocery bag’s needs. How much does she need to make the lunch bag?
\(\frac{3}{4}\) × \(\frac{2}{3}\) = \(\frac{1}{2}\)
Thus Kyle needs \(\frac{1}{2}\) of the grocery bag to make the lunch bag.

Question 7.
Test Prep Albert’s photograph has an area of 80 square inches. The length of the photo is 1 \(\frac{1}{4}\) the width. Which of the following could be the dimensions of the photograph?
Options:
a. 5 inches by 16 inches
b. 12 inches by 10 inches
c. 6 inches by 5 inches
d. 10 inches by 8 inches

Answer: 10 inches by 8 inches

Explanation:
Albert’s photograph has an area of 80 square inches.
The length of the photo is 1 \(\frac{1}{4}\) the width.
My guess for 80 square inches is 10 inches by 8 inches.
Now let us check the numbers.
8 × 1 \(\frac{1}{4}\) = 8 × \(\frac{5}{4}\) = 10
Thus the correct answer is option D.

Chapter Review/Test – Page No. 333

Concepts and Skills

Question 1.
When you multiply 3 \(\frac{1}{4}\) by a number greater than one, how does the product compare to 3 \(\frac{1}{4}\)? Explain.
Type below:
__________

Answer:
Your product will be greater than 3 1/4 because anytime you multiply a fraction times a whole number less than 1 you get a fraction less than one and any time you multiply by a fraction and a whole number greater than 1 your answer is greater than 1.

Use a model to solve.

Question 4.
\(\frac{5}{8}\) × 24 = _____

Answer: 15

Explanation:

\(\frac{5}{8}\) × 24
8 divides 24 three times.
5 × 3 = 15
\(\frac{5}{8}\) × 24 = 15

Find the product. Write the product in simplest form.

Question 5.
\(\frac{3}{5}\) × 8 = _____ \(\frac{□}{□}\)

Answer: 4 \(\frac{4}{5}\)

Explanation:
\(\frac{3}{5}\) × 8 = \(\frac{24}{5}\)
The mixed fraction of \(\frac{24}{5}\) is 4 \(\frac{4}{5}\)
\(\frac{3}{5}\) × 8 = 4 \(\frac{4}{5}\)

Question 6.
\(\frac{1}{4}\) × 10 = _____ \(\frac{□}{□}\)

Answer: 2 \(\frac{1}{2}\)

Explanation:
\(\frac{1}{4}\) × 10
2 divides 10 five times.
\(\frac{1}{2}\) × 5 = \(\frac{5}{2}\)
The mixed fraction of \(\frac{5}{2}\) is 2 \(\frac{1}{2}\)
\(\frac{1}{4}\) × 10 = 2 \(\frac{1}{2}\)

Question 9.
\(\frac{1}{5}\) × \(\frac{5}{7}\) = \(\frac{□}{□}\)

Answer: \(\frac{1}{7}\)

Explanation:
\(\frac{1}{5}\) × \(\frac{5}{7}\)
5 in the numerator and 5 in the denominator gets canceled.
= \(\frac{1}{7}\)
Thus \(\frac{1}{5}\) × \(\frac{5}{7}\) = \(\frac{1}{7}\)

Question 10.
\(\frac{3}{8}\) × \(\frac{1}{6}\) = \(\frac{□}{□}\)

Answer: \(\frac{1}{16}\)

Explanation:
\(\frac{3}{8}\) × \(\frac{1}{6}\)
3 divides 6 two times
\(\frac{3}{8}\) × \(\frac{1}{6}\) = \(\frac{1}{8}\) × \(\frac{1}{2}\)
Multiply the denominators.
= \(\frac{1}{16}\)
Thus \(\frac{3}{8}\) × \(\frac{1}{6}\) = \(\frac{1}{16}\)

Complete the statement with equal to, greater than, or less than.

Chapter Review/Test – Page No. 334

Fill in the bubble completely to show your answer.

Question 13.
Wolfgang wants to enlarge a picture he developed. Which factor listed below would scale up (enlarge) his picture the most if he used it to multiply its current dimensions?
Options:
a. \(\frac{7}{8}\)
b. \(\frac{14}{14}\)
c. 1 \(\frac{4}{9}\)
d. \(\frac{3}{2}\)

Answer: 1 \(\frac{4}{9}\)

Explanation:
The greatest fraction among all the fractions is 1 \(\frac{4}{9}\).
1 \(\frac{4}{9}\) is greater than 1.
Thus the correct answer is option C.

Question 14.
Rachel wants to reduce the size of her photo. Which factor listed below would scale down (reduce) the size of her picture the most?
Options:
a. \(\frac{5}{8}\)
b. \(\frac{11}{16}\)
c. 1 \(\frac{3}{4}\)
d. \(\frac{8}{5}\)

Answer: \(\frac{5}{8}\)

Explanation:
Compare to all the fractions \(\frac{5}{8}\) is smaller.
So, Rachel would reduce the size of her picture to \(\frac{5}{8}\)
So, the correct answer is option A.

Question 15.
Marteen wants to paint \(\frac{2}{3}\) of her room today. She wants to paint \(\frac{1}{4}\) of that before lunch. How much of her room will she paint today before lunch?
Options:
a. \(\frac{1}{12}\)
b. \(\frac{1}{6}\)
c. 1 \(\frac{5}{12}\)
d. \(\frac{11}{12}\)

Answer: \(\frac{1}{6}\)

Explanation:
Marteen wants to paint \(\frac{2}{3}\) of her room today.
She wants to paint \(\frac{1}{4}\) of that before lunch.
\(\frac{2}{3}\) × \(\frac{1}{4}\) = \(\frac{1}{6}\)
So, the answer is option B.

Chapter Review/Test – Page No. 335

Fill in the bubble completely to show your answer.

Question 16.
Gia’s bus route to school is 5 \(\frac{1}{2}\) miles. The bus route home is 1 \(\frac{3}{5}\) times as long. How long is Gia’s bus route home?
Options:
a. 5 \(\frac{3}{10}\) miles
b. 8 miles
c. 8 \(\frac{4}{5}\) miles
d. 17 \(\frac{3}{5}\) miles

Answer: 8 \(\frac{4}{5}\) miles

Explanation:
Gia’s bus route to school is 5 \(\frac{1}{2}\) miles. The bus route home is 1 \(\frac{3}{5}\) times as long.
5 \(\frac{1}{2}\) × 1 \(\frac{3}{5}\)
Convert the mixed fractions to improper fractions.
\(\frac{11}{2}\) × \(\frac{8}{5}\) = \(\frac{88}{10}\) = \(\frac{44}{5}\)
The mixed fraction of \(\frac{44}{5}\) is 8 \(\frac{4}{5}\) miles
Therefore the answer is option C.

Question 17.
Carl’s dog weighs 2 \(\frac{1}{3}\) times what Judy’s dog weighs. If Judy’s dog weighs 35 \(\frac{1}{2}\) pounds, how much does Carl’s dog weigh?
Options:
a. 88 \(\frac{3}{4}\) pounds
b. 82 \(\frac{5}{6}\) pounds
c. 81 \(\frac{2}{3}\) pounds
d. 71 pounds

Answer: 82 \(\frac{5}{6}\) pounds

Explanation:
Carl’s dog weighs 2 \(\frac{1}{3}\) times what Judy’s dog weighs.
To find the weigh of Carl’s dog we need to multiply the fractions
2 \(\frac{1}{3}\) and 35 \(\frac{1}{2}\)
\(\frac{7}{3}\) × \(\frac{71}{2}\) = \(\frac{497}{6}\)
The mixed fraction of \(\frac{497}{6}\) is 82 \(\frac{5}{6}\) pounds.
Thus the correct answer is option B.

Question 18.
In a fifth grade class, \(\frac{4}{5}\) of the girls have brown hair. Of the brown-haired girls, \(\frac{3}{5}\) of the girls have long hair. What fraction of the girls in the class have long brown hair?
Options:
a. \(\frac{1}{20}\)
b. \(\frac{1}{5}\)
c. \(\frac{3}{5}\)
d. \(\frac{1}{4}\)

Answer: \(\frac{1}{5}\)

Explanation:
In a fifth grade class, \(\frac{4}{5}\) of the girls have brown hair. Of the brown-haired girls, \(\frac{3}{5}\) of the girls have long hair.
\(\frac{4}{5}\) – \(\frac{3}{5}\) = \(\frac{1}{5}\)
The correct answer is option B.

Chapter Review/Test – Page No. 336

Constructed Response

Question 19.
Tasha plans to tile the floor in her room with square tiles that are \(\frac{1}{4}\) foot long. Will she use more or fewer tiles if she is only able to purchase square tiles that are \(\frac{1}{3}\) foot long? Explain.
_________ tiles

Answer: Fewer

Explanation:
\(\frac{1}{4}\) is less than \(\frac{1}{3}\)
So, Tasha will use fewer tiles if she is only able to purchase square tiles that are \(\frac{1}{3}\) long.

Performance Task

Question 20.
For a bake sale, Violet wants to use the recipe below.

A). If she wants to double the recipe, how much flour will she need?
_____ \(\frac{□}{□}\) cups flour

Answer: 5 \(\frac{1}{2}\) cups flour

Explanation:
To bake the sugar cookies she needs 2 \(\frac{3}{4}\) cups flour.
If she wants to double the recipe, she needs to multiply the 2 \(\frac{3}{4}\) cups flour by 2.
2 \(\frac{3}{4}\) + 2 \(\frac{3}{4}\) = 5 \(\frac{1}{2}\) cups flour

Question 20.
B). Baxter wants to make 1 \(\frac{1}{2}\) times the recipe. Will he need more or less sugar than Violet needs if she doubles the recipe? Explain.
__________ sugar

Answer: less

Explanation:
If violet doubles the recipe he need 2 × 1 \(\frac{1}{2}\) = 3 cups of sugar
Baxter wants to make 1 \(\frac{1}{2}\) times the recipe.
1 \(\frac{1}{2}\) × \(\frac{1}{2}\) = 1 \(\frac{3}{4}\)
Baxter needs less sugar when compared to Violet’s recipe.

Question 20.
C). As shown, the recipe makes 60 cookies. Jorge wants to bring 150 cookies. How much flour will he need to make 150 cookies? Explain how you got your answer. (Hint: what can you multiply 60 by to get 150?)
_____ \(\frac{□}{□}\) cups flour

Answer: 2 \(\frac{1}{2}\) cups flour

Explanation:
The recipe makes 60 cookies. Jorge wants to bring 150 cookies.
Let the amount of flour be x.
60 × x = 150
x = 150/60 = 5/2
The mixed fraction of \(\frac{5}{2}\) is 2 \(\frac{1}{2}\)
Thus, Jorge need 2 \(\frac{1}{2}\) cups flour to make 150 cookies.

Conclusion

Make the most out of the Go Math Answer Key for Grade 5 Ch 7 Multiply Fractions. Our guide on 5th Grade Chapter 7 Multiply Fractions helps you to clarify the concepts related in no time. Keep in touch with our site to avail Go Math Answer Key for different Grades all in one place.

Go Math Grade 5 Answer Key Chapter 6 Add and Subtract Fractions with Unlike Denominators can be downloaded free of cost from here. HMH Go Math Grade 5 Answer Key includes topics such as Addition and Subtraction with unlike denominators, Estimate fraction sums and differences, Least Common Denominators, etc. Begin your preparation from Go Math Grade 5 Chapter 6 Solution Key Add and Subtract Fractions with Unlike Denominators and score better grades in your exams.

Go Math Grade 5 Answer Key Chapter 6 Add and Subtract Fractions with Unlike Denominators

The Go Math Grade 5 Answer Key Add and Subtract Fractions with Unlike Denominators covers all the Questions in Chapter Tests, Practice Tests, etc., and has detailed solutions for all of them. Try to solve as many problems as possible from the  Go Math Answer Key for Grade 5 Chapter 6 Add and Subtract Fractions with Unlike Denominators and know the concepts behind them easily. Access the Topics in 5th Grade Go Math Ch 6 Add and Subtract Fractions with Unlike Denominators via quick links available in the forthcoming modules.

Lesson 1: Investigate • Addition with Unlike Denominators

• Share and Show – Page No. 244
• Investigate • Addition with Unlike Denominators Page No. 245
• Problem Solving – Page No. 246

Lesson 2: Investigate • Subtraction with Unlike Denominators

• Share and Show – Page No. 248
• Investigate • Subtraction with Unlike Denominators Page No. 249
• UNLOCK the Problem – Page No. 250

Lesson 3: Estimate Fraction Sums and Differences

• Share and Show – Page No. 253
• Problem Solving – Page No. 254

Lesson 4: Common Denominators and Equivalent Fractions

• Share and Show – Page No. 256
• Common Denominators and Equivalent Fractions Page No. 257
• UNLOCK the Problem – Page No. 258

Lesson 5: Add and Subtract Fractions

• Share and Show – Page No. 260
• On Your Own – Page No. 261
• Problem Solving – Page No. 262

Mid-Chapter Checkpoint

• Mid-Chapter Checkpoint – Vocabulary – Page No. 263
• Mid-Chapter Checkpoint – Page No. 264

Lesson 6: Add and Subtract Mixed Numbers

• Share and Show – Page No. 266
• Page No. 267
• Problem Solving – Page No. 268

Lesson 7: Subtraction with Renaming

• Share and Show – Page No. 270
• Subtraction with Renaming Page No. 271
• Subtraction with Renaming Page No. 272

Lesson 8: Algebra • Patterns with Fractions

• Share and Show – Page No. 275
• Problem Solving – Page No. 276

Lesson 9: Problem Solving • Practice Addition and Subtraction

• Share and Show – Page No. 279
• On Your Own – Page No. 280

Lesson 10: Algebra • Use Properties of Addition

• Share and Show – Page No. 283
• Problem Solving – Page No. 284

Chapter 6 Review/Test

• Chapter Review/Test – Vocabulary – Page No. 285
• Chapter Review/Test – Page No. 286
• Chapter Review/Test – Page No. 287
• Chapter Review/Test – Page No. 288

Share and Show – Page No. 244

Use fraction strips to find the sum. Write your answer in simplest form.

Question 1.

\(\frac{1}{2}+\frac{3}{8}=\)
\(\frac{□}{□}\)

Answer: \(\frac{7}{8}\)
Explanation:
Step 1:
Place three \(\frac{1}{8}\) fractions strips under the 1 whole strip on your Mathboard. Then place a \(\frac{1}{2}\) fraction strip beside the three \(\frac{1}{8}\) strips.
Step 2:
Find fraction strips, all with the same denominator, that are equivalent to \(\frac{1}{2}\) and \(\frac{3}{8}\). Place the fraction strips under the sum. At the right, draw a picture of the model and write the equivalent fractions.
\(\frac{1}{2}\) = \(\frac{1}{2}\) × \(\frac{4}{4}\) = \(\frac{4}{8}\)
\(\frac{3}{8}\)
Step 3:
Add the fractions with like denominators. Use the 1 whole strip to rename the sum in the simplest form.
\(\frac{4}{8}\) + \(\frac{3}{8}\) = \(\frac{7}{8}\)

Question 2.

\(\frac{1}{2}+\frac{2}{5}=\)
\(\frac{□}{□}\)

Answer: \(\frac{9}{10}\)
Explanation:
Step 1:
Place two \(\frac{1}{5}\) fractions strips under the 1 whole strip on your Mathboard. Then place a \(\frac{1}{2}\) fraction strip beside the two \(\frac{1}{5}\) strips.
Step 2:
Find fraction strips, all with the same denominator, that are equivalent to \(\frac{1}{2}\) and \(\frac{2}{5}\). Place the fraction strips under the sum. At the right, draw a picture of the model and write the equivalent fractions.
\(\frac{1}{2}\) = \(\frac{1}{2}\) × \(\frac{5}{5}\) = \(\frac{5}{10}\)
\(\frac{2}{5}\) = \(\frac{2}{5}\) × \(\frac{2}{2}\) = \(\frac{4}{10}\)
Step 3:
Add the fractions with like denominators. Use the 1 whole strip to rename the sum in the simplest form.
\(\frac{5}{10}\) + \(\frac{4}{10}\) = \(\frac{9}{10}\)
Thus, \(\frac{1}{2}\) + \(\frac{2}{5}\) = \(\frac{9}{10}\)

Page No. 245

Use fraction strips to find the sum. Write your answer in simplest form.

Question 3.

\(\frac{3}{8}+\frac{1}{4}=\)
\(\frac{□}{□}\)

Answer: \(\frac{5}{8}\)
Explanation:
Step 1:
Place three \(\frac{1}{8}\) fractions strips under the 1 whole strip on your Mathboard. Then place a \(\frac{1}{4}\) fraction strip beside the three \(\frac{1}{8}\) strips.
Step 2:
Find fraction strips, all with the same denominator, that are equivalent to \(\frac{1}{4}\) and \(\frac{3}{8}\). Place the fraction strips under the sum. At the right, draw a picture of the model and write the equivalent fractions.
\(\frac{1}{4}\) × \(\frac{2}{2}\) = \(\frac{2}{8}\)
Step 3:
Add the fractions with like denominators. Use the 1 whole strip to rename the sum in the simplest form.
\(\frac{2}{8}\) + \(\frac{3}{8}\) = \(\frac{5}{8}\)

Question 4.

\(\frac{3}{4}+\frac{1}{3}=\)
______ \(\frac{□}{□}\)

Answer: 1 \(\frac{1}{12}\)
Explanation:
Step 1:
Place three \(\frac{3}{4}\) fractions strips under the 1 whole strip on your Mathboard. Then place a \(\frac{1}{3}\) fraction strip beside the three \(\frac{1}{4}\) strips.
Step 2:
Find fraction strips, all with the same denominator, that are equivalent to \(\frac{1}{3}\) and \(\frac{3}{4}\). Place the fraction strips under the sum. At the right, draw a picture of the model and write the equivalent fractions.
\(\frac{1}{3}\) × \(\frac{4}{4}\) = \(\frac{4}{12}\)
\(\frac{3}{4}\) × \(\frac{3}{3}\) = \(\frac{9}{12}\)
Step 3:
Add the fractions with like denominators. Use the 1 whole strip to rename the sum in the simplest form.
\(\frac{4}{12}\) + \(\frac{9}{12}\) = \(\frac{13}{12}\) = 1 \(\frac{1}{12}\)

Use fraction strips to find the sum. Write your answer in simplest form.

Question 6.
\(\frac{1}{4}+\frac{1}{12}=\)
\(\frac{□}{□}\)

Answer: \(\frac{4}{12}\)
Explanation:
Step 1:
Place \(\frac{1}{12}\) fractions strips under the 1 whole strip on your Mathboard. Then place a \(\frac{1}{4}\) fraction strip beside the \(\frac{1}{12}\) strips.
Step 2:
Find fraction strips, all with the same denominator, that are equivalent to \(\frac{1}{12}\) and \(\frac{1}{4}\). Place the fraction strips under the sum. At the right, draw a picture of the model and write the equivalent fractions.
\(\frac{1}{4}\) • \(\frac{3}{3}\) = \(\frac{3}{12}\)
\(\frac{1}{12}\)
Step 3:
Add the fractions with like denominators. Use the 1 whole strip to rename the sum in the simplest form.
\(\frac{3}{12}\) + \(\frac{1}{12}\) = \(\frac{4}{12}\)

Question 7.
\(\frac{1}{2}+\frac{3}{10}=\)
\(\frac{□}{□}\)

Answer: \(\frac{8}{10}\)
Explanation:
Step 1:
Place three \(\frac{1}{10}\) fractions strips under the 1 whole strip on your Mathboard. Then place a \(\frac{1}{2}\) fraction strip beside the three \(\frac{1}{10}\) strips.
Step 2:
Find fraction strips, all with the same denominator, that are equivalent to \(\frac{1}{2}\) and \(\frac{3}{10}\). Place the fraction strips under the sum. At the right, draw a picture of the model and write the equivalent fractions.
\(\frac{1}{2}\) • \(\frac{5}{5}\) = \(\frac{5}{10}\)
Step 3:
Add the fractions with like denominators. Use the 1 whole strip to rename the sum in the simplest form.
\(\frac{5}{10}\) + \(\frac{3}{10}\) = \(\frac{8}{10}\)

Question 8.
\(\frac{2}{3}+\frac{1}{6}=\)
\(\frac{□}{□}\)

Answer: \(\frac{5}{6}\)
Explanation:
Step 1:
Place two \(\frac{1}{3}\) fractions strips under the 1 whole strip on your Mathboard. Then place a \(\frac{1}{6}\) fraction strip beside the two \(\frac{1}{3}\) strips.
Step 2:
Find fraction strips, all with the same denominator, that are equivalent to \(\frac{1}{6}\) and \(\frac{2}{3}\). Place the fraction strips under the sum. At the right, draw a picture of the model and write the equivalent fractions.
\(\frac{2}{3}\) = \(\frac{2}{3}\) • \(\frac{2}{2}\) = \(\frac{4}{6}\)
Step 3:
Add the fractions with like denominators. Use the 1 whole strip to rename the sum in the simplest form.
\(\frac{4}{6}\) + \(\frac{1}{6}\) = \(\frac{5}{6}\)

Question 9.
\(\frac{5}{8}+\frac{1}{4}=\)
\(\frac{□}{□}\)

Answer: \(\frac{7}{8}\)
Explanation:
Step 1:
Place five \(\frac{1}{8}\) fractions strips under the 1 whole strip on your Mathboard. Then place a \(\frac{1}{4}\) fraction strip beside the five \(\frac{1}{8}\) strips.
Step 2:
Find fraction strips, all with the same denominator, that are equivalent to \(\frac{1}{4}\) and \(\frac{5}{8}\). Place the fraction strips under the sum. At the right, draw a picture of the model and write the equivalent fractions.
\(\frac{1}{4}\) • \(\frac{2}{2}\) = \(\frac{2}{8}\)
Step 3:
Add the fractions with like denominators. Use the 1 whole strip to rename the sum in the simplest form.
\(\frac{2}{8}\) + \(\frac{5}{8}\) = \(\frac{7}{8}\)

Question 11.
\(\frac{3}{4}+\frac{1}{6}=\)
\(\frac{□}{□}\)

Answer: \(\frac{11}{12}\)
Explanation:
Find fraction strips, all with the same denominator, that are equivalent to \(\frac{3}{4}\) and \(\frac{1}{6}\). Place the fraction strips under the sum. At the right, draw a picture of the model and write the equivalent fractions.
\(\frac{3}{4}\) • \(\frac{3}{3}\) = \(\frac{9}{12}\)
\(\frac{1}{6}\) • \(\frac{2}{2}\)  = \(\frac{2}{12}\)
Add the fractions with like denominators. Use the 1 whole strip to rename the sum in the simplest form.
\(\frac{9}{12}\) + \(\frac{2}{12}\) = \(\frac{11}{12}\)

Question 12.
\(\frac{1}{2}+\frac{2}{3}=\)
______ \(\frac{□}{□}\)

Answer: 1 \(\frac{1}{6}\)
Explanation:
Find fraction strips, all with the same denominator, that are equivalent to \(\frac{2}{3}\) and \(\frac{1}{2}\). Place the fraction strips under the sum. At the right, draw a picture of the model and write the equivalent fractions.
\(\frac{2}{3}\) • \(\frac{2}{2}\) = \(\frac{4}{6}\)
\(\frac{1}{2}\) • \(\frac{3}{3}\) = \(\frac{3}{6}\)
Add the fractions with like denominators. Use the 1 whole strip to rename the sum in the simplest form.
\(\frac{4}{6}\) + \(\frac{3}{6}\) = \(\frac{7}{6}\)
\(\frac{7}{6}\) is greater than 1.
Convert the fraction into the mixed fraction
\(\frac{7}{6}\) = 1 \(\frac{1}{6}\)

Question 13.
\(\frac{7}{8}+\frac{1}{4}=\)
______ \(\frac{□}{□}\)

Answer: 1 \(\frac{1}{8}\)
Explanation:
Find fraction strips, all with the same denominator, that are equivalent to \(\frac{7}{8}\) and \(\frac{1}{4}\). Place the fraction strips under the sum. At the right, draw a picture of the model and write the equivalent fractions.
\(\frac{1}{4}\) • \(\frac{2}{2}\) = \(\frac{2}{8}\)
Add the fractions with like denominators. Use the 1 whole strip to rename the sum in the simplest form.
\(\frac{7}{8}\) + \(\frac{2}{8}\) = \(\frac{9}{8}\)
Convert \(\frac{9}{8}\) into the mixed fraction.
\(\frac{9}{8}\) = 1 \(\frac{1}{8}\)

Question 14.
Explain how using fraction strips with like denominators makes it possible to add fractions with unlike denominators.
Type below:
_________

Answer: The strips for both fractions need to be the same size. Finding like denominators is done by trying smaller strips so they can all be the same size.

Problem Solving – Page No. 246

Question 15.
Maya makes trail mix by combining \(\frac{1}{3}\) cup of mixed nuts and \(\frac{1}{4}\) cup of dried fruit. What is the total amount of ingredients in her trail mix?
\(\frac{1}{3}+\frac{1}{4}=\frac{7}{12}\)
Maya uses \(\frac{1}{12}\) cup of ingredients.
Write a new problem using different amounts for each ingredient. Each amount should be a fraction with a denominator of 2, 3, or 4. Then use fraction strips to solve your problem.
Pose a problem                          Solve your problem. Draw a picture of the
fraction strips you use to solve the problem.
Explain why you chose the amounts you did for your problem.
Type below:
_________

Answer:
\(\frac{1}{3}+\frac{1}{4}=\frac{7}{12}\)
Maya uses \(\frac{1}{12}\) cup of ingredients.
Maya makes trail mix by combining \(\frac{1}{2}\) cup of mixed nuts and \(\frac{1}{3}\) cup of dried fruit and \(\frac{1}{4}\) cup of chocolate morsels. What is the total amount of ingredients in her trail mix?
\(\frac{1}{2}\) + \(\frac{1}{3}\) + \(\frac{1}{4}\) = x
2 • \(\frac{1}{2}\) +  2 • \(\frac{1}{3}\) +  2 • \(\frac{1}{4}\) =  2 • x
1 + \(\frac{2}{3}\) + \(\frac{1}{2}\) = 2x
Now multiply with 3 on both sides
3 • 1 + 3 • \(\frac{2}{3}\) + 3 • \(\frac{1}{2}\) = 3 • 2x
3 + 2 + \(\frac{3}{2}\) = 6x
6 + 4 + 1 = 12 x
11 = 12x
x = \(\frac{11}{12}\)
\(\frac{1}{2}\) + \(\frac{1}{3}\) + \(\frac{1}{4}\) = \(\frac{11}{12}\)

Share and Show – Page No. 248

Use fraction strips to find the difference. Write your answer in simplest form.

Question 1.

\(\frac{7}{10}-\frac{2}{5}=\)
\(\frac{□}{□}\)

Answer:
\(\frac{7}{10}\) – \(\frac{2}{5}\)
\(\frac{7}{10}\) – \(\frac{2}{5}\) • \(\frac{2}{2}\)
\(\frac{7}{10}\) – \(\frac{4}{10}\) = \(\frac{3}{10}\)

Question 2.

\(\frac{2}{3}-\frac{1}{4}=\)
\(\frac{□}{□}\)

Answer:
\(\frac{2}{3}\) – \(\frac{1}{4}\)
Now we have to make the fractions like denominators
\(\frac{2}{3}\) • \(\frac{4}{4}\) – \(\frac{1}{4}\) • \(\frac{3}{3}\)
\(\frac{8}{12}\) – \(\frac{3}{12}\) = \(\frac{5}{12}\)

Page No. 249

Use fraction strips to find the difference. Write your answer in simplest form.

Question 3.

\(\frac{5}{6}-\frac{1}{4}=\)
\(\frac{□}{□}\)

Answer:
Step 1:
Find fraction strips, all with the same denominator, that fit exactly under the difference \(\frac{5}{6}-\frac{1}{4}\)
Step 2:
Find another set of fraction strips, all with the same the denominator, that fit exactly under the difference \(\frac{5}{6}-\frac{1}{4}\)
Step 3:
Find other fraction strips, all with the same denominator, that fit exactly under the difference \(\frac{5}{6}-\frac{1}{4}\)
\(\frac{5}{6}\) • \(\frac{4}{4}\) – \(\frac{1}{4}\) • \(\frac{6}{6}\)
\(\frac{20}{24}\) – \(\frac{6}{24}\) = \(\frac{14}{24}\) = \(\frac{7}{12}\)
Thus, \(\frac{5}{6}-\frac{1}{4}\) = \(\frac{7}{12}\)

Question 4.

\(\frac{1}{2}-\frac{3}{10}=\)
\(\frac{□}{□}\)

Answer:
\(\frac{1}{2}-\frac{3}{10}\)
\(\frac{1}{2}\) • \(\frac{5}{5}\) – \(\frac{3}{10}\)
\(\frac{5}{10}\) – \(\frac{3}{10}\) = \(\frac{2}{10}\)

Question 5.

\(\frac{3}{8}-\frac{1}{4}=\)
\(\frac{□}{□}\)

Answer:
\(\frac{3}{8}-\frac{1}{4}\)
\(\frac{3}{8}\) – \(\frac{1}{4}\) • \(\frac{2}{2}\)
= \(\frac{3}{8}\) – \(\frac{2}{8}\) = \(\frac{1}{8}\)

Question 6.

\(\frac{2}{3}-\frac{1}{2}=\)
\(\frac{□}{□}\)

Answer:
\(\frac{2}{3}-\frac{1}{2}\)
\(\frac{2}{3}\) • \(\frac{2}{2}\) – \(\frac{1}{2}\) • \(\frac{3}{3}\)
\(\frac{4}{6}-\frac{3}{6}\) = \(\frac{1}{6}\)

Use fraction strips to find the difference. Write your answer in simplest form.

Question 9.
\(\frac{1}{2}-\frac{1}{10}=\) \(\frac{□}{□}\)

Answer:
\(\frac{1}{2}-\frac{1}{10}\)
Make the denominators equal and then subtract the subtract the fraction with lide denominators.
\(\frac{1}{2}\) • \(\frac{5}{5}\) – \(\frac{1}{10}\)
\(\frac{5}{10}\) – \(\frac{1}{10}\) = \(\frac{4}{10}\)

Question 10.
\(\frac{3}{5}-\frac{1}{2}=\) \(\frac{□}{□}\)

Answer:
\(\frac{3}{5}-\frac{1}{2}\)
Make the denominators equal and then subtract the subtract the fraction with lide denominators.
\(\frac{3}{5}\) • \(\frac{2}{2}\) – \(\frac{1}{2}\) • \(\frac{5}{5}\)
\(\frac{6}{10}-\frac{5}{10}\) = \(\frac{1}{10}\)

Question 11.
\(\frac{7}{8}-\frac{1}{4}=\) \(\frac{□}{□}\)

Answer:
\(\frac{7}{8}-\frac{1}{4}\)
Make the denominators equal and then subtract the subtract the fraction with lide denominators.
\(\frac{7}{8}\) – \(\frac{1}{4}\) • \(\frac{2}{2}\)
\(\frac{7}{8}\) – \(\frac{2}{8}\) = \(\frac{5}{8}\)

Question 12.
\(\frac{5}{6}-\frac{2}{3}=\) \(\frac{□}{□}\)

Answer:
\(\frac{5}{6}-\frac{2}{3}\)
Make the denominators equal and then subtract the subtract the fraction with lide denominators.
\(\frac{5}{6}\) – \(\frac{2}{3}\) • \(\frac{2}{2}\)
\(\frac{5}{6}\) – \(\frac{4}{6}\)
\(\frac{1}{6}\)

Question 13.
\(\frac{3}{4}-\frac{1}{3}=\) \(\frac{□}{□}\)

Answer:
\(\frac{3}{4}-\frac{1}{3}\)
\(\frac{3}{4}\) • \(\frac{3}{3}\) – \(\frac{1}{3}\) • \(\frac{4}{4}\)
\(\frac{9}{12}\) – \(\frac{4}{12}\) = \(\frac{5}{12}\)

UNLOCK the Problem – Page No. 250

Question 17.
The picture at the right shows how much pizza was left over from lunch. Jason eats \(\frac{1}{4}\) of the whole pizza for dinner. Which subtraction sentence represents the amount of pizza that is remaining after dinner?

a. What problem are you being asked to solve?
Type below:
_________

Answer: I am asked to solve which subtraction sentence represents the amount of pizza that is remaining after dinner.

Question 17.
b. How will you use the diagram to solve the problem?
Type below:
_________

Answer: I will use number of slices left in the pizza to solve the problem.

Question 17.
c. Jason eats \(\frac{1}{4}\) of the whole pizza. How many slices does he eat?
______ slices

Answer: 2 slices

Explanation:
Given that, Jason eats \(\frac{1}{4}\) of the whole pizza.
The pizza is cut into 8 slices.
So, 8 × \(\frac{1}{4}\) = 2 slices.
Thus Jason ate 2 slices.

Question 17.
d. Redraw the diagram of the pizza. Shade the sections of pizza that are remaining after Jason eats his dinner.
Type below:
_________

Question 17.
e. Write a fraction to represent the amount of pizza that is remaining.
\(\frac{□}{□}\) of a pizza

Answer: \(\frac{3}{8}\) of a pizza

Explanation:
The fraction of pizzz Jason ate = \(\frac{1}{4}\)
Number of slices left = \(\frac{5}{8}\)
Now subtract \(\frac{5}{8}\) – \(\frac{1}{4}\)
= \(\frac{3}{8}\)
Thus the fraction to represent the amount of pizza that is remaining is \(\frac{3}{8}\)

Question 17.
f. Fill in the bubble for the correct answer choice above.
Options:
a. 1 – \(\frac{1}{4}\) = \(\frac{3}{4}\)
b. \(\frac{5}{8}\) – \(\frac{1}{4}\) = \(\frac{3}{8}\)
c. \(\frac{3}{8}\) – \(\frac{1}{4}\) = \(\frac{2}{8}\)
d. 1 – \(\frac{3}{8}\) = \(\frac{5}{8}\)

Answer: B
The fraction of pizzz Jason ate = \(\frac{1}{4}\)
Number of slices left = \(\frac{5}{8}\)
Now subtract \(\frac{5}{8}\) – \(\frac{1}{4}\) = \(\frac{3}{8}\)
Thus the correct answer is option B.

Question 18.
The diagram shows what Tina had left from a yard of fabric. She now uses \(\frac{2}{3}\) yard of fabric for a project. How much of the original yard of fabric does Tina have left after the project?

Options:
a. \(\frac{2}{3}\) yard
b. \(\frac{1}{2}\) yard
c. \(\frac{1}{3}\) yard
d. \(\frac{1}{6}\) yard

Answer: \(\frac{1}{3}\) yard

Explanation:
The original yard of fabric is 6
Tina uses \(\frac{2}{3}\) yard of fabric for a project.
\(\frac{1}{1}\) – \(\frac{2}{3}\)
\(\frac{3}{3}\) – \(\frac{2}{3}\) = \(\frac{1}{3}\) yard

Share and Show – Page No. 253

Estimate the sum or difference.

Question 1.
\(\frac{5}{6}+\frac{3}{8}\)
a. Round \(\frac{5}{6}\) to its closest benchmark. ____
b. Round \(\frac{3}{8}\) to its closest benchmark. ____
c. Add to find the estimate. ____ + ____ = ____
_____ \(\frac{□}{□}\)

Answer:
a. Round \(\frac{5}{6}\) to its closest benchmark. \(\frac{6}{6}\) or 1.
b. Round \(\frac{3}{8}\) to its closest benchmark. \(\frac{4}{8}\) or \(\frac{1}{2}\)
c. Add to find the estimate. ____ + ____ = ____
1 + \(\frac{1}{2}\) = \(\frac{3}{2}\) = 1 \(\frac{1}{2}\)

Question 2.
\(\frac{5}{9}-\frac{3}{8}\)
_____

Answer: 0

Explanation:
Step 1: Place a point at \(\frac{5}{9}\) on the number line.
The fraction is between 0 and \(\frac{1}{2}\).
The fraction rounded to \(\frac{5}{9}\) is \(\frac{1}{2}\)
Step 2: Place a point at \(\frac{3}{8}\) on the number line.
The fraction is between 0 and \(\frac{1}{2}\).
The fraction rounded to \(\frac{3}{8}\) is \(\frac{1}{2}\).
\(\frac{1}{2}\) – \(\frac{1}{2}\) = 0

Question 5.
\(3 \frac{9}{10}-1 \frac{2}{9}\)
_____

Answer: 3

Explanation:

Step 1: Place a point at \(\frac{9}{10}\) on the number line.
The fraction is between \(\frac{1}{2}\) and 1.
Step 2: Place a point at \(\frac{2}{9}\) on the number line.
The fraction is between 0 and \(\frac{1}{2}\).
3 × 1 – 1 × 0 = 3 – 0 = 3
\(3 \frac{9}{10}-1 \frac{2}{9}\) = 3

Question 6.
\(\frac{4}{6}+\frac{1}{9}\)
\(\frac{□}{□}\)

Answer: \(\frac{1}{2}\)

Explanation:

Step 1: Place a point at \(\frac{4}{6}\) on the number line.
The fraction is between \(\frac{1}{2}\) and 1.
Step 2: Place a point at \(\frac{1}{9}\) on the number line.
The fraction is between 0 and \(\frac{1}{2}\).
So, \(\frac{1}{2}\) + 0 = \(\frac{1}{2}\)
\(\frac{4}{6}+\frac{1}{9}\) = \(\frac{1}{2}\)

On Your Own

Estimate the sum or difference.

Question 8.
\(\frac{5}{8}-\frac{1}{5}\)
\(\frac{□}{□}\)

Answer: \(\frac{1}{2}\)

Explanation:
Step 1: Place a point at \(\frac{5}{8}\) on the number line.
The fraction is between \(\frac{1}{2}\) and 1.
Step 2: Place a point at \(\frac{1}{5}\) on the number line.
The fraction is between 0 and \(\frac{1}{2}\).
1 – \(\frac{1}{2}\) = \(\frac{1}{2}\)

Question 9.
\(\frac{1}{6}+\frac{3}{8}\)
\(\frac{□}{□}\)

Answer: \(\frac{1}{2}\)

Explanation:
Step 1: Place a point at \(\frac{1}{6}\) on the number line.
The fraction is between 0 and \(\frac{1}{2}\)
Step 2: Place a point at \(\frac{3}{8}\) on the number line.
The fraction is between 0 and \(\frac{1}{2}\)
0 + \(\frac{1}{2}\) = \(\frac{1}{2}\)

Question 10.
\(\frac{6}{7}-\frac{1}{5}\)
_____

Answer: 1

Explanation:
Step 1: Place a point at \(\frac{6}{7}\) on the number line.
The fraction is between \(\frac{1}{2}\) and 1.
Step 2: Place a point at \(\frac{1}{5}\) on the number line.
The fraction is between 0 and \(\frac{1}{2}\)
1 – 0 = 1
\(\frac{6}{7}-\frac{1}{5}\) = 1

Question 11.
\(\frac{11}{12}+\frac{6}{10}\)
_____ \(\frac{□}{□}\)

Answer: 1 \(\frac{1}{2}\)

Explanation:
Step 1: Place a point at \(\frac{11}{12}\) on the number line.
The fraction is between \(\frac{1}{2}\) and 1
Step 2: Place a point at \(\frac{6}{10}\) on the number line.
The fraction is between \(\frac{1}{2}\) and 1
1 + \(\frac{1}{2}\) = \(\frac{3}{2}\) = 1 \(\frac{1}{2}\)
\(\frac{11}{12}+\frac{6}{10}\) = 1 \(\frac{1}{2}\)

Question 12.
\(\frac{9}{10}-\frac{1}{2}\)
\(\frac{□}{□}\)

Answer: \(\frac{1}{2}\)

Explanation:
Step 1: Place a point at \(\frac{9}{10}\) on the number line.
The fraction is between \(\frac{1}{2}\) and 1.
Step 2: Place a point at \(\frac{1}{2}\) on the number line.
The fraction is between 0 and \(\frac{1}{2}\)
1 – \(\frac{1}{2}\) = \(\frac{1}{2}\)
\(\frac{9}{10}-\frac{1}{2}\) = \(\frac{1}{2}\)

Question 13.
\(\frac{3}{6}+\frac{4}{5}\)
_____ \(\frac{□}{□}\)

Answer: 1 \(\frac{1}{2}\)

Explanation:
Step 1: Place a point at \(\frac{3}{6}\) on the number line.
The fraction is between 0 and \(\frac{1}{2}\)
Step 2: Place a point at \(\frac{4}{5}\) on the number line.
The fraction is between \(\frac{1}{2}\) and 1
\(\frac{1}{2}\) + 1 = \(\frac{3}{2}\) = 1 \(\frac{1}{2}\)
\(\frac{3}{6}+\frac{4}{5}\) = 1 \(\frac{1}{2}\)

Question 14.
\(\frac{5}{6}-\frac{3}{8}\)
\(\frac{□}{□}\)

Answer: \(\frac{1}{2}\)

Explanation:
Step 1: Place a point at \(\frac{5}{6}\) on the number line.
The fraction is between \(\frac{1}{2}\) and 1.
Step 2: Place a point at \(\frac{3}{8}\) on the number line.
The fraction is between 0 and \(\frac{1}{2}\)
1 – \(\frac{1}{2}\) = \(\frac{1}{2}\)
\(\frac{5}{6}-\frac{3}{8}\) = \(\frac{1}{2}\)

Problem Solving – Page No. 254

Question 17.
Lisa and Valerie are picnicking in Trough Creek State Park in Pennsylvania. Lisa has brought a salad that she made with \(\frac{3}{4}\) cup of strawberries, \(\frac{7}{8}\) cup of peaches, and \(\frac{1}{6}\) cup of blueberries. About how many total cups of fruit are in the salad?
_____ cups

Answer: 2 cups

Explanation:
Lisa and Valerie are picnicking in Trough Creek State Park in Pennsylvania.
Lisa has brought a salad that she made with \(\frac{3}{4}\) cup of strawberries, \(\frac{7}{8}\) cup of peaches, and \(\frac{1}{6}\) cup of blueberries.
Step 1: Place \(\frac{3}{4}\) on the number line.
The fraction is between \(\frac{1}{2}\) and 1.
Step 2: Place \(\frac{7}{8}\) on the number line.
The fraction is between \(\frac{1}{2}\) and 1.
Step 3: Place \(\frac{1}{6}\) on the number line.
The fraction is between 0 and \(\frac{1}{2}\).
1 + 1 + 0 = 2
Thus 2 cups of fruit are in the salad.

Question 18.
At Trace State Park in Mississippi, there is a 25-mile mountain bike trail. If Tommy rode \(\frac{1}{2}\) of the trail on Saturday and \(\frac{1}{5}\) of the trail on Sunday, about what fraction of the trail did he ride?
\(\frac{□}{□}\)

Answer: \(\frac{1}{2}\)

Explanation:
At Trace State Park in Mississippi, there is a 25-mile mountain bike trail.
If Tommy rode \(\frac{1}{2}\) of the trail on Saturday and \(\frac{1}{5}\) of the trail on Sunday
Step 1: Place \(\frac{1}{2}\) on the number line.
\(\frac{1}{2}\) lies between 0 and \(\frac{1}{2}\)
Step 2: Place \(\frac{1}{5}\) on the number line.
\(\frac{1}{5}\) 0 and \(\frac{1}{2}\)
The number closer to \(\frac{1}{5}\) is 0
\(\frac{1}{2}\) – 0 = \(\frac{1}{2}\)
The estimated fraction of the trail he ride is \(\frac{1}{2}\)

Question 21.
Test Prep Jake added \(\frac{1}{8}\) cup of sunflower seeds and \(\frac{4}{5}\) cup of banana chips to his sundae. Which is the best estimate of the total amount of toppings Jake added to his sundae?
Options:
a. about \(\frac{1}{2}\) cup
b. about 1 cup
c. about 1 \(\frac{1}{2}\) cups
d. about 2 cups

Answer: about 1 cup

Explanation:
Given, Test Prep Jake added \(\frac{1}{8}\) cup of sunflower seeds and \(\frac{4}{5}\) cup of banana chips to his sundae
Step 1: Place \(\frac{1}{8}\) on the number line.
\(\frac{1}{8}\) lies between 0 and \(\frac{1}{2}\)
Step 2: Place \(\frac{4}{5}\) on the number line.
\(\frac{4}{5}\) lies between \(\frac{1}{2}\) and 1.
0 + 1 = 1
The best estimate of the total amount of toppings Jake added to his sundae is about 1 cup.

Share and Show – Page No. 256

Question 1.
Find a common denominator of \(\frac{1}{6}\) and \(\frac{1}{9}\) . Rewrite the pair of fractions using the common denominator.
• Multiply the denominators.
A common denominator of \(\frac{1}{6}\) and \(\frac{1}{9}\) is ____.
• Rewrite the pair of fractions using the common denominator.
Type below:
_________

Answer:
Common denominator is 18.
\(\frac{1}{6}\) × \(\frac{3}{3}\) = \(\frac{3}{18}\)
\(\frac{1}{9}\) × \(\frac{2}{2}\) = \(\frac{2}{18}\)
The pair of fractions using the common denominator is \(\frac{3}{18}\), \(\frac{2}{18}\)

Use a common denominator to write an equivalent fraction for each fraction.

Question 2.
\(\frac{1}{3}, \frac{1}{5}\)
common denominator: _________
Type below:
_________

Answer: 15

Explanation:
Multiply the denominators of the fraction.
\(\frac{1}{3}\) × \(\frac{1}{5}\) = \(\frac{1}{15}\)
Thus the common denominator is 15.

Question 4.
\(\frac{2}{9}, \frac{1}{15}\)
common denominator: _________
Type below:
_________

Answer: 45

Explanation:
Multiply the denominators
\(\frac{2}{9}\) × \(\frac{1}{15}\)
The least common denominator of 15 and 9 is 45.
So, the common denominator of \(\frac{2}{9}, \frac{1}{15}\) is 45.

Page No. 257

Use the least common denominator to write an equivalent fraction for each fraction.

Question 5.
\(\frac{1}{4}, \frac{3}{8}\)
least common denominator: ______
Type below:
_________

Answer: 8

Explanation:

First multiply the denominators of the fractions \(\frac{1}{4}, \frac{3}{8}\)
4 × 8 = 32
The least common denominator is 8
The equivalent fractions with LCD
\(\frac{1}{4}\) = \(\frac{2}{8}\)
\(\frac{3}{8}\) = \(\frac{3}{8}\)

Question 7.
\(\frac{4}{5}, \frac{1}{6}\)
least common denominator: ______
Type below:
_________

Answer: 30

Explanation:
First, multiply the denominators of the fractions.
5 × 6 = 30
The least common denominator (LCD) = 30
\(\frac{4}{5}\) × \(\frac{6}{6}\)= \(\frac{24}{30}\)
\(\frac{1}{6}\) × \(\frac{5}{5}\) = \(\frac{5}{30}\)

On Your Own

Use a common denominator to write an equivalent fraction for each fraction.

Question 8.
\(\frac{3}{5}, \frac{1}{4}\)
common denominator: ______
Type below:
_________

Answer: 20

Explanation:
Multiply the denominators of the fractions to find the common denominator.
5 × 4 = 20
So, the common denominator of \(\frac{3}{5}, \frac{1}{4}\) is 20.

Question 9.
\(\frac{5}{8}, \frac{1}{5}\)
common denominator: ______
Type below:
_________

Answer: 40

Explanation:
Multiply the denominators of the fractions to find the common denominator.
8 × 5 = 40
So, the common denominator of \(\frac{5}{8}, \frac{1}{5}\) is 40.

Practice: Copy and Solve Use the least common denominator to write an equivalent fraction for each fraction.

Question 11.
\(\frac{1}{6}, \frac{4}{9}\)
Type below:
_________

Answer: \(\frac{3}{18}, \frac{8}{18}\)

Explanation:
Multiply the denominators of the fractions.
The Least Common Denominator = 18
Now rewrite the fractions
\(\frac{1}{6}\) × \(\frac{3}{3}\) = \(\frac{3}{18}\)
\(\frac{4}{9}\) × \(\frac{2}{2}\) = \(\frac{8}{18}\)

Question 12.
\(\frac{7}{9}, \frac{8}{27}\)
Type below:
_________

Answer: \(\frac{21}{27}, \frac{8}{27}\)

Explanation:
Multiply the denominators of the fractions.
The Least Common Denominator = 27
Now rewrite the fractions
\(\frac{7}{9}\) × \(\frac{3}{3}\) = \(\frac{21}{27}\)
\(\frac{8}{27}\) × \(\frac{1}{1}\) = \(\frac{8}{27}\)

Question 13.
\(\frac{7}{10}, \frac{3}{8}\)
Type below:
_________

Answer: \(\frac{28}{40}, \frac{15}{40}\)

Explanation:
Multiply the denominators of the fractions.
The Least Common Denominator = 40
Now rewrite the fractions
\(\frac{7}{10}\) × \(\frac{4}{4}\) = \(\frac{28}{40}\)
\(\frac{3}{8}\) × \(\frac{5}{5}\) = \(\frac{15}{40}\)

Question 14.
\(\frac{1}{3}, \frac{5}{11}\)
Type below:
_________

Answer: \(\frac{11}{33}, \frac{15}{33}\)

Explanation:
Multiply the denominators of the fractions.
The Least Common Denominator = 33
Now rewrite the fractions
\(\frac{1}{3}\) × \(\frac{11}{11}\) = \(\frac{11}{33}\)
\(\frac{5}{11}\) × \(\frac{3}{3}\) = \(\frac{15}{33}\)

Question 16.
\(\frac{1}{6}, \frac{4}{21}\)
Type below:
_________

Answer: \(\frac{7}{42}, \frac{8}{42}\)

Explanation:
Multiply the denominators of the fractions.
The Least Common Denominator = 42
Now rewrite the fractions
\(\frac{1}{6}\) × \(\frac{7}{7}\) = \(\frac{7}{42}\)
\(\frac{4}{21}\) × \(\frac{2}{2}\) = \(\frac{8}{42}\)

Question 17.
\(\frac{5}{14}, \frac{8}{42}\)
Type below:
_________

Answer: \(\frac{15}{42}, \frac{8}{42}\)

Explanation:
Multiply the denominators of the fractions.
The Least Common Denominator = 42
Now rewrite the fractions
\(\frac{5}{14}\) × \(\frac{3}{3}\) = \(\frac{15}{42}\)
\(\frac{8}{42}\) × \(\frac{1}{1}\) = \(\frac{8}{42}\)

Question 18.
\(\frac{7}{12}, \frac{5}{18}\)
Type below:
_________

Answer: \(\frac{21}{36}, \frac{10}{36}\)

Explanation:
Multiply the denominators of the fractions.
The Least Common Denominator = 36
Now rewrite the fractions
\(\frac{7}{12}\) × \(\frac{3}{3}\) = \(\frac{21}{36}\)
\(\frac{5}{18}\) × \(\frac{2}{2}\) = \(\frac{10}{36}\)

Algebra Write the unknown number for each ■.

Question 19.
\(\frac{1}{5}, \frac{1}{8}\)
least common denominator: ■
■ = ______

Answer: 40

Explanation:
Multiply the denominators of the fractions.
5 × 8 = 40
Therefore, ■ = 40

Question 20.
\(\frac{2}{5}, \frac{1}{■}\)
least common denominator: 15
■ = ______

Answer: 3

Explanation:
Multiply the denominators of the fractions.
5 × ■ = 15
■ = 15/5 = 3
Thus ■ = 3

Question 21.
\(\frac{3}{■}, \frac{5}{6}\)
least common denominator: 42
■ = ______

Answer: 7

Explanation:
\(\frac{3}{■}, \frac{5}{6}\)
■ × 6 = 42
■ = 42/6
■ = 7

UNLOCK the Problem – Page No. 258

Question 22.
Katie made two pies for the bake sale. One was cut into three equal slices and the other into 5 equal slices. She will continue to cut the pies so each one has the same number of equal-sized slices. What is the least number of equal-sized slices each pie could have?
a. What information are you given?
Type below:
_________

Answer: I have the information about the two pies for the bake sale. One was cut into three equal slices and the other into 5 equal slices. She will continue to cut the pies so each one has the same number of equal-sized slices.

Question 22.
b. What problem are you being asked to solve?
Type below:
_________

Answer: I am asked to solve the least number of equal-sized slices each pie could have.

Question 22.
c. When Katie cuts the pies more, can she cut each pie the same number of times and have all the slices the same size? Explain.
Type below:
_________

Answer: Yes she can cut into more equal pieces. Katie can cut the pie into 6 equal pieces and 10 equal pieces. But the least number of equal-sized slices each pie could have is 3 and 5.

Question 22.
d. Use the diagram to show the steps you use to solve the problem.

Type below:
_________

Answer:
There are 2 pies. One pie is cut into 3 equal pieces and the second pie is cut into 5 equal pieces.
So, there are 15 pieces of pies.

Question 22.
e. Complete the sentences.
The least common denominator of \(\frac{1}{3}\) and \(\frac{1}{5}\) is ____.
Katie can cut each piece of the first pie into ____ and each piece of the second pie into ____ .
That means that Katie can cut each pie into pieces that are ____ of the whole pie.
Type below:
_________

Answer:
The least common denominator of \(\frac{1}{3}\) and \(\frac{1}{5}\) is 15
5 × 3 = 15
Katie can cut each piece of the first pie into three and each piece of the second pie into five.
That means that Katie can cut each pie into pieces that are 15 of the whole pie.

Question 23.
A cookie recipe calls for \(\frac{1}{3}\) cup of brown sugar and \(\frac{1}{8}\) cup of walnuts. Find the least common denominator of the fractions used in the recipe.
____

Answer: 24

Explanation:

A cookie recipe calls for \(\frac{1}{3}\) cup of brown sugar and \(\frac{1}{8}\) cup of walnuts.
We can calculate the LCD by multiplying the denominators of the fraction.
3 × 8 = 24.

Question 24.
Test Prep Which fractions use the least common denominator and are equivalent to \(\frac{5}{8}\) and \(\frac{7}{10}\) ?
Options:
a. \(\frac{10}{40} \text { and } \frac{14}{40}\)
b. \(\frac{25}{40} \text { and } \frac{28}{40}\)
c. \(\frac{25}{80} \text { and } \frac{21}{80}\)
d. \(\frac{50}{80} \text { and } \frac{56}{80}\)

Answer: \(\frac{50}{80} \text { and } \frac{56}{80}\)

Explanation:
The least common denominator of \(\frac{5}{8}\) and \(\frac{7}{10}\) is 80.
\(\frac{5}{8}\) × \(\frac{10}{10}\) and \(\frac{7}{10}\) × \(\frac{8}{8}\)
= \(\frac{50}{80} \text { and } \frac{56}{80}\)
Thus the correct answer is option D.

Share and Show – Page No. 260

Find the sum or difference. Write your answer in simplest form.

Question 1.
\(\frac{5}{12}+\frac{1}{3}\)
\(\frac{□}{□}\)

Answer:
Find a common denominator by multiplying the denominators.
\(\frac{5}{12}+\frac{1}{3}\)
\(\frac{5}{12}\) + \(\frac{1}{3}\) × \(\frac{4}{4}\)
\(\frac{5}{12}\) + \(\frac{4}{12}\)
\(\frac{9}{12}\)

Question 2.
\(\frac{2}{5}+\frac{3}{7}\)
\(\frac{□}{□}\)

Answer:
Find a common denominator by multiplying the denominators.
Use the common denominator to write equivalent fractions with like denominators. Then add, and write your answer in simplest form.
\(\frac{2}{5}+\frac{3}{7}\)
\(\frac{2}{5}\) × \(\frac{7}{7}\) + \(\frac{3}{7}\) × \(\frac{5}{5}\)
\(\frac{14}{35}+\frac{15}{35}\)
= \(\frac{29}{35}\)
\(\frac{2}{5}+\frac{3}{7}\) = \(\frac{29}{35}\)

Question 5.
\(\frac{1}{4}-\frac{1}{7}\)
\(\frac{□}{□}\)

Answer:
First, find a common denominator by multiplying the denominators.
Use the common denominator to write equivalent fractions with like denominators. Then add, and write your answer in simplest form.
\(\frac{1}{4}-\frac{1}{7}\)
\(\frac{1}{4}\) × \(\frac{7}{7}\) – \(\frac{1}{7}\) × \(\frac{4}{4}\)
\(\frac{7}{28}\) – \(\frac{4}{28}\) = \(\frac{3}{28}\)
\(\frac{1}{4}-\frac{1}{7}\) = \(\frac{3}{28}\)

Question 6.
\(\frac{9}{10}-\frac{1}{4}\)
\(\frac{□}{□}\)

Answer:
First, find a common denominator by multiplying the denominators.
Use the common denominator to write equivalent fractions with like denominators. Then add, and write your answer in simplest form.
\(\frac{9}{10}-\frac{1}{4}\)
\(\frac{9}{10}\) × \(\frac{4}{4}\) – \(\frac{1}{4}\) × \(\frac{10}{10}\)
\(\frac{36}{40}\) – \(\frac{10}{40}\) = \(\frac{26}{40}\)
\(\frac{9}{10}-\frac{1}{4}\) = \(\frac{26}{40}\)

On Your Own – Page No. 261

Find the sum or difference. Write your answer in simplest form.

Question 7.
\(\frac{3}{8}+\frac{1}{4}\)
\(\frac{□}{□}\)

Answer: \(\frac{5}{8}\)

Explanation:
\(\frac{3}{8}+\frac{1}{4}\) = \(\frac{3}{8}\) + \(\frac{1}{4}\)
LCD = 8
\(\frac{3}{8}\) + \(\frac{1}{4}\) × \(\frac{2}{2}\)
\(\frac{3}{8}\) + \(\frac{2}{8}\) = \(\frac{5}{8}\)
Thus \(\frac{3}{8}+\frac{1}{4}\) = \(\frac{5}{8}\)

Question 8.
\(\frac{7}{8}+\frac{1}{10}\)
\(\frac{□}{□}\)

Answer:
\(\frac{7}{8}+\frac{1}{10}\)
First, find the Least Common Denominator and rewrite the fractions with the common denominator.
LCD = 40
\(\frac{7}{8}\) × \(\frac{5}{5}\) + \(\frac{1}{10}\) × \(\frac{4}{4}\)
\(\frac{35}{40}\) + \(\frac{4}{40}\) = \(\frac{39}{40}\)
\(\frac{7}{8}+\frac{1}{10}\) = \(\frac{39}{40}\)

Question 9.
\(\frac{2}{7}+\frac{3}{10}\)
\(\frac{□}{□}\)

Answer:
\(\frac{2}{7}+\frac{3}{10}\)
First, find the Least Common Denominator and rewrite the fractions with the common denominator.
LCD = 70
\(\frac{2}{7}\) × \(\frac{10}{10}\) + \(\frac{3}{10}\) × \(\frac{7}{7}\)
\(\frac{20}{70}\) + \(\frac{21}{70}\) = \(\frac{41}{70}\)
\(\frac{2}{7}+\frac{3}{10}\) = \(\frac{41}{70}\)

Question 12.
\(\frac{7}{16}+\frac{1}{4}\)
\(\frac{□}{□}\)

Answer:
\(\frac{7}{16}+\frac{1}{4}\)
First, find the Least Common Denominator and rewrite the fractions with the common denominator.
LCD = 16
\(\frac{7}{16}\) + \(\frac{1}{4}\) = \(\frac{7}{16}\) + \(\frac{1}{4}\) × \(\frac{4}{4}\)
\(\frac{7}{16}\) + \(\frac{4}{16}\) = \(\frac{11}{16}\)

Question 13.
\(\frac{5}{6}+\frac{3}{8}\)
\(\frac{□}{□}\)

Answer:
\(\frac{5}{6}+\frac{3}{8}\)
First, find the Least Common Denominator and rewrite the fractions with the common denominator.
\(\frac{5}{6}\) + \(\frac{3}{8}\)
LCD = 24
\(\frac{5}{6}\) × \(\frac{4}{4}\) + \(\frac{3}{8}\) × \(\frac{3}{3}\)
= \(\frac{20}{24}\) + \(\frac{9}{24}\) = \(\frac{29}{24}\)
\(\frac{5}{6}+\frac{3}{8}\) = \(\frac{29}{24}\)

Question 14.
\(\frac{3}{4}+\frac{1}{2}\)
\(\frac{□}{□}\)

Answer:
\(\frac{3}{4}+\frac{1}{2}\)
First, find the Least Common Denominator and rewrite the fractions with the common denominator.
\(\frac{3}{4}\) + \(\frac{1}{2}\)
LCD = 4
\(\frac{3}{4}\) + \(\frac{1}{2}\) × \(\frac{2}{2}\)
= \(\frac{3}{4}\) + \(\frac{2}{4}\) = \(\frac{5}{4}\)
The miced fractiion of \(\frac{5}{4}\) is 1 \(\frac{1}{4}\)

Question 15.
\(\frac{5}{12}+\frac{1}{4}\)
\(\frac{□}{□}\)

Answer:
\(\frac{5}{12}+\frac{1}{4}\)
First, find the Least Common Denominator and rewrite the fractions with the common denominator.
\(\frac{5}{12}\) + \(\frac{1}{4}\)
LCD = 12
\(\frac{5}{12}\) + \(\frac{1}{4}\) × \(\frac{3}{3}\)
\(\frac{5}{12}\) + \(\frac{3}{12}\) = \(\frac{8}{12}\) = \(\frac{2}{3}\)

Practice: Copy and Solve Find the sum or difference. Write your answer in simplest form.

Question 16.
\(\frac{1}{3}+\frac{4}{18}\)
\(\frac{□}{□}\)

Answer:
\(\frac{1}{3}+\frac{4}{18}\)
First, find the Least Common Denominator and rewrite the fractions with the common denominator.
LCD = 18
\(\frac{1}{3}\) + \(\frac{4}{18}\)
\(\frac{1}{3}\) × \(\frac{6}{6}\) + \(\frac{4}{18}\)
\(\frac{6}{18}\) + \(\frac{4}{18}\) = \(\frac{10}{18}\) = \(\frac{5}{9}\)
\(\frac{1}{3}+\frac{4}{18}\) = \(\frac{5}{9}\)

Question 17.
\(\frac{3}{5}+\frac{1}{3}\)
\(\frac{□}{□}\)

Answer:
\(\frac{3}{5}+\frac{1}{3}\)
First, find the Least Common Denominator and rewrite the fractions with the common denominator.
LCD = 15
\(\frac{3}{5}\) + \(\frac{1}{3}\)
\(\frac{3}{5}\) × \(\frac{3}{3}\) + \(\frac{1}{3}\) × \(\frac{5}{5}\)
\(\frac{9}{15}\) + \(\frac{5}{15}\) = \(\frac{14}{15}\)
\(\frac{3}{5}+\frac{1}{3}\) = \(\frac{14}{15}\)

Question 18.
\(\frac{3}{10}+\frac{1}{6}\)
\(\frac{□}{□}\)

Answer:
\(\frac{3}{10}+\frac{1}{6}\)
First, find the Least Common Denominator and rewrite the fractions with the common denominator.
LCD = 30
\(\frac{3}{10}\) + \(\frac{1}{6}\)
\(\frac{3}{10}\) × \(\frac{3}{3}\) + \(\frac{1}{6}\) × \(\frac{5}{5}\)
\(\frac{9}{30}\) + \(\frac{5}{30}\) = \(\frac{14}{30}\)
\(\frac{3}{10}+\frac{1}{6}\) = \(\frac{14}{30}\)

Question 19.
\(\frac{1}{2}+\frac{4}{9}\)
\(\frac{□}{□}\)

Answer:
\(\frac{1}{2}+\frac{4}{9}\)
First, find the Least Common Denominator and rewrite the fractions with the common denominator.
LCD = 18
\(\frac{1}{2}\) + \(\frac{4}{9}\)
\(\frac{1}{2}\) × \(\frac{9}{9}\) + \(\frac{4}{9}\) × \(\frac{2}{2}\)
= \(\frac{9}{18}\) + \(\frac{8}{18}\) = \(\frac{17}{18}\)
\(\frac{1}{2}+\frac{4}{9}\) = \(\frac{17}{18}\)

Question 20.
\(\frac{1}{2}-\frac{3}{8}\)
\(\frac{□}{□}\)

Answer:
\(\frac{1}{2}-\frac{3}{8}\)
First, find the Least Common Denominator and rewrite the fractions with the common denominator.
LCD = 8
\(\frac{1}{2}\) – \(\frac{3}{8}\)
\(\frac{1}{2}\) × \(\frac{4}{4}\) – \(\frac{3}{8}\)
\(\frac{4}{8}\) – \(\frac{3}{8}\) = \(\frac{1}{8}\)
\(\frac{1}{2}-\frac{3}{8}\) = \(\frac{1}{8}\)

Question 23.
\(\frac{11}{12}-\frac{7}{15}\)
\(\frac{□}{□}\)

Answer:
\(\frac{11}{12}-\frac{7}{15}\)
First, find the Least Common Denominator and rewrite the fractions with the common denominator.
LCD = 60
\(\frac{11}{12}\) – \(\frac{7}{15}\)
\(\frac{11}{12}\) × \(\frac{5}{5}\) – \(\frac{7}{15}\) × \(\frac{4}{4}\)
\(\frac{55}{60}\) – \(\frac{28}{60}\) = \(\frac{27}{60}\)
\(\frac{11}{12}-\frac{7}{15}\) = \(\frac{27}{60}\) = \(\frac{9}{20}\)

Algebra Find the unknown number.

Question 24.
\(\frac{9}{10}\) − ■ = \(\frac{1}{5}\)
■ = \(\frac{□}{□}\)

Answer:
\(\frac{9}{10}\) – \(\frac{1}{5}\) = ■
■ = \(\frac{9}{10}\) – \(\frac{1}{5}\)
■ = \(\frac{9}{10}\) – \(\frac{2}{10}\) = \(\frac{7}{10}\)
■ = \(\frac{7}{10}\)

Question 25.
\(\frac{5}{12}\) + ■ = \(\frac{1}{2}\)
■ = \(\frac{□}{□}\)

Answer:
\(\frac{5}{12}\) + ■ = \(\frac{1}{2}\)
\(\frac{5}{12}\) − \(\frac{1}{2}\) = – ■
– ■ = \(\frac{5}{12}\) − \(\frac{1}{2}\)
– ■ = \(\frac{5}{12}\) − \(\frac{1}{2}\) × \(\frac{6}{6}\)
– ■ = \(\frac{5}{12}\) − \(\frac{6}{12}\) = – \(\frac{1}{12}\)
■ = \(\frac{1}{12}\)

Problem Solving – Page No. 262

Use the picture for 26–27.

Question 26.
Sara is making a key chain using the bead design shown. What fraction of the beads in her design are either blue or red?
\(\frac{□}{□}\)

Answer: \(\frac{11}{15}\)

Explanation:
Total number of red beads = 6
Total number of blue beads = 5
Total number of beads = 6 + 5 = 11
The fraction of beads = \(\frac{11}{15}\)

Question 27.
In making the key chain, Sara uses the pattern of beads 3 times. After the key chain is complete, what fraction of the beads in the key chain are either white or blue?
______ \(\frac{□}{□}\)

Answer: 1 \(\frac{4}{5}\)

Explanation:
In making the key chain, Sara uses the pattern of beads 3 times.
Given that Sara uses the pattern of beads 3 times.
Total number of blue beads = 5
5 × 3 = 15
Number of white beads = 4
4 × 3 = 12
15 + 12 = 27
Actual number of beads = 15
So, the fraction is \(\frac{27}{15}\) = \(\frac{9}{5}\)
The mixed fraction of \(\frac{9}{5}\) is 1 \(\frac{4}{5}\)

Question 28.
Jamie had \(\frac{4}{5}\) of a spool of twine. He then used \(\frac{1}{2}\) of a spool of twine to make friendship knots. He claims to have \(\frac{3}{10}\) of the original spool of twine left over. Explain how you know whether Jamie’s claim is reasonable.
Type below:
_________

Answer: Jamie’s claim is reasonable

Explanation:
Jamie had \(\frac{4}{5}\) of a spool of twine. He then used \(\frac{1}{2}\) of a spool of twine to make friendship knots. He claims to have \(\frac{3}{10}\) of the original spool of twine left over.
To know whether his estimation is reasonable or not we have to subtract the total spool of twine from used spool of twine.
\(\frac{4}{5}\) – \(\frac{1}{2}\)
LCD = 10
\(\frac{4}{5}\) × \(\frac{2}{2}\)  – \(\frac{1}{2}\) × \(\frac{5}{5}\)
\(\frac{8}{10}\) – \(\frac{5}{10}\) = \(\frac{3}{10}\)
By this is can that Jamie’s claim is reasonable.

Question 29.
Does Test Prep Which equation represent the fraction of beads that are green or yellow?

Options:
a. \(\frac{1}{4}+\frac{1}{8}=\frac{3}{8}\)
b. [atex]\frac{1}{2}+\frac{1}{4}=\frac{3}{4}[/latex]
c. \(\frac{1}{2}+\frac{1}{8}=\frac{5}{8}\)
d. \(\frac{3}{4}+\frac{2}{8}=1\)

Answer: [atex]\frac{1}{2}+\frac{1}{4}=\frac{3}{4}[/latex]

Explanation:
Number of green beads = 4 = [atex]\frac{1}{2}[/latex]
Number of blue beads = 3 = [atex]\frac{3}{4}[/latex]
Number of yellow beads = 1 [atex]\frac{1}{4}[/latex]
The fraction of beads that are green or yellow is [atex]\frac{1}{2}+\frac{1}{4}=\frac{3}{4}[/latex]
The correct answer is option B.

Mid-Chapter Checkpoint – Vocabulary – Page No. 263

Choose the best term from the box.

Question 1.
A ________ is a number that is a multiple of two or more numbers.
________

Answer: Common Multiple
A Common Multiple is a number that is a multiple of two or more numbers.

Question 2.
A ________ is a common multiple of two or more denominators.
________

Answer: Common denominator
A Common denominator is a common multiple of two or more denominators.

Concepts and Skills

Estimate the sum or difference.

Question 3.
\(\frac{8}{9}+\frac{4}{7}\)
about ______ \(\frac{□}{□}\)

Answer: 1 \(\frac{1}{2}\)

Place \(\frac{8}{9}\) on the number line.
\(\frac{8}{9}\) lies between \(\frac{1}{2}\) and 1.
\(\frac{8}{9}\) is closer to 1.
Place \(\frac{4}{7}\) on the number line.
\(\frac{4}{7}\) lies between \(\frac{1}{2}\) and 1.
\(\frac{4}{7}\) is closer to \(\frac{1}{2}\).
1 + \(\frac{1}{2}\) = 1 \(\frac{1}{2}\)

Use a common denominator to write an equivalent fraction for each fraction.

Question 6.
\(\frac{1}{6}, \frac{1}{9}\)
common denominator:
Type below:
__________

Answer: 54
Multiply the denominators
6 × 9 = 54
Thus the common denominator of \(\frac{1}{6}, \frac{1}{9}\) is 54

Question 7.
\(\frac{3}{8}, \frac{3}{10}\)
common denominator:
Type below:
__________

Answer: 80
Multiply the denominators
8 × 10 = 80
The common denominator of \(\frac{3}{8}, \frac{3}{10}\) is 80

Question 8.
\(\frac{1}{9}, \frac{5}{12}\)
common denominator:
Type below:
__________

Answer: 36
Multiply the denominators
9 × 12 = 108
The common denominator of \(\frac{1}{9}, \frac{5}{12}\) is 108

Use the least common denominator to write an equivalent fraction for each fraction.

Question 9.
\(\frac{2}{5}, \frac{1}{10}\)
least common denominator: ______
Explain:
__________

Answer: 10

Explanation:
Multiply the denominators
5 × 10 = 50
The least common denominators of \(\frac{2}{5}, \frac{1}{10}\) is 10.

Question 11.
\(\frac{1}{3}, \frac{2}{7}\)
least common denominator: ______
Explain:
__________

Answer: 21

Explanation:
Multiply the denominators
The least common denominator of 3 and 7 is 21.
Thus the LCD of \(\frac{1}{3}, \frac{2}{7}\) is 21.

Find the sum or difference. Write your answer in simplest form.

Question 12.
\(\frac{11}{18}-\frac{1}{6}\)
\(\frac{□}{□}\)

Answer: \(\frac{8}{18}\)

Explanation:
Make the fractions like denominators.
\(\frac{11}{18}\) – \(\frac{1}{6}\)
\(\frac{1}{6}\) × \(\frac{3}{3}\) = \(\frac{3}{18}\)
\(\frac{11}{18}\) – \(\frac{3}{18}\) = \(\frac{8}{18}\)

Question 13.
\(\frac{2}{7}+\frac{2}{5}\)
\(\frac{□}{□}\)

Answer: \(\frac{24}{35}\)

Explanation:
Make the fractions like denominators.
\(\frac{2}{7}\) × \(\frac{5}{5}\) = \(\frac{10}{35}\)
\(\frac{2}{5}\) × \(\frac{7}{7}\) = \(\frac{14}{35}\)
\(\frac{10}{35}\) + \(\frac{14}{35}\) = \(\frac{24}{35}\)
Thus \(\frac{2}{7}+\frac{2}{5}\) = \(\frac{24}{35}\)

Question 14.
\(\frac{3}{4}-\frac{3}{10}\)
\(\frac{□}{□}\)

Answer: \(\frac{18}{40}\)

Explanation:
Make the fractions like denominators.
\(\frac{3}{4}\) × \(\frac{10}{10}\) = \(\frac{30}{40}\)
\(\frac{3}{10}\) × \(\frac{4}{4}\) = \(\frac{12}{40}\)
\(\frac{30}{40}\) – \(\frac{12}{40}\) = \(\frac{18}{40}\)

Mid-Chapter Checkpoint – Page No. 264

Question 15.
Mrs. Vargas bakes a pie for her book club meeting. The shaded part of the diagram below shows the amount of pie left after the meeting. That evening, Mr. Vargas eats \(\frac{1}{4}\) of the whole pie. What fraction represents the amount of pie remaining?

\(\frac{□}{□}\)

Answer: \(\frac{1}{4}\)

Explanation:
Mrs. Vargas bakes a pie for her book club meeting. The shaded part of the diagram below shows the amount of pie left after the meeting.
So, the fraction of the pie is \(\frac{1}{2}\)
That evening, Mr. Vargas eats \(\frac{1}{4}\) of the whole pie.
\(\frac{1}{2}\) – \(\frac{1}{4}\) = \(\frac{1}{4}\)
Thus the fraction represents the amount of pie remaining is \(\frac{1}{4}\)

Question 16.
Keisha makes a large sandwich for a family picnic. She takes \(\frac{1}{2}\) of the sandwich to the picnic. At the picnic, her family eats \(\frac{3}{8}\) of the whole sandwich. What fraction of the whole sandwich does Keisha bring back from the picnic?
\(\frac{□}{□}\)

Answer: \(\frac{1}{8}\)

Explanation:
Keisha makes a large sandwich for a family picnic. She takes \(\frac{1}{2}\) of the sandwich to the picnic.
At the picnic, her family eats \(\frac{3}{8}\) of the whole sandwich.
\(\frac{1}{2}\) – \(\frac{3}{8}\)
\(\frac{1}{2}\) × \(\frac{4}{4}\) – \(\frac{3}{8}\)
\(\frac{4}{8}\) – \(\frac{3}{8}\) = \(\frac{1}{8}\)
Thus Keisha brought \(\frac{1}{8}\) of the sandwich from the picnic.

Share and Show – Page No. 266

Question 1.
Use a common denominator to write equivalent fractions with like denominators and then find the sum. Write your answer in the simplest form.
7 \(\frac{2}{5}\) = ■
+ 4 \(\frac{3}{4}\) = + ■
—————————
■
_____ \(\frac{□}{□}\)

Answer: 12 \(\frac{3}{20}\)

Explanation:
First convert the mixed fraction to proper fraction.
7 \(\frac{2}{5}\) = \(\frac{37}{5}\)
4 \(\frac{3}{4}\) = \(\frac{19}{4}\)
\(\frac{37}{5}\) + \(\frac{19}{4}\)
= \(\frac{37}{5}\) × \(\frac{4}{4}\) = \(\frac{148}{20}\)
\(\frac{19}{4}\) × \(\frac{5}{5}\) = \(\frac{95}{20}\)
\(\frac{148}{20}\) + \(\frac{95}{20}\) = \(\frac{243}{20}\)
Now convert it into mixed fraction = 12 \(\frac{3}{20}\)

Find the sum. Write your answer in simplest form.

Question 2.
\(2 \frac{3}{4}+3 \frac{3}{10}\)
_____ \(\frac{□}{□}\)

Answer: 6 \(\frac{1}{20}\)

Explanation:
First convert the mixed fraction to proper fraction.
\(2 \frac{3}{4}\) = \(\frac{11}{4}\)
3 \(\frac{3}{10}\) = \(\frac{33}{10}\)
Now make the common denominators of the above fractions.
\(\frac{11}{4}\) × \(\frac{10}{10}\) = \(\frac{110}{40}\)
\(\frac{33}{10}\) × \(\frac{4}{4}\) = \(\frac{132}{40}\) = \(\frac{121}{20}\)
Now convert the fraction into mixed fraction.
\(\frac{121}{20}\) = 6 \(\frac{1}{20}\)

Page No. 267

Find the difference. Write your answer in simplest form.

Question 5.
\(9 \frac{5}{6}-2 \frac{1}{3}\)
_____ \(\frac{□}{□}\)

Answer: 7 \(\frac{1}{2}\)

Explanation:
\(9 \frac{5}{6}-2 \frac{1}{3}\) = \(\frac{59}{6}\) – \(\frac{14}{6}\)
= \(\frac{45}{6}\) = \(\frac{15}{2}\) = 7 \(\frac{1}{2}\)

Question 6.
\(10 \frac{5}{9}-9 \frac{1}{6}\)
_____ \(\frac{□}{□}\)

Answer: 1 \(\frac{7}{18}\)

Explanation:
\(10 \frac{5}{9}-9 \frac{1}{6}\) = \(\frac{95}{9}\) – \(\frac{55}{6}\)
= \(\frac{190}{18}\) – \(\frac{165}{18}\) = \(\frac{25}{18}\)
= 1 \(\frac{7}{18}\)
\(10 \frac{5}{9}-9 \frac{1}{6}\) = 1 \(\frac{7}{18}\)

Question 7.
\(7 \frac{2}{3}-3 \frac{1}{6}\)
_____ \(\frac{□}{□}\)

Answer: 4 \(\frac{1}{2}\)

Explanation:
\(7 \frac{2}{3}-3 \frac{1}{6}\)
\(\frac{23}{3}\) – \(\frac{19}{6}\) = \(\frac{46}{6}\) – \(\frac{19}{6}\)
= \(\frac{27}{6}\) = 4 \(\frac{1}{2}\)
\(7 \frac{2}{3}-3 \frac{1}{6}\) = 4 \(\frac{1}{2}\)

On Your Own

Find the sum or difference. Write your answer in simplest form.

Question 10.
\(2 \frac{1}{2}+2 \frac{1}{3}\)
_____ \(\frac{□}{□}\)

Answer: 4 \(\frac{5}{6}\)

Explanation:
\(2 \frac{1}{2}+2 \frac{1}{3}\) = \(\frac{5}{2}\) + \(\frac{7}{3}\)
\(\frac{15}{6}\) + \(\frac{14}{6}\)= \(\frac{29}{6}\)
The mixed fraction of \(\frac{29}{6}\) is 4 \(\frac{5}{6}\)

Question 11.
\(5 \frac{1}{4}+9 \frac{1}{3}\)
_____ \(\frac{□}{□}\)

Answer: 14 \(\frac{7}{12}\)

Explanation:
\(5 \frac{1}{4}+9 \frac{1}{3}\) = \(\frac{21}{4}\) + \(\frac{28}{3}\)
\(\frac{63}{12}\) + \(\frac{112}{12}\) = \(\frac{175}{12}\)
The mixed fraction of \(\frac{175}{12}\) is 14 \(\frac{7}{12}\)

Question 12.
\(8 \frac{1}{6}+7 \frac{3}{8}\)
_____ \(\frac{□}{□}\)

Answer: 15 \(\frac{13}{24}\)

Explanation:
\(8 \frac{1}{6}+7 \frac{3}{8}\) = \(\frac{49}{6}\) + \(\frac{59}{8}\)
\(\frac{196}{24}\) + \(\frac{177}{24}\) = \(\frac{373}{24}\)
The mixed fraction of \(\frac{373}{24}\) is 15 \(\frac{13}{24}\)

Question 13.
\(14 \frac{7}{12}-5 \frac{1}{4}\)
_____ \(\frac{□}{□}\)

Answer: 9 \(\frac{1}{3}\)

Explanation:
\(14 \frac{7}{12}-5 \frac{1}{4}\) = \(\frac{175}{12}\) – \(\frac{21}{4}\)
\(\frac{175}{12}\) – \(\frac{63}{12}\) = \(\frac{112}{12}\)
The mixed fraction of \(\frac{112}{12}\) is 9 \(\frac{1}{3}\)

Question 14.
\(12 \frac{3}{4}-6 \frac{1}{6}\)
_____ \(\frac{□}{□}\)

Answer: 6 \(\frac{7}{12}\)

Explanation:
\(12 \frac{3}{4}-6 \frac{1}{6}\) = \(\frac{51}{4}\) – \(\frac{37}{6}\)
\(\frac{153}{12}\) – \(\frac{74}{12}\) = \(\frac{79}{12}\)
The mixed fraction of \(\frac{79}{12}\) is 6 \(\frac{7}{12}\)

Question 15.
\(2 \frac{5}{8}-1 \frac{1}{4}\)
_____ \(\frac{□}{□}\)

Answer: 1 \(\frac{3}{8}\)

Explanation:
\(2 \frac{5}{8}-1 \frac{1}{4}\)
\(\frac{21}{8}\) – \(\frac{5}{4}\) = \(\frac{21}{8}\) – \(\frac{10}{8}\)
= \(\frac{11}{8}\)
The mixed fraction of \(\frac{11}{8}\) is 1 \(\frac{3}{8}\)

Question 16.
\(10 \frac{1}{2}-2 \frac{1}{5}\)
_____ \(\frac{□}{□}\)

Answer: 8 \(\frac{3}{10}\)

Explanation:
\(10 \frac{1}{2}-2 \frac{1}{5}\) = \(\frac{21}{2}\) – \(\frac{11}{5}\)
\(\frac{105}{10}\) – \(\frac{22}{10}\) = \(\frac{83}{10}\)
The mixed fraction of \(\frac{83}{10}\) is 8 \(\frac{3}{10}\)

Practice: Copy and Solve Find the sum or difference. Write your answer in simplest form.

Question 19.
\(2 \frac{1}{6}+4 \frac{5}{9}\)
_____ \(\frac{□}{□}\)

Answer: 6 \(\frac{13}{18}\)

Explanation:
\(2 \frac{1}{6}+4 \frac{5}{9}\) = \(\frac{13}{6}\) + \(\frac{41}{9}\)
\(\frac{39}{18}\) + \(\frac{82}{18}\) = \(\frac{121}{18}\)
The mixed fraction of \(\frac{121}{18}\) is 6 \(\frac{13}{18}\)
\(2 \frac{1}{6}+4 \frac{5}{9}\) = 6 \(\frac{13}{18}\)

Question 20.
\(20 \frac{5}{8}+\frac{5}{12}\)
_____ \(\frac{□}{□}\)

Answer: 21 \(\frac{1}{24}\)

Explanation:
\(20 \frac{5}{8}+\frac{5}{12}\) = \(\frac{165}{8}\) + \(\frac{5}{12}\)
\(\frac{495}{24}\) + \(\frac{10}{24}\) = \(\frac{505}{24}\)
The mixed fraction of \(\frac{505}{24}\) is 21 \(\frac{1}{24}\)
\(20 \frac{5}{8}+\frac{5}{12}\) = 21 \(\frac{1}{24}\)

Question 21.
\(3 \frac{2}{3}-1 \frac{1}{6}\)
_____ \(\frac{□}{□}\)

Answer: 2 \(\frac{1}{2}\)

Explanation:
\(3 \frac{2}{3}-1 \frac{1}{6}\) = \(\frac{11}{3}\) – \(\frac{7}{6}\)
\(\frac{22}{6}\) – \(\frac{7}{6}\) = \(\frac{15}{6}\) = \(\frac{5}{2}\)
The mixed fraction of \(\frac{5}{2}\) is 2 \(\frac{1}{2}\)
\(3 \frac{2}{3}-1 \frac{1}{6}\) = 2 \(\frac{1}{2}\)

Question 22.
\(5 \frac{6}{7}-1 \frac{2}{3}\)
_____ \(\frac{□}{□}\)

Answer: 4 \(\frac{4}{21}\)

Explanation:
\(5 \frac{6}{7}-1 \frac{2}{3}\) = \(\frac{41}{7}\) – \(\frac{5}{3}\)
\(\frac{123}{21}\) – \(\frac{35}{21}\) = \(\frac{88}{21}\)
The mixed fraction of \(\frac{88}{21}\) is 4 \(\frac{4}{21}\)

Problem Solving – Page No. 268

Use the table to solve 25–28.

Question 25.
Gavin is mixing a batch of Sunrise Orange paint for an art project. How much paint does Gavin mix?
_____ \(\frac{□}{□}\) ounces

Answer: 5 \(\frac{7}{8}\) ounces

Explanation:
Gavin is mixing a batch of Sunrise Orange paint for an art project.
2 \(\frac{5}{8}\) + 3 \(\frac{1}{4}\)
Solving the whole numbers
2 + 3 = 5
Add the fraction parts
\(\frac{5}{8}\) + \(\frac{1}{4}\)
LCD = 8
\(\frac{5}{8}\) + \(\frac{2}{8}\) = \(\frac{7}{8}\)
5 + \(\frac{7}{8}\) = 5 \(\frac{7}{8}\) ounces

Question 26.
Gavin plans to mix a batch of Tangerine paint. He expects to have a total of 5 \(\frac{3}{10}\) ounces of paint after he mixes the amounts of red and yellow. Explain how you can tell if Gavin’s expectation is reasonable.
Type below:
_________

Answer:
Gavin plans to mix a batch of Tangerine paint. He expects to have a total of 5 \(\frac{3}{10}\) ounces of paint after he mixes the amounts of red and yellow.
To mix a batch of Tangerine paint he need 3 \(\frac{9}{10}\) red and 2 \(\frac{3}{8}\) yellow paint.
Add the fractions
3 + \(\frac{9}{10}\) + 2 + \(\frac{3}{8}\)
Solving the whole numbers
3 + 2 = 5
\(\frac{9}{10}\) + \(\frac{3}{8}\)
LCD = 40
\(\frac{9}{10}\) + \(\frac{3}{8}\) = \(\frac{36}{40}\) + \(\frac{15}{40}\) = \(\frac{51}{40}\) = 1 \(\frac{11}{40}\)
5 + 1 \(\frac{11}{40}\) = 6 \(\frac{11}{40}\)

Question 27.
For a special project, Gavin mixes the amount of red from one shade of paint with the amount of yellow from a different shade. He mixes the batch so he will have the greatest possible amount of paint. What amounts of red and yellow from which shades are used in the mixture for the special project? Explain your answer.
Type below:
_________

Answer:
Gavin used red paint from mango and yellow paint from Sunrise Orange.
5 \(\frac{5}{6}\) + 3 \(\frac{1}{4}\)
Solving the whole numbers parts
5 + 3 = 8
Solving the fraction part
\(\frac{5}{6}\) + \(\frac{1}{4}\)
LCD = 12
\(\frac{10}{12}\) + \(\frac{3}{12}\) = \(\frac{13}{12}\)
\(\frac{13}{12}\) = 1 \(\frac{1}{12}\)

Question 28.
Gavin needs to make 2 batches of Mango paint. Explain how you could find the total amount of paint Gavin mixed.
Type below:
_________

Answer:
Gavin used Red paint and Yellow Paint to make Mango shade.
For one batch he need to add 5 \(\frac{5}{6}\) + 5 \(\frac{5}{6}\)
Foe 2 batches
5 \(\frac{5}{6}\)+ 5 \(\frac{5}{6}\) + 5 \(\frac{5}{6}\) + 5 \(\frac{5}{6}\)
Solving the whole numbers
5 + 5 + 5 + 5 = 20
Solving the fractions part
\(\frac{5}{6}\) + \(\frac{5}{6}\) + \(\frac{5}{6}\) + \(\frac{5}{6}\) = \(\frac{20}{6}\)
= \(\frac{10}{3}\)
Gavin mixed \(\frac{10}{3}\) of paint to make 2 batches of Mango Paint.

Question 29.
Test Prep Yolanda walked 3 \(\frac{6}{10}\) miles. Then she walked 4 \(\frac{1}{2}\) more miles. How many miles did Yolanda walk?
Options:
a. 7 \(\frac{1}{10}\) miles
b. 7 \(\frac{7}{10}\) miles
c. 8 \(\frac{1}{10}\) miles
d. 8 \(\frac{7}{10}\) miles

Answer: 8 \(\frac{1}{10}\) miles

Explanation:
Test Prep Yolanda walked 3 \(\frac{6}{10}\) miles.
Then she walked 4 \(\frac{1}{2}\) more miles.
3 \(\frac{6}{10}\) + 4 \(\frac{1}{2}\) = 3 + \(\frac{6}{10}\) + 4 + \(\frac{1}{2}\)
Add whole numbers
3 + 4 = 7
Add the fractions
\(\frac{6}{10}\) + \(\frac{1}{2}\)
LCD = 10
\(\frac{6}{10}\) + \(\frac{5}{10}\) = \(\frac{11}{10}\)
\(\frac{11}{10}\) = 8 \(\frac{1}{10}\) miles
Thus the correct answer is option C.

Share and Show – Page No. 270

Estimate. Then find the difference and write it in simplest form.

Question 1.
Estimate: ______
1 \(\frac{3}{4}-\frac{7}{8}\)
Estimate: _____ \(\frac{□}{□}\)
Difference: _____ \(\frac{□}{□}\)

Answer:
Estimate: 1
Difference: \(\frac{7}{8}\)

Explanation:
Estimation: 1 + \(\frac{3}{4}\) – \(\frac{7}{8}\)
\(\frac{7}{8}\) is close to 1.
\(\frac{3}{4}\) is close to 1.
1 + 1 – 1 = 1
Difference: 1 \(\frac{3}{4}-\frac{7}{8}\)
1 + \(\frac{3}{4}\) – \(\frac{7}{8}\)
\(\frac{3}{4}\) – \(\frac{7}{8}\)
\(\frac{3}{4}\) × \(\frac{8}{8}\) – \(\frac{7}{8}\) × \(\frac{4}{4}\)
\(\frac{24}{32}\) – \(\frac{28}{32}\) = – \(\frac{1}{8}\)
1 – \(\frac{1}{8}\) = \(\frac{7}{8}\)

Page No. 271

Estimate. Then find the difference and write it in simplest form.

Question 3.
Estimate: ________
\(4 \frac{1}{2}-3 \frac{4}{5}\)
Estimate: _____ \(\frac{□}{□}\)
Difference: _____ \(\frac{□}{□}\)

Answer:
Estimate: \(\frac{1}{2}\)
Difference: \(\frac{7}{10}\)

Explanation:
\(4 \frac{1}{2}-3 \frac{4}{5}\)
4 – \(\frac{1}{2}\) – 3 – 1
= \(\frac{1}{2}\)
Difference:
\(4 \frac{1}{2}-3 \frac{4}{5}\)
4 \(\frac{1}{2}\) – 3 \(\frac{4}{5}\)
Solving the whole number parts
4 – 3 = 1
Solving the fraction parts
\(\frac{1}{2}\) – \(\frac{4}{5}\)
LCD = 10
\(\frac{5}{10}\) – \(\frac{8}{10}\) = – \(\frac{3}{10}\)
1 – \(\frac{3}{10}\) = \(\frac{7}{10}\)

On Your Own

Estimate. Then find the difference and write it in simplest form.

Question 5.
Estimate: ________
\(3 \frac{2}{3}-1 \frac{11}{12}\)
Estimate: _____ \(\frac{□}{□}\)
Difference: _____ \(\frac{□}{□}\)

Answer:
Estimate: 2
Difference: 1 \(\frac{3}{4}\)

Explanation:
Estimate:
\(3 \frac{2}{3}-1 \frac{11}{12}\)
\(\frac{2}{3}\) is close to 1.
\(\frac{11}{12}\) is close to 1.
3 + 1 – 1 – 1 = 2
Difference:
\(3 \frac{2}{3}-1 \frac{11}{12}\)
3 + \(\frac{2}{3}\) – 1 – \(\frac{11}{12}\)
3 – 1 = 2
Solving the fractions part
\(\frac{2}{3}\) – \(\frac{11}{12}\)
LCD = 12
\(\frac{8}{12}\) – \(\frac{11}{12}\) = – \(\frac{3}{12}\) = – \(\frac{1}{4}\)
3 – \(\frac{1}{4}\) = 1 \(\frac{3}{4}\)
\(3 \frac{2}{3}-1 \frac{11}{12}\) = 1 \(\frac{3}{4}\)

Question 6.
Estimate: ________
\(4 \frac{1}{4}-2 \frac{1}{3}\)
Estimate: _____ \(\frac{□}{□}\)
Difference: _____ \(\frac{□}{□}\)

Answer:
Estimate: 2
Difference: 1 \(\frac{11}{12}\)

Explanation:
\(4 \frac{1}{4}-2 \frac{1}{3}\)
\(\frac{1}{4}\) is close to 0.
\(\frac{1}{3}\) is close to 0.
4 – 2 = 2
Solving the fractions part
\(\frac{1}{4}\) – \(\frac{1}{3}\)
LCD = 12
\(\frac{1}{4}\) × \(\frac{3}{3}\) – \(\frac{1}{3}\) × \(\frac{4}{4}\)
\(\frac{3}{12}\) – \(\frac{4}{12}\) = – \(\frac{1}{12}\)
2 – \(\frac{1}{12}\) = 1 \(\frac{11}{12}\)

Question 7.
Estimate: ________
\(5 \frac{2}{5}-1 \frac{1}{2}\)
Estimate: _____ \(\frac{□}{□}\)
Difference: _____ \(\frac{□}{□}\)

Answer:
Estimate: 4
Difference: 3 \(\frac{9}{10}\)

Explanation:
Estimate:
\(5 \frac{2}{5}-1 \frac{1}{2}\)
5 + \(\frac{1}{2}\) – 1 – \(\frac{1}{2}\)
5 – 1 = 4
Solving the fractions part
\(5 \frac{2}{5}-1 \frac{1}{2}\)
LCD = 10
\(\frac{4}{10}\) – \(\frac{5}{10}\) = – \(\frac{1}{10}\)
4 – \(\frac{1}{10}\) = 3 \(\frac{9}{10}\)

Question 8.
\(7 \frac{5}{9}-2 \frac{5}{6}\)
Estimate: _____ \(\frac{□}{□}\)
Difference: _____ \(\frac{□}{□}\)

Answer:
Estimate: 4 \(\frac{1}{2}\)
Difference: 4 \(\frac{13}{18}\)

Explanation:
Estimate:
\(7 \frac{5}{9}-2 \frac{5}{6}\)
\(\frac{5}{9}\) is close to \(\frac{1}{2}\)
\(\frac{5}{6}\) is close to 1.
7 + \(\frac{1}{2}\) – 2 – 1
4 \(\frac{1}{2}\)
Difference:
\(7 \frac{5}{9}-2 \frac{5}{6}\)
7 + \(\frac{5}{9}\) – 2 – \(\frac{5}{6}\)
Solving the whole numbers
7 – 2 = 5
Solving the fraction part
\(\frac{5}{9}\) – \(\frac{5}{6}\)
LCD = 18
\(\frac{10}{18}\) – \(\frac{15}{18}\) = – \(\frac{5}{18}\)
5 – \(\frac{5}{18}\) = 4 \(\frac{13}{18}\)

Question 9.
Estimate: ________
\(7-5 \frac{2}{3}\)
Estimate: _____ \(\frac{□}{□}\)
Difference: _____ \(\frac{□}{□}\)

Answer:
Estimate: 1
Difference: 1 \(\frac{1}{3}\)

Explanation:
Estimate:
\(7-5 \frac{2}{3}\)
7 – 5 – \(\frac{2}{3}\)
7 – 5 – 1 = 1
Difference:
\(7-5 \frac{2}{3}\)
7 – 5 = 2
2 – \(\frac{2}{3}\) = 1 \(\frac{1}{3}\)
Thus \(7-5 \frac{2}{3}\) = 1 \(\frac{1}{3}\)

Practice: Copy and Solve Find the difference and write it in simplest form.

Question 12.
\(6-3 \frac{1}{2}\)
_____ \(\frac{□}{□}\)

Answer: 2 \(\frac{1}{2}\)

Explanation:
Rewriting our equation with parts separated
6 – 3 – \(\frac{1}{2}\)
3 – \(\frac{1}{2}\) = 2 \(\frac{1}{2}\)

Question 13.
\(4 \frac{3}{8}-3 \frac{1}{2}\)
\(\frac{□}{□}\)

Answer: \(\frac{7}{8}\)

Explanation:
Rewriting our equation with parts separated
4 + \(\frac{3}{8}\) – 3 – \(\frac{1}{2}\)
Solving the whole number parts
4 – 3 = 1
Solving the fraction parts
\(\frac{3}{8}\) – \(\frac{1}{2}\) = \(\frac{3}{8}\) – \(\frac{4}{8}\)
= – \(\frac{1}{8}\)
1 – \(\frac{1}{8}\) = \(\frac{7}{8}\)

Question 14.
\(9 \frac{1}{6}-3 \frac{5}{8}\)
_____ \(\frac{□}{□}\)

Answer: 5 \(\frac{13}{24}\)

Explanation:
Rewriting our equation with parts separated
9 + \(\frac{1}{6}\) – 3 – \(\frac{5}{8}\)
Solving the whole number parts
9 – 3 = 6
Solving the fraction parts
\(\frac{1}{6}\) – \(\frac{5}{8}\)
\(\frac{4}{24}\) – \(\frac{15}{24}\) = – \(\frac{11}{24}\)
6 – \(\frac{11}{24}\) = 5 \(\frac{13}{24}\)

Question 15.
\(1 \frac{1}{5}-\frac{1}{2}\)
\(\frac{□}{□}\)

Answer: \(\frac{7}{10}\)

Explanation:
Rewriting our equation with parts separated
1 + \(\frac{1}{5}\) – \(\frac{1}{2}\)
Solving the whole number parts
1 + 0 = 1
Solving the fraction parts
\(\frac{1}{5}\) – \(\frac{1}{2}\)
LCD = 10
\(\frac{2}{10}\) – \(\frac{5}{10}\) = – \(\frac{3}{10}\)
1 – \(\frac{3}{10}\) = \(\frac{7}{10}\)

Question 16.
\(13 \frac{1}{6}-3 \frac{4}{5}\)
_____ \(\frac{□}{□}\)

Answer: 9 \(\frac{11}{30}\)

Explanation:
Rewriting our equation with parts separated
13 + \(\frac{1}{6}\) – 3 – \(\frac{4}{5}\)
Solving the whole number parts
13 – 3 = 10
Solving the fraction parts
\(\frac{1}{6}\) – \(\frac{4}{5}\)
LCD = 30
\(\frac{5}{30}\) – \(\frac{24}{30}\) = – \(\frac{19}{30}\)
10 – \(\frac{19}{30}\) = 9 \(\frac{11}{30}\)

Question 17.
\(12 \frac{2}{5}-5 \frac{3}{4}\)
_____ \(\frac{□}{□}\)

Answer: 6 \(\frac{13}{20}\)

Explanation:
Rewriting our equation with parts separated
12 + \(\frac{2}{5}\) – 5 – \(\frac{3}{4}\)
Solving the whole number parts
12 – 5 = 7
Solving the fraction parts
\(\frac{2}{5}\) – \(\frac{3}{4}\)
LCD = 20
\(\frac{8}{20}\) – \(\frac{15}{20}\) = – \(\frac{7}{20}\)
7 – \(\frac{7}{20}\) = 6 \(\frac{13}{20}\)

Question 18.
\(7 \frac{3}{8}-2 \frac{7}{9}\)
_____ \(\frac{□}{□}\)

Answer: 4 \(\frac{43}{72}\)

Explanation:
7 + \(\frac{3}{8}\) – 2 – \(\frac{7}{9}\)
7 – 2 = 5
\(\frac{3}{8}\) – \(\frac{7}{9}\) = \(\frac{27}{72}\) – \(\frac{56}{72}\)
– \(\frac{29}{72}\)
5 – \(\frac{29}{72}\) = 4 \(\frac{43}{72}\)

Page No. 272

Connect to Reading

Summarize
An amusement park in Sandusky, Ohio, offers 17 amazing roller coasters for visitors to ride. One of the roller coasters runs at 60 miles per hour and has 3,900 feet of twisting track. This coaster also has 3 trains with 8 rows per train. Riders stand in rows of 4, for a total of 32 riders per train.

The operators of the coaster recorded the number of riders on each train during a run. On the first train, the operators reported that 7 \(\frac{1}{4}\) rows were filled. On the second train, all 8 rows were filled, and on the third train, 5 \(\frac{1}{2}\) rows were filled. How many more rows were filled on the first train than on the third train?

When you summarize, you restate the most important information in a shortened form to more easily understand what you have read.
Summarize the information given.
______________________
Use the summary to solve.

Question 19.
Solve the problem above.
Type below:
_________

Answer:
On the first train, the operators reported that 7 \(\frac{1}{4}\) rows were filled.
On the third train, 5 \(\frac{1}{2}\) rows were filled.
7 \(\frac{1}{4}\) – 5 \(\frac{1}{2}\)
Solving the whole numbers
7 – 5 = 2
Solving the fractions
\(\frac{1}{4}\) – \(\frac{1}{2}\) = – \(\frac{1}{4}\)
2 – \(\frac{1}{4}\) = 1 \(\frac{3}{4}\)
1 \(\frac{3}{4}\) more rows were filled on the first train than on the third train.

Question 20.
How many rows were empty on the third train? How many additional riders would it take to fill the empty rows? Explain your answer.
Type below:
_________

Answer:
The coaster also has 3 trains with 8 rows per train.
The third train has 8 rows.
On the third train, 5 \(\frac{1}{2}\) rows were filled.
8 – 5 \(\frac{1}{2}\)
8 – 5 – \(\frac{1}{2}\) = 2 \(\frac{1}{2}\)
2 \(\frac{1}{2}\) rows are empty.
So, it takes 10 additional riders to fill the empty rows on the third train.

Share and Show – Page No. 275

Write a rule for the sequence.

Question 1.
\(\frac{1}{4}, \frac{1}{2}, \frac{3}{4}, \cdots\)
Think: Is the sequence increasing or decreasing?
Rule: _________
Type below:
_________

Answer: The sequence is increasing order with difference \(\frac{1}{4}\)

Question 2.
\(\frac{1}{9}, \frac{1}{3}, \frac{5}{9}, \ldots\)
Type below:
_________

Answer: The sequence is increasing in order with a difference of 2 in the numerator.

Write a rule for the sequence. Then, find the unknown term.

Question 4.
\(10 \frac{2}{3}, 9 \frac{11}{18}, 8 \frac{5}{9}\), ______ \(\frac{□}{□}\) , \(6 \frac{4}{9}\)

Answer: 7 \(\frac{1}{2}\)

Explanation:
\(\frac{32}{3}\), \(\frac{173}{18}\), \(\frac{77}{9}\), x, \(\frac{58}{9}\)
LCD = 54
\(\frac{576}{54}\), \(\frac{519}{54}\), \(\frac{462}{54}\), x, \(\frac{348}{54}\)
According to the series x = \(\frac{405}{54}\) = \(\frac{15}{2}\)
The mixed fraction of \(\frac{15}{2}\) is 7 \(\frac{1}{2}\)

Question 5.
\(1 \frac{1}{6}\), ______ \(\frac{□}{□}\) , \(1, \frac{11}{12}, \frac{5}{6}\)

Answer: 1 \(\frac{1}{12}\)

Explanation:
\(1 \frac{1}{6}\), ______ \(\frac{□}{□}\) , \(1, \frac{11}{12}, \frac{5}{6}\)
The LCD of the above fractons is 12
Convert them into improper fractions
\(\frac{14}{12}\), x, \(\frac{12}{12}\), \(\frac{11}{12}\), \(\frac{10}{12}\)
According to the series x = \(\frac{13}{12}\)
The mixed fraction of \(\frac{13}{12}\) is 1 \(\frac{1}{12}\)

Question 6.
\(2 \frac{3}{4}, 4,5 \frac{1}{4}, 6 \frac{1}{2}\), ______ \(\frac{□}{□}\)

Answer: 7 \(\frac{3}{4}\)

Explanation:
\(2 \frac{3}{4}, 4,5 \frac{1}{4}, 6 \frac{1}{2}\), ______ \(\frac{□}{□}\)
Convert the mixed fractions into improper fractions
\(\frac{11}{4}\), \(\frac{4}{1}\), \(\frac{21}{4}\), \(\frac{13}{2}\), x
\(\frac{11}{4}\), \(\frac{16}{4}\), \(\frac{21}{4}\), \(\frac{26}{4}\), x
According to the series x = \(\frac{31}{4}\)
The mixed fraction of \(\frac{31}{4}\) is 7 \(\frac{3}{4}\)

On Your Own

Write a rule for the sequence. Then, find the unknown term.

Question 7.
\(\frac{1}{8}, \frac{1}{2}\), \(\frac{□}{□}\) , \(1 \frac{1}{4}, 1 \frac{5}{8}\)

Answer: \(\frac{7}{8}\)

Explanation:
\(\frac{1}{8}, \frac{1}{2}\), \(1 \frac{1}{4}, 1 \frac{5}{8}\), x
LCD = 8
\(\frac{1}{8}, \frac{4}{8}\), \(\frac{10}{8}, \frac{26}{8}\), x
\(\frac{1}{8}\), \(\frac{4}{8}\), x, \(\frac{10}{8}\), \(\frac{26}{8}\)
The difference between the series is 3 in numerator.
x = \(\frac{7}{8}\)

Question 10.
\(9 \frac{1}{3}\), ______ \(\frac{□}{□}\) , \(6 \frac{8}{9}, 5 \frac{2}{3}, 4 \frac{4}{9}\)

Answer: 8 \(\frac{1}{9}\)

Explanation:
\(9 \frac{1}{3}\), x , \(6 \frac{8}{9}, 5 \frac{2}{3}, 4 \frac{4}{9}\)
Convert the mixed fractions into improper fractions
\(\frac{28}{3}\), x, \(\frac{62}{9}\), \(\frac{17}{3}\), \(\frac{40}{9}\)
LCD = 9
\(\frac{84}{9}\), x, \(\frac{62}{9}\), \(\frac{51}{9}\), \(\frac{40}{9}\)
According to the series x =  \(\frac{73}{9}\) = 8 \(\frac{1}{9}\)

Write the first four terms of the sequence.

Question 11.
Rule: start at 5 \(\frac{3}{4}\), subtract \(\frac{5}{8}\)
First term: ______ \(\frac{□}{□}\)
Second term: ______ \(\frac{□}{□}\)
Third term: ______ \(\frac{□}{□}\)
Fourth term: ______ \(\frac{□}{□}\)

Answer:
Let the first term be 5 \(\frac{3}{4}\)
Second term = 5 \(\frac{3}{4}\) – \(\frac{5}{8}\) = \(\frac{41}{8}\) = 5 \(\frac{1}{8}\)
Third term = 5 \(\frac{1}{8}\) – \(\frac{5}{8}\) = \(\frac{36}{8}\) = 4 \(\frac{1}{2}\)
Fourth term = \(\frac{36}{8}\) – \(\frac{5}{8}\) = \(\frac{31}{8}\) = 3 \(\frac{7}{8}\)

Question 12.
Rule: start at \(\frac{3}{8}\), add \(\frac{3}{16}\)
Type below:
_________

Answer:
Let the first term be \(\frac{3}{8}\)
Second term = \(\frac{3}{8}\) + \(\frac{3}{16}\) = \(\frac{9}{16}\)
Third term = \(\frac{9}{16}\) + \(\frac{3}{16}\) = \(\frac{12}{16}\)
Fourth term = \(\frac{12}{16}\) + \(\frac{3}{16}\) = \(\frac{15}{16}\)

Question 14.
Rule: start at \(\frac{8}{9}\), subtract \(\frac{1}{18}\)
Type below:
_________

Answer:
Let the first term be \(\frac{8}{9}\)
Second term = \(\frac{8}{9}\) – \(\frac{1}{18}\) = \(\frac{15}{18}\) = \(\frac{5}{6}\)
Third term = \(\frac{15}{18}\) – \(\frac{1}{18}\) = \(\frac{14}{18}\) = \(\frac{7}{9}\)
Fourth term = \(\frac{14}{18}\) – \(\frac{1}{18}\) = \(\frac{13}{18}\)

Problem Solving – Page No. 276

Question 15.
When Bill bought a marigold plant, it was \(\frac{1}{4}\) inch tall. After the first week, it measured 1 \(\frac{1}{12}\) inches tall. After the second week, it was 1 \(\frac{11}{12}\) inches. After week 3, it was 2 \(\frac{3}{4}\) inches tall. Assuming the growth of the plant was constant, what was the height of the plant at the end of week 4?
______ \(\frac{□}{□}\) inches

Answer: 3 \(\frac{7}{12}\) inches

The sequence is the increasing where the first term is \(\frac{1}{4}\)
LCD = 12
First week is \(\frac{3}{12}\)
Second week = \(\frac{13}{12}\) = 1 \(\frac{1}{12}\)
Third week = 1 \(\frac{11}{12}\) = \(\frac{23}{12}\)
Fourth week = \(\frac{33}{12}\) = 2 \(\frac{3}{4}\)
At the end of fourth week = \(\frac{43}{12}\) = 3 \(\frac{7}{12}\) inches
The height of the plant at the end of the week is 3 \(\frac{7}{12}\) inches.

Question 16.
What if Bill’s plant grew at the same rate but was 1 \(\frac{1}{2}\) inches when he bought it? How tall would the plant be after 3 weeks?
______ inches

Answer: 4 inches

Explanation:
The sequence is increasing.
First week 1 \(\frac{1}{2}\)
Let the first term is \(\frac{6}{12}\)
Second term is 1 \(\frac{16}{12}\)
Third term is 1 \(\frac{26}{12}\)
Fourth week is 1 \(\frac{36}{12}\)
1 \(\frac{36}{12}\) = 1 \(\frac{3}{1}\) = 1 + 3 = 4
After 4 weeks the plant grew 4 inches.

Question 18.
Mr. Conners drove 78 \(\frac{1}{3}\) miles on Monday, 77 \(\frac{1}{12}\) miles on Tuesday, and 75 \(\frac{5}{6}\) miles on Wednesday. If he continues this pattern on Thursday and Friday, how many miles will he drive on Friday?
______ \(\frac{□}{□}\) miles

Answer:
Given that,
Mr. Conners drove 78 \(\frac{1}{3}\) miles on Monday, 77 \(\frac{1}{12}\) miles on Tuesday, and 75 \(\frac{5}{6}\) miles on Wednesday.
The sequence is decreasing where the first term is 78 \(\frac{4}{12}\)
78 \(\frac{4}{12}\) – 77 \(\frac{1}{12}\) = 1 \(\frac{3}{12}\)
The difference between the term is 1 \(\frac{3}{12}\)
On thursday, 75 \(\frac{5}{6}\) – 1 \(\frac{3}{12}\) = 74 \(\frac{7}{12}\)
On friday, 74 \(\frac{7}{12}\) – 1 \(\frac{3}{12}\) = 73 \(\frac{4}{12}\) = 73 \(\frac{1}{3}\)

Question 19.
Test Prep Zack watered his garden with 1 \(\frac{3}{8}\) gallons of water the first week he planted it. He watered it with 1 \(\frac{3}{4}\) gallons the second week, and 2 \(\frac{1}{8}\) gallons the third week. If he continued watering in this pattern, how much water did he use on the fifth week?
Options:
a. 2 \(\frac{1}{2}\) gallons
b. 2 \(\frac{7}{8}\) gallons
c. 3 \(\frac{1}{4}\) gallons
d. 6 \(\frac{7}{8}\) gallons

Answer: 2 \(\frac{7}{8}\) gallons

Explanation:
First term = 1 \(\frac{3}{8}\)
The difference is \(\frac{3}{4}\) – \(\frac{3}{8}\) = \(\frac{3}{8}\)
Second term is 1 \(\frac{3}{8}\) + \(\frac{3}{8}\) = 1 \(\frac{3}{4}\)
Third term = 1 \(\frac{3}{4}\) + \(\frac{3}{8}\) = 1 + 1 \(\frac{1}{8}\) = 2 \(\frac{1}{8}\)
Fourth term = 2 \(\frac{1}{8}\) + \(\frac{3}{8}\) = 2 \(\frac{1}{2}\)
Fifth term = 2 \(\frac{1}{2}\) + \(\frac{3}{8}\) = 2 \(\frac{7}{8}\) gallons
Thus the correct answer is option B.

Share and Show – Page No. 279

Question 1.
Caitlin has 4 \(\frac{3}{4}\) pounds of clay. She uses 1 \(\frac{1}{10}\) pounds to make a cup, and another 2 pounds to make a jar. How many pounds are left?
First, write an equation to model the problem.
Type below:
_________

Answer: 4 \(\frac{3}{4}\) – 1 \(\frac{1}{10}\) – 2

Explanation:
Subtract the total pound of clay from used clay.
So, the equation of the clay leftover is 4 \(\frac{3}{4}\) – 1 \(\frac{1}{10}\) – 2

Question 1.
Next, work backwards and rewrite the equation to find x.
Type below:
_________

Answer: 4 \(\frac{3}{4}\) – 1 \(\frac{1}{10}\) – 2 = x

Explanation:
Let the leftover clay be x
4 \(\frac{3}{4}\) – 1 \(\frac{1}{10}\) – 2 = x
x = 4 \(\frac{3}{4}\) – 1 \(\frac{1}{10}\) – 2

Question 1.
Solve.
_____________________
So, ________ pounds of clay remain.
Type below:
_________

Answer: 1 \(\frac{13}{20}\) pounds

Explanation:
4 \(\frac{3}{4}\) – 1 \(\frac{1}{10}\) – 2
4 + \(\frac{3}{4}\) – 1 – \(\frac{1}{10}\) – 2
4 – 3 = 1
\(\frac{3}{4}\) – \(\frac{1}{10}\) = \(\frac{13}{20}\)
1 + \(\frac{13}{20}\) = 1 \(\frac{13}{20}\) pounds

Question 2.
What if Caitlin had used more than 2 pounds of clay to make a jar? Would the amount remaining have been more or less than your answer to Exercise 1?
Type below:
_________

Answer:
Let us assume that Catlin used 2 \(\frac{1}{4}\) pounds of clay to make a jar and 1 \(\frac{1}{10}\) pounds to make a cup.
4 \(\frac{3}{4}\) – 1 \(\frac{1}{10}\) – 2 \(\frac{1}{4}\) = 2 \(\frac{1}{20}\)

Question 3.
A pet store donated 50 pounds of food for adult dogs, puppies, and cats to an animal shelter. 19 \(\frac{3}{4}\) pounds was adult dog food and 18 \(\frac{7}{8}\) pounds was puppy food. How many pounds of cat food did the pet store donate?
______ \(\frac{□}{□}\) pounds of cat food

Answer: 11 \(\frac{3}{8}\) pounds of cat food

Explanation:
A pet store donated 50 pounds of food for adult dogs, puppies, and cats to an animal shelter.
19 \(\frac{3}{4}\) pounds was adult dog food and 18 \(\frac{7}{8}\) pounds was puppy food.
19 \(\frac{3}{4}\) + 18 \(\frac{7}{8}\) = 38 \(\frac{5}{8}\)
50 – 38 \(\frac{5}{8}\) = 11 \(\frac{3}{8}\) pounds of cat food
Thus the pet store donate 11 \(\frac{3}{8}\) pounds of cat food

On Your Own – Page No. 280

Question 5.
Martin is making a model of a Native American canoe. He has 5 \(\frac{1}{2}\) feet of wood. He uses 2 \(\frac{3}{4}\) feet for the hull and 1 \(\frac{1}{4}\) feet for the paddles and struts. How much wood does he have left?
______ \(\frac{□}{□}\) feet

Answer: 1 \(\frac{1}{2}\) feet

Explanation:
Martin is making a model of a Native American canoe.
He has 5 \(\frac{1}{2}\) feet of wood.
He uses 2 \(\frac{3}{4}\) feet for the hull and 1 \(\frac{1}{4}\) feet for the paddles and struts.
2 \(\frac{3}{4}\) + 1 \(\frac{1}{4}\)
2 + \(\frac{3}{4}\) + 1 + \(\frac{1}{4}\)
2 + 1 = 3
\(\frac{3}{4}\) + \(\frac{1}{4}\) = 1
3 + 1 = 4
5 \(\frac{1}{2}\) – 4 = 1 \(\frac{1}{2}\)

Question 7.
Beth’s summer vacation lasted 87 days. At the beginning of her vacation, she spent 3 weeks at soccer camp, 5 days at her grandmother’s house, and 13 days visiting Glacier National Park with her parents. How many vacation days remained?
______ days

Answer: 48 days

Explanation:
Given,
Beth’s summer vacation lasted 87 days.
At the beginning of her vacation, she spent 3 weeks at soccer camp, 5 days at her grandmother’s house, and 13 days visiting Glacier National Park with her parents.
87 – 21 – 5 – 13 = 48 days
The remaining vacation days are 48.

Question 8.
You can buy 2 DVDs for the same price you would pay for 3 CDs selling for $13.20 apiece. Explain how you could find the price of 1 DVD.
$ ______

Answer: $19.8

Explanation:
To find what is the price of 1 DVD we will find what is the price of 3 DVDs and then because 2 DVDs price is the same than 3 CDs we can easily find the price of 1 DVD.
$13.20 × 3 = $39.6
We will divide $39.6 by 2.
$39.6 ÷ 2 = $19.8
The price of 1 DVD is $19.8

Question 9.
Test Prep During the 9 hours between 8 A.M. and 5 P.M., Bret spent 5 \(\frac{3}{4}\) hours in class and 1 \(\frac{1}{2}\) hours at band practice. How much time did he spend on other activities?
Options:
a. \(\frac{3}{4}\) hour
b. 1 \(\frac{1}{4}\) hour
c. 1 \(\frac{1}{2}\) hour
d. 1 \(\frac{3}{4}\) hour

Answer: 1 \(\frac{3}{4}\) hour

Explanation:
Test Prep During the 9 hours between 8 A.M. and 5 P.M., Bret spent 5 \(\frac{3}{4}\) hours in class and 1 \(\frac{1}{2}\) hours at band practice.
5 \(\frac{3}{4}\) + 1 \(\frac{1}{2}\) = 7 \(\frac{1}{4}\) hour
9 – 7 \(\frac{1}{4}\) hour
8 + 1 – 7 – \(\frac{1}{4}\)
1 \(\frac{3}{4}\) hour
The correct answer is option D.

Share and Show – Page No. 283

Use the properties and mental math to solve. Write your answer in simplest form.

Question 1.
\(\left(2 \frac{5}{8}+\frac{5}{6}\right)+1 \frac{1}{8}\)
______ \(\frac{□}{□}\)

Answer:
\(\left(2 \frac{5}{8}+\frac{5}{6}\right)+1 \frac{1}{8}\)
2 \(\frac{5}{8}\) + \(\frac{5}{6}\)
2 + \(\frac{5}{8}\) + \(\frac{5}{6}\)
LCD = 24
\(\frac{15}{24}\) + \(\frac{20}{24}\) = \(\frac{35}{24}\)
\(\frac{35}{24}\) = 1 \(\frac{11}{24}\)
2 + 1 \(\frac{11}{24}\) = 3 \(\frac{11}{24}\)
3 \(\frac{11}{24}\) + 1 \(\frac{1}{8}\) = 4 \(\frac{7}{12}\)

Question 2.
\(\frac{5}{12}+\left(\frac{5}{12}+\frac{3}{4}\right)\)
______ \(\frac{□}{□}\)

Answer:
\(\frac{5}{12}+\left(\frac{5}{12}+\frac{3}{4}\right)\)
\(\frac{5}{12}\) + \(\frac{3}{4}\)
LCD = 12
\(\frac{5}{12}\) + \(\frac{3}{4}\) × \(\frac{3}{3}\)
\(\frac{5}{12}\) + \(\frac{9}{12}\) = \(\frac{14}{12}\)
\(\frac{5}{12}\) + \(\frac{14}{12}\) = \(\frac{19}{12}\)
\(\frac{19}{12}\) = 1 \(\frac{7}{12}\)

On Your Own

Use the properties and mental math to solve. Write your answer in simplest form.

Question 4.
\(\left(\frac{2}{7}+\frac{1}{3}\right)+\frac{2}{3}\)
______ \(\frac{□}{□}\)

Answer:
\(\left(\frac{2}{7}+\frac{1}{3}\right)+\frac{2}{3}\)
\(\left(\frac{2}{7}+\frac{1}{3}\right)\)
LCD = 21
\(\left(\frac{6}{21}+\frac{7}{21}\right)\) = \(\frac{13}{21}\)
\(\frac{13}{21}\) + \(\frac{2}{3}\)
LCD = 21
\(\frac{13}{21}\) + \(\frac{14}{21}\)
\(\frac{27}{21}\) = \(\frac{9}{7}\)
= 1 \(\frac{2}{7}\)

Question 5.
\(\left(\frac{1}{5}+\frac{1}{2}\right)+\frac{2}{5}\)
______ \(\frac{□}{□}\)

Answer:
\(\left(\frac{1}{5}+\frac{1}{2}\right)\)
\(\frac{1}{5}\) + \(\frac{1}{2}\)
LCD = 10
\(\frac{2}{10}\) + \(\frac{5}{10}\) = \(\frac{7}{10}\)
\(\frac{7}{10}\) + \(\frac{2}{5}\)
\(\frac{7}{10}\) + \(\frac{4}{10}\) = \(\frac{11}{10}\)
\(\frac{11}{10}\) = 1 \(\frac{1}{10}\)

Question 6.
\(\left(\frac{1}{6}+\frac{3}{7}\right)+\frac{2}{7}\)
\(\frac{□}{□}\)

Answer:
\(\left(\frac{1}{6}+\frac{3}{7}\right)\)
LCD = 42
\(\left(\frac{7}{42}+\frac{18}{42}\right)\) = \(\frac{25}{42}\)
\(\frac{25}{42}\) + \(\frac{2}{7}\)
LCD = 42
\(\frac{25}{42}\) + \(\frac{12}{42}\) = \(\frac{37}{42}\)
\(\left(\frac{1}{6}+\frac{3}{7}\right)+\frac{2}{7}\) = \(\frac{37}{42}\)

Question 7.
\(\left(2 \frac{5}{12}+4 \frac{1}{4}\right)+\frac{1}{4}\)
______ \(\frac{□}{□}\)

Answer:
\(\left(2 \frac{5}{12}+4 \frac{1}{4}\right)\)
2 \(\frac{5}{12}\) + 4 \(\frac{1}{4}\)
2 + \(\frac{5}{12}\) + 4 + \(\frac{1}{4}\)
2 + 4 = 6
\(\frac{5}{12}\) + \(\frac{1}{4}\) = \(\frac{8}{12}\)
6 \(\frac{8}{12}\) = 6 \(\frac{2}{3}\)
6 \(\frac{2}{3}\) + \(\frac{1}{4}\) = 6 \(\frac{11}{12}\)

Question 8.
\(1 \frac{1}{8}+\left(5 \frac{1}{2}+2 \frac{3}{8}\right)\)
______

Answer:
5 \(\frac{1}{2}\) + 2 \(\frac{3}{8}\)
5 + 2 = 7
\(\frac{1}{2}\) + \(\frac{3}{8}\)
LCD = 8
\(\frac{4}{8}\) + \(\frac{3}{8}\) = \(\frac{7}{8}\)
= 7 \(\frac{7}{8}\)
1 \(\frac{1}{8}\) + 7 \(\frac{7}{8}\) = 9

Problem Solving – Page No. 284

Use the map to solve 10–12.

Question 10.
In the morning, Julie rides her bike from the sports complex to the school. In the afternoon, she rides from the school to the mall, and then to Kyle’s house. How far does Julie ride her bike?
______ \(\frac{□}{□}\) miles

Answer: 1 \(\frac{13}{15}\) miles

Explanation:
Julie rides her bike from the sports complex to the school = \(\frac{2}{3}\) mile
In the afternoon, she rides from the school to the mall, and then to Kyle’s house. = \(\frac{2}{5}\) + \(\frac{4}{5}\) = \(\frac{6}{5}\) = 1 \(\frac{1}{5}\)
1 \(\frac{1}{5}\) + \(\frac{2}{3}\) mile = 1 \(\frac{13}{15}\) miles

Question 11.
On one afternoon, Mario walks from his house to the library. That evening, Mario walks from the library to the mall, and then to Kyle’s house. Describe how you can use the properties to find how far Mario walks.
______ \(\frac{□}{□}\) miles

Answer:
Mario walks from his house to the library = 1 \(\frac{3}{5}\) miles
Mario walks from the library to the mall, and then to Kyle’s house = 1 \(\frac{1}{3}\) and \(\frac{4}{5}\)
1 \(\frac{3}{5}\) + (1 \(\frac{1}{3}\) + \(\frac{4}{5}\))
1 \(\frac{3}{5}\) + 2 \(\frac{2}{15}\) = 3 \(\frac{11}{15}\) miles

Question 12.
Pose a Problem Write and solve a new problem that uses the distances between four locations.
Type below:
_________

Answer:
In the evening Kyle rides his bike from the sports complex to school. Then he rides from School to the mall and then to his house. How far does Kyle ride his bike?
The distance from Sports complex to School is \(\frac{2}{3}\) mile
The distance from School to the mall is \(\frac{2}{5}\)
The distance from the mall to Kyle house is \(\frac{4}{5}\)
\(\frac{2}{3}\) + (\(\frac{2}{5}\) + \(\frac{4}{5}\))
\(\frac{2}{3}\) + \(\frac{6}{5}\) = 1 \(\frac{13}{15}\) miles

Question 13.
Do Test Prep Which property or properties do the problem below using?
\(\frac{1}{9}+\left(\frac{4}{9}+\frac{1}{6}\right)=\left(\frac{1}{9}+\frac{4}{9}\right)+\frac{1}{6}\)
Options:
a. Commutative Property
b. Associative Property
c. Commutative Property and Associative Property
d. Distributive Property

Answer: Associative Property
The associative property states that you can add or multiply regardless of how the numbers are grouped. By ‘grouped’ we mean ‘how you use parenthesis’. In other words, adding or multiplying does not matter where you put the parenthesis.

Chapter Review/Test – Vocabulary – Page No. 285

Choose the best term from the box.

Question 1.
A _________ is a number that is a common multiple of two or more denominators.
_________

Answer: Common Denominator

Concepts and Skills

Use a common denominator to write an equivalent fraction for each fraction.

Question 4.
\(\frac{2}{3}, \frac{1}{6}\)
common denominator: ______
Explain:
_________

Answer: 18
Multiply the denominators of the fractions
3 × 6 = 18

Find the sum or difference. Write your answer in simplest form

Estimate. Then find the difference and write it in simplest form.

Question 8.
\(1 \frac{2}{5}-\frac{2}{3}\)
Type below:
________

Answer:
Estimate: \(\frac{1}{2}\)
Difference:
Rewriting our equation with parts separated
1 + \(\frac{2}{5}\) – \(\frac{2}{3}\)
\(\frac{7}{5}\) – \(\frac{2}{3}\)
\(\frac{7}{5}\) × \(\frac{3}{3}\) – \(\frac{2}{3}\) × \(\frac{5}{5}\)
= \(\frac{21}{15}\) – \(\frac{10}{15}\)
= \(\frac{11}{15}\)

Question 9.
\(7-\frac{3}{7}\)
Type below:
________

Answer: 6 \(\frac{4}{7}\)

Explanation:
\(7-\frac{3}{7}\) = \(\frac{49}{7}\) – \(\frac{3}{7}\)
\(\frac{46}{7}\) = 6 \(\frac{4}{7}\)
\(7-\frac{3}{7}\) = 6 \(\frac{4}{7}\)

Question 10.
\(5 \frac{1}{9}-3 \frac{5}{6}\)
Type below:
________

Answer: 1 \(\frac{5}{18}\)

Explanation:
\(5 \frac{1}{9}-3 \frac{5}{6}\) = 5 + \(\frac{1}{9}\) – 3 – \(\frac{5}{6}\)
5 – 3 = 2
\(\frac{1}{9}\) – \(\frac{5}{6}\) = \(\frac{2}{18}\) – \(\frac{15}{18}\) = – \(\frac{13}{18}\)
2 – \(\frac{13}{18}\) = 1 \(\frac{5}{18}\)

Use the properties and mental math to solve. Write your answer in simplest form.

Question 13.
\(3 \frac{5}{9}+\left(1 \frac{7}{9}+2 \frac{5}{12}\right)\)
______ \(\frac{□}{□}\)

Answer: 7 \(\frac{3}{4}\)

Explanation:
1 \(\frac{7}{9}\) + 2 \(\frac{5}{12}\)
1 + 2 = 3
\(\frac{7}{9}\) + \(\frac{5}{12}\)
LCD is 36
\(\frac{28}{36}\) + \(\frac{15}{36}\) = \(\frac{43}{36}\)
\(\frac{43}{36}\) = 1 \(\frac{7}{36}\)
3 + 1 + \(\frac{7}{36}\) = 4 \(\frac{7}{36}\)
4 \(\frac{7}{36}\) + 3 \(\frac{5}{9}\)
4 + \(\frac{7}{36}\) + 3 + \(\frac{5}{9}\)
4 + 3 = 7
\(\frac{7}{36}\) + \(\frac{5}{9}\)
= \(\frac{7}{36}\) + \(\frac{20}{36}\) = \(\frac{27}{36}\) = \(\frac{3}{4}\)
7 + \(\frac{3}{4}\) = 7 \(\frac{3}{4}\)

Chapter Review/Test – Page No. 286

Question 14.
Ursula mixed 3 \(\frac{1}{8}\) cups of dry ingredients with 1 \(\frac{2}{5}\) cups of liquid ingredients. Which answer represents the best estimate of the total amount of ingredients Ursula mixed?
Options:
a. about 4 cups
b. about 4 \(\frac{1}{2}\) cups
c. about 5 cups
d. about 5 \(\frac{1}{2}\) cups

Answer: about 4 \(\frac{1}{2}\) cups

Explanation:
Ursula mixed 3 \(\frac{1}{8}\) cups of dry ingredients with 1 \(\frac{2}{5}\) cups of liquid ingredients.
3 + 1 = 4
\(\frac{1}{8}\) is closer to 0.
\(\frac{2}{5}\) is closer to \(\frac{1}{2}\)
4 + \(\frac{1}{2}\) = 4 \(\frac{1}{2}\)
Thus the correct answer is option B.

Question 15.
Samuel walks in the Labor Day parade. He walks 3 \(\frac{1}{4}\) miles along the parade route and 2 \(\frac{5}{6}\) miles home. How many miles does Samuel walk?
Options:
a. \(\frac{5}{10}\) mile
b. 5 \(\frac{1}{12}\) miles
c. 5 \(\frac{11}{12}\) miles
d. 6 \(\frac{1}{12}\) miles

Answer: 6 \(\frac{1}{12}\) miles

Explanation:
Samuel walks in the Labor Day parade.
He walks 3 \(\frac{1}{4}\) miles along the parade route and 2 \(\frac{5}{6}\) miles home.
3 + \(\frac{1}{4}\) + 2 + \(\frac{5}{6}\)
3 + 2 =5
\(\frac{5}{6}\) + \(\frac{1}{4}\) = \(\frac{10}{12}\) + \(\frac{3}{12}\) = \(\frac{13}{12}\)
\(\frac{13}{12}\) = 6 \(\frac{1}{12}\) miles
Thus the correct answer is option D.

Question 16.
A gardener has a container with 6 \(\frac{1}{5}\) ounces of liquid plant fertilizer. On Sunday, the gardener uses 2 \(\frac{1}{2}\) ounces on a flower garden. How many ounces of liquid plant fertilizer are left?
Options:
a. 3 \(\frac{7}{10}\) ounces
b. 5 \(\frac{7}{10}\) ounces
c. 6 \(\frac{7}{10}\) ounces
d. 9 \(\frac{7}{10}\) ounces

Answer: 9 \(\frac{7}{10}\) ounces

Explanation:
A gardener has a container with 6 \(\frac{1}{5}\) ounces of liquid plant fertilizer.
On Sunday, the gardener uses 2 \(\frac{1}{2}\) ounces on a flower garden.
6 + \(\frac{1}{5}\) + 2 + \(\frac{1}{2}\)
6 + 2 = 8
\(\frac{1}{5}\) + \(\frac{1}{2}\)
LCD = 10
\(\frac{2}{10}\) + \(\frac{5}{10}\) = \(\frac{7}{10}\)
8 \(\frac{7}{10}\)

Question 17.
Aaron is practicing for a triathlon. On Sunday, he bikes 12 \(\frac{5}{8}\) miles and swims 5 \(\frac{2}{3}\) miles. On Monday, he runs 6 \(\frac{3}{8}\) miles. How many total miles does Aaron cover on the two days?
Options:
a. 23 \(\frac{1}{6}\) miles
b. 24 \(\frac{7}{12}\) miles
c. 24 \(\frac{2}{3}\) miles
d. 25 \(\frac{7}{12}\) miles

Answer: 24 \(\frac{2}{3}\) miles

Explanation:
Aaron is practicing for a triathlon.
On Sunday, he bikes 12 \(\frac{5}{8}\) miles and swims 5 \(\frac{2}{3}\) miles.
On Monday, he runs 6 \(\frac{3}{8}\) miles.
5 \(\frac{2}{3}\) + 6 \(\frac{3}{8}\) = 12 \(\frac{1}{24}\)
12 \(\frac{1}{24}\) + 12 \(\frac{5}{8}\) miles
12 + \(\frac{1}{24}\) + 12 + \(\frac{5}{8}\)
12 + 12 = 24
\(\frac{1}{24}\) + \(\frac{5}{8}\) = \(\frac{1}{24}\) + \(\frac{15}{24}\) = \(\frac{16}{24}\) = \(\frac{2}{3}\)
24 + \(\frac{2}{3}\) = 24 \(\frac{2}{3}\) mile
The correct answer is option D.

Chapter Review/Test – Page No. 287

Fill in the bubble completely to show your answer.

Question 18.
Mrs. Friedmon baked a walnut cake for her class. The pictures below show how much cake she brought to school and how much she had left at the end of the day.

Which fraction represents the difference between the amounts of cake Mrs. Friedmon had before school and after school?
Options:
a. \(\frac{5}{8}\)
b. 1 \(\frac{1}{2}\)
c. 1 \(\frac{5}{8}\)
d. 2 \(\frac{1}{2}\)

Answer: 1 \(\frac{5}{8}\)

Explanation:
The fraction for the above figure is 1 \(\frac{7}{8}\)
The fraction for the second figure is \(\frac{1}{4}\)
1 + \(\frac{7}{8}\) – \(\frac{1}{4}\)
\(\frac{7}{8}\) – \(\frac{1}{4}\) = \(\frac{7}{8}\) – \(\frac{2}{8}\)
\(\frac{7}{8}\) – \(\frac{2}{8}\) = \(\frac{5}{8}\)
1 + \(\frac{5}{8}\) = 1 \(\frac{5}{8}\)
The correct answer is option C.

Question 19.
Cody is designing a pattern for a wood floor. The length of the pieces of wood are 1 \(\frac{1}{2}\) inches, 1 \(\frac{13}{16}\) inches, and 2 \(\frac{1}{8}\) inches. What is the length of the 5th piece of wood if the pattern continues?
Options:
a. 2 \(\frac{7}{6}\) inches
b. 2 \(\frac{3}{4}\) inches
c. 3 \(\frac{1}{2}\) inches
d. 4 inches

Answer: 2 \(\frac{3}{4}\) inches

Explanation:
The length of the pieces of wood are 1 \(\frac{1}{2}\) inches, 1 \(\frac{13}{16}\) inches, and 2 \(\frac{1}{8}\) inches
1 \(\frac{1}{2}\) = \(\frac{3}{2}\)
1 \(\frac{13}{16}\) inches = \(\frac{29}{16}\)
\(\frac{29}{16}\) – \(\frac{3}{2}\) = latex]\frac{5}{16}[/latex]
5th piece = \(\frac{3}{2}\) + latex]\frac{5}{16}[/latex] (5 – 1)
= \(\frac{3}{2}\) + latex]\frac{5}{16}[/latex] 4
= \(\frac{3}{2}\) + latex]\frac{20}{16}[/latex]
= \(\frac{3}{2}\) × latex]\frac{8}{8}[/latex] + latex]\frac{20}{16}[/latex]
= latex]\frac{44}{16}[/latex] = 2 latex]\frac{3}{4}[/latex]
Thus the correct answer is option B.

Question 20.
Julie spends \(\frac{3}{4}\) hour studying on Monday and \(\frac{1}{6}\) hour studying on Tuesday. How many hours does Julie study on those two days?
Options:
a. \(\frac{1}{3}\) hour
b. \(\frac{2}{5}\) hour
c. \(\frac{5}{6}\) hour
d. \(\frac{11}{12}\) hour

Answer: \(\frac{11}{12}\) hour

Explanation:
Julie spends \(\frac{3}{4}\) hour studying on Monday and \(\frac{1}{6}\) hour studying on Tuesday.
\(\frac{3}{4}\) + \(\frac{1}{6}\)
LCD = 12
\(\frac{9}{12}\) + \(\frac{2}{12}\) = \(\frac{11}{12}\) hour
So, the correct answer is option D.

Chapter Review/Test – Page No. 288

Constructed Response

Performance Task

Question 22.
For a family gathering, Marcos uses the recipe below to make a lemon-lime punch.

A). How would you decide the size of a container you need for one batch of the Lemon-Lime Punch?
Type below:
________

Answer: He may use \(\frac{1}{4}\) gallon lime juice for one batch of the lemon-lime punch.

Question 22.
B). If Marcos needs to make two batches of the recipe, how much of each ingredient will he need? How many gallons of punch will he have? Show your math solution and explain your thinking when you solve both questions.
Type below:
________

Answer: \(\frac{2}{3}\) gallon lime juice

Question 22.
C). Marcos had 1 \(\frac{1}{3}\) gallons of punch left over. He poured all of it into several containers for family members to take home. Use fractional parts of a gallon to suggest a way he could have shared the punch in three different-sized containers.
Type below:
________

Answer: 1 \(\frac{1}{12}\)

Conclusion

Answering the concepts in Go Math Grade 5 Chapter 6 Solution Key helps students to attempt the exam with confidence and prepare accordingly. Once you get to know the concept better you can solve any kind of question framed on Addition and Subtraction of Fractions with Unlike Denominators. Check your knowledge by taking an Assessment Test on Chapter 6 available.

Students who are unable to solve the textbook problems can go through the Go Math Grade 6 Answer Key Chapter 12 Data Displays and Measures of Center. Refer to HMH Go Math 6th Grade Solution Key of Chapter 12 Data Displays and Measures of Center to understand the problem-solving methods in depth. Go Math Grade 6 Answer Key was explained by the experts in a unique and simple way.

Go Math Grade 6 Answer Key Chapter 12 Data Displays and Measures of Center

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Lesson 1: Recognize Statistical Questions

• Share and Show – Page No. 651
• Problem Solving + Applications – Page No. 652
• Recognize Statistical Questions – Page No. 653
• Lesson Check – Page No. 654

Lesson 2: Describe Data Collection

• Share and Show – Page No. 657
• Summarize – Page No. 658
• Describe Data Collection – Page No. 659
• Lesson Check – Page No. 660

Lesson 3: Dot Plots and Frequency Tables

• Share and Show – Page No. 663
• Unlock the Problem – Page No. 664
• Dot Plots and Frequency Tables – Page No. 665
• Lesson Check – Page No. 666

Lesson 4: Histograms

• Share and Show – Page No. 669
• Problem Solving + Applications – Page No. 670
• Histograms – Page No. 671
• Lesson Check – Page No. 672

Mid-Chapter Checkpoint

• Mid-Chapter Checkpoint – Vocabulary – Page No. 673
• Page No. 674

Lesson 5: Investigate • Mean as Fair Share and Balance Point

• Share and Show – Page No. 677
• Page No. 678
• Mean as Fair Share and Balance Point – Page No. 679
• Lesson Check – Page No. 680

Lesson 6: Measures of Center

• Share and Show – Page No. 683
• Problem Solving + Applications – Page No. 684
• Measures of Center – Page No. 685
• Lesson Check – Page No. 686

Lesson 7: Effects of Outliers

• Share and Show – Page No. 689
• Problem Solving + Applications – Page No. 690
• Effects of Outliers – Page No. 691
• Lesson Check – Page No. 692

Lesson 8: Problem-Solving • Data Displays

• Share and Show – Page No. 695
• On Your Own – Page No. 696
• Problem Solving Data Displays – Page No. 697
• Lesson Check – Page No. 698

Chapter 12 Review/Test

• Chapter 12 Review/Test – Page No. 699
• Page No. 700
• Page No. 701
• Page No. 702
• Page No. 703
• Page No. 704

Share and Show – Page No. 651

Identify the statistical question. Explain your reasoning.

Question 1.
A. What was the low temperature in Chicago each day in March?
B. What was the low temperature in Chicago on March 7?

Answer: A is the statistical question.

Explanation: As in A temperature was asked for each day and in B temperature was asked for only one day.

Question 2.
A. How long did it take you to get to school this morning?
B. How long did it take you to get to school each morning this week?

Answer: B is the statistical question.

Explanation: In B it was asked for each morning in a week and in A It was asked for only for this morning.

Write a statistical question you could ask in the situation.

Question 3.
A student recorded the number of pets in the households of 50 sixth-graders.

Answer: How many households have one or more pets?

On Your Own

Identify the statistical question. Explain your reasoning.

Question 4.
A. How many gold medals has Finland won at each of the last 10 Winter Olympics?
B. How many gold medals did Finland win at the 2008 Winter Olympics?

Answer: A is the statistical question.

Explanation: A asks about the no.of medals won at 10 different times and in B asks about no.of medals won at 1 time.

Write a statistical question you could ask in the situation.

Question 5.
A wildlife biologist measured the length of time that 17 grizzly bears hibernated.

Answer: What was the least amount of time grizzly bears hibernated?

Question 6.
A doctor recorded the birth weights of 48 babies.

Answer: What was the highest birth weight recorded?

Problem Solving + Applications – Page No. 652

Use the table for 7 and 8.

Question 7.
Give a statistical question that you could ask about the data recorded in the table.

Answer: Which Roller coaster reaches the maximum height?

Question 8.
What statistical question could “92 mi/hr” be the answer to?

Answer: What is the maximum speed of Roller Coasters?

Question 9.
Explain A video game company will make a new game. The manager must choose between a roleplaying game and an action game. He asks his sales staff which of the last 10 released games sold the most copies. Explain why this is a statistical question.

Answer: As the manager asks his sales staff about the last 10 released games and most sold-out copies, so it is a statistical question.

Question 10.
Think of a topic. Record a set of data for the topic. Write a statistical question that you could ask about your data.

Answer: John spend 10 hours to watch TV in each week.

Explanation: What was the time spent by the John to watch TV in each week?

Question 11.
For numbers 11a–11d, choose Yes or No to indicate whether the question is a statistical question.
11a. How many minutes did it take Ethan to complete his homework last night?
11b. How many minutes did it take Madison to complete her homework each night this week?
11c. How many more minutes did Andrew spend on homework on Tuesday than on Thursday?
11d. What was the longest amount of time Abigail spent on homework this week?
11a. ____________
11b. ____________
11c. ____________
11d. ____________

Answer:
11a. No.

Explanation: The question asked only about last night’s homework, so it is not a statistical question.

11b. Yes

Explanation: As the question was asked for each night in a week, so it is a statistical question.

11c. No

Explanation: The question is about the differences in the duration of homework at a time, so it is not a statistical question.

11d. Yes

Explanation: As the question is on the longest amount of time on homework, so it is a statistical question.

Recognize Statistical Questions – Page No. 653

Identify the statistical question. Explain your reasoning.

Question 1.
A. How many touchdowns did the quarterback throw during the last game of the season?
B. How many touchdowns did the quarterback throw each game of the season?

Answer: B is the statistical question.

Explanation: In A asks for no.of touchdowns in the last game and in B asks for no.of touchdowns in each game.

Question 2.
A. What was the score in the first frame of a bowling game?
B. What are the scores in 10 frames of a bowling game?

Answer: B is the statistical question.

Explanation: In A, asks for only for the first frame, and in B asks for 10 frames and score in each frame.

Question 3.
A. How many hours of television did you watch each day this week?
B. How many hours of television did you watch on Saturday?

Answer: A is the statistical question.

Explanation: In A, the question was asked for no.of hour’s television for each day in a week. And in B the question was asked for only for Saturday.

Write a statistical question you could ask in the situation.

Question 4.
A teacher recorded the test scores of her students.

Answer: What was the highest test score recorded?

Question 5.
A car salesman knows how many of each model of a car was sold in a month.

Answer: What was the least sold model of the car?

Problem Solving

Question 6.
The city tracked the amount of waste that was recycled from 2000 to 2007. Write a statistical question about the situation.

Answer: What was the amount of waste that was recycled for each year from 2000 to 2007?

Question 7.
The daily low temperature is recorded for a week. Write a statistical question about the situation.

Answer: What was the daily low temperature recorded each day this week?

Question 8.
Write three statistical questions that you could use to gather data about your family. Explain why the questions are statistical.

Answer:
Which family member was oldest?
Which family member was tallest?
Which family member has the highest income?

Lesson Check – Page No. 654

Question 1.
Elise says that the question “Do you have any siblings?” is a statistical question. Mark says that “How many siblings do you have?” is a statistical question. Who is correct?

Answer: How many siblings do you have? is a statistical question. So Mark is correct.

Question 2.
Kate says that “What was the lowest amount of precipitation in one month last year?” is a statistical question. Mike says that “What is the speed limit?” is a statistical question. Who is correct?

Answer: What was the lowest amount of precipitation in one month last year?. Is a statistical question. So Kate is correct.

Spiral Review

Question 4.
Mikki uses the net shown to make a solid figure.

What solid figure does Mikki make?

Answer: Triangular pyramid.

Explanation: Mikki makes a Triangular pyramid.

Question 5.
A prism is filled with 30 cubes with \(\frac{1}{2}\)-unit side lengths. What is the volume of the prism in cubic units?
_______ \(\frac{□}{□}\) cubic units.

Answer: 3.75 cubic units.

Explanation: As it takes 8 cubes with a side length of ½ to form a unit cube, so the volume of the cube is 308= 3.75 cubic units.

Share and Show – Page No. 657

Describe the data set by listing the attribute measured, the unit of measure, the likely means of measurement, and the number of observations.

Question 1.
Greg’s 100-meter race results.

Answer:
The attribute is the Duration of run data.
The unit of measure is Seconds.
The likely means of measurement was taken on Stopwatch.
No.of observations are 7.

Question 2.
The Andrews family’s water use.

Answer:
The attribute is the amount of water used daily.
The unit of measure is Gallons.
The likely means of measurement was taken on Water meter.
No.of observations are 14.

On Your Own

Question 3.
Practice: Copy and Solve Collect data on one of the topics listed below. You may wish to work with other students. Make a chart of your results. Then describe the data set.

• Weights of cereal boxes, soup cans, or other items
• Numbers of family members
• Lengths of time to multiply two 2-digit numbers
• Numbers of pets in families
• Lengths of forearm (elbow to fingertip)
• Numbers of pages in books

Answer:
The attribute is the Number of pages in books.
The unit of measure is Numbers.
The likely means of measurement were counting.
No.of observations are 6.

Question 4.
Describe the data set by writing the attribute measured, the unit of measure, the likely means of measurement, and the number of observations in the correct location on the chart.

Answer:

Summarize – Page No. 658

When you summarize a reading passage, you restate the most important information in a shortened form. This allows you to understand more easily what you have read. Read the following passage:

A biologist is studying green anacondas. The green anaconda is the largest snake in the world. Finding the length of any snake is difficult because the snake can curl up or stretch out while being measured. Finding the length of a green anaconda is doubly difficult because of the animal’s great size and strength.

The standard method for measuring a green anaconda is to calm the snake, lay a piece of string along its entire length, and then measure the length of the string. The table at the right gives data collected by the biologist using the string method.

Question 5.
Analyze Summarize the passage in your own words

Answer: The world’s largest snake in the world is green anacondas. Finding the length of these green anacondas is very difficult because of the animal’s great size and strength. So there is a standard method for measuring. Firstly calm the snake, then lay a piece of string along its entire length, and then measure the length of the string.

Question 6.
Use your summary to name the attribute the biologist was measuring. Describe how the biologist measured this attribute.

Answer:
The attribute biologist was measuring green anacondas. And the biologist measured by laying a piece of string along its entire length, and then measuring the length of the string.

Question 8.
Write the greatest green anaconda length that the biologist measured in feet. Round your answer to the nearest foot. (Hint: 1 foot is equal to about 30 centimeters.)

Answer: 507.5 cm, 17 feet

Explanation: The greatest green anaconda length that the biologist measured was 507.5 cm. As 1 foot= 30 cm, so 507.5÷30= 16.9 feet round off to 17 feet.

Describe Data Collection – Page No. 659

Describe the data set by listing the attribute measured, the unit of measure, the likely means of measurement, and the number of observations.

Question 1.
Daily temperature

Answer:
The attribute is Daily Temperature.
The unit of measure is Fahrenheit.
The likely means of measurement is Thermometer
No.of observations are 25.

Question 2.
Plant heights

Answer:
The attribute is the Height of plants
The unit of measure in inches.
The likely means of measurement is the Ruler.
No.of observations are 10.

Question 3.
Cereal in boxes

Answer:
The attribute is the Amount of Cereal in Boxes
The unit of measure is Cup.
The likely means of measurement is measuring cup
No.of observations are 16.

Question 4.
Dog weights

Answer:
The attribute is Dog weights.
The unit of measure is Pounds.
The likely means of measurement is scale.
No.of observations are 8.

Problem Solving

Question 5.
The table below gives the amount of time Preston spends on homework. Name the likely means of measurement.

Answer:
The likely means of measurement is Clock.

Question 6.
The table below shows the speed of cars on a highway. Name the unit of measure.

Answer: The unit of measure is miles per hour.

Question 7.
Gather data about the heights of your family members or friends. Then describe how you collected the data set.

Answer:

Lesson Check – Page No. 660

Question 1.
What is the attribute of the data set shown in the table?

Answer: The attribute is the Mass of produce.

Question 2.
What is the number of observations of the data set shown below?

Answer:
No.of observation: 6

Spiral Review

Question 3.
What is the area of the figure shown below?

_______ cm²

Answer: 23 cm²

Explanation:
Area of the rectangle= Length×width
= 7×4.5
= 31.5 cm²
Area of trapezoid= 1/2 ×(b1+b2)×h
= 1/2 ×(7+4.5)×4
= 11.5×2
= 23 cm²

Question 5.
How much sand can this container hold?

_______ in.³

Answer: 225 in.³

Explanation:
Volume= Length×width×Height
= 5×10×4 1/2
= 5 × 10 × 5/2
= 5×5×9
= 225 in.³

Question 6.
Jay says that “How much does Rover weigh today?” is a statistical question. Kim says that “How long are the puppies’ tails in the pet store?” is a statistical question. Who is NOT correct?

Answer: “How much does Rover weigh today?” is not correct as it is not a statistical question.

Share and Show – Page No. 663

For 1−4, use the data at right.

Question 1.
Complete the dot plot.

Answer:

Question 2.
What was the most common distance Lionel biked? How do you know?

Answer: The most common distance Lionel biked is 6 km.

Question 3.
Make a frequency table. Use the intervals 1−3 km, 4−6 km, 7−9 km, and 10−12 km.

Answer:

Question 4.
Make a relative frequency table. Use the same intervals as in Exercise 3.

Answer:
As there are 25 data values, so
1-3 km 8÷25= 0.32= 32% relative frequency.
4-6 km 9÷25= 0.36= 36% relative frequency.
7-9 km 4÷25= 0.16= 16% relative frequency.
10-12 km 4÷25= 0.16= 16% relative frequency.

On Your Own

Practice: Copy and Solve For 5−9, use the table.

Question 5.
Make a dot plot of the data.

Answer:

Question 6.
Make a frequency table of the data with three intervals.

Answer:

Question 7.
Make a relative frequency table of the data with three intervals.

Answer:
As there are 25 data values, so
3-7    3÷25= 0.12= 12% relative frequency.
8-12  7÷25= 0.28= 28% relative frequency.
13-17 15÷25= 0.6= 60% relative frequency.

Question 8.
Describe how you decided on the intervals for the frequency table.

Answer: As we found 3 intervals of equal size that cover the full range of data values.

Question 9.
Could someone use the information in the frequency table to make a dot plot? Explain.

Answer: No. Because the data is grouped in intervals, but a dot plot requires individual data values.

Unlock the Problem – Page No. 664

Question 10.
The manager of a fitness center asked members to rate the fitness center. The results of the survey are shown in the frequency table. What percent of members in the survey rated the center as excellent or good?

a. What do you need to find?

Answer: We need to find what percent of members in the survey rated the center as excellent or good.

Question 10.
b. How can you use relative frequency to help you solve the problem?

Answer: We can solve by adding the relative frequencies of excellent and good responses.

Question 10.
c. Show the steps you use to solve the problem.

Answer: 30%, 25%.

Explanation:
The total no.of responses are 18+15+21+6= 60. So the percent for excellent and good responses are
18÷60= 0.3= 30%
15÷60= 0.25= 25%

Question 10.
d. Complete the sentences.

Answer:
The percent of members who were rated excellent is 30%
The percent of members who were rated good is 25%
So total members rated excellent and good are 30%+25%= 55%

Question 11.
Use the table above. What is the difference in percent of the members in the survey that rated the fitness center as poor versus excellent?
_________ %

Answer: 20%.

Explanation: The percents for poor responses are 6÷60= 0.1= 10%, so the difference in the percent of the members in the survey that rated the fitness center as poor versus excellent is 30%-10%= 20%.

Question 12.
Julie kept a record of the number of minutes she spent reading for 20 days. Complete the frequency table by finding the frequency and the relative frequency (%).

Answer:
As there are 20 data values, so
30 mins 8÷20= 0.4= 40% relative frequency.
45 mins 4÷20= 0.2= 20% relative frequency.
60 mins 3÷20= 0.15= 15% relative frequency.

Dot Plots and Frequency Tables – Page No. 665

For 1–4, use the chart.

Question 1.
The chart shows the number of pages of a novel that Julia reads each day. Complete the dot plot using the data in the table.

Answer:

Question 2.
What number of pages does Julia read most often? Explain.

Answer: Julia reads most often 15 pages because we can see in the dot plot as 15 was the highest.

Question 3.
Make a frequency table in the space below. Use the intervals 10–13, 14–17, and 18–21.

Answer:

Question 4.
Make a relative frequency table in the space below.

Answer:
As there are 20 data values, so
10-13  7÷20= 0.35= 35% relative frequency.
14-17  9÷20= 0.45= 45% relative frequency.
18-21  4÷20= 0.2=  20% relative frequency.

Problem Solving

Question 5.
The frequency table shows the ages of the actors in a youth theater group. What percent of the actors are 10 to 12 years old?

_______ %

Answer: 55%

Explanation:
As there are 8+22+10= 40 data values, so the percent of the actors are 10 to 12 years old is
22÷40= 0.55= 55%.

Lesson Check – Page No. 666

Question 1.
The dot plot shows the number of hours Mai babysat each week. How many hours is Mai most likely to babysit?

_______ hours

Answer: Mai is most likely to babysit for 9 hours.

Explanation: As we can see dot plot with the highest dots is 9 hours. So Mai is most likely to babysit for 9 hours.

Question 2.
The frequency table shows the ratings that a movie received from online reviewers. What percent of the reviewers gave the movie a 4-star rating?

_______ %

Answer: 30%.

Explanation: The percent of reviewers gave the movie a 4-star rating are 6/20
= 0.30
= 30%

Spiral Review

Question 6.
For a science experiment, Juanita records the height of a plant every day in centimeters. What is the attribute measured in her experiment?

Answer: The attribute measured in her experiment was height.

Share and Show – Page No. 669

For 1–4, use the data at right.

Question 1.
Complete the frequency table for the age data in the table at right.

Answer:

Question 2.
Complete the histogram for the data.

Answer:

Question 3.
Use your histogram to find the number of people at the health club who are 30 or older.
_______ people

Answer: 7 people.

Explanation: The people from 30-39 are 5 and from 40-49 are 2 people.

Question 4.
Use your histogram to determine the percentage of the people at the health club who are 20–29 years old.
_______ %

Answer: 40%.

Explanation: The data value is 2+6+5+2= 15, so the percent of the people at the health club who are 20–29 years old are
6÷15= 0.4= 40%.

On Your Own

Practice: Copy and Solve For 5–7, use the table.

Question 5.
Make a histogram of the data using the intervals 10–19, 20–29, and 30–39.

Answer:

Question 6.
Make a histogram of the data using the intervals 10–14, 15–19, 20–24, 25–29, 30–34, and 35–39.

Answer:

Question 7.
Compare Explain how using different intervals changed the appearance of your histogram.

Answer: In the histogram, smaller intervals shows that most of the data are clustered between 15 and 24. And larger intervals show that the data is evenly spread out.

Problem Solving + Applications – Page No. 670

The histogram shows the hourly salaries, to the nearest dollar, of the employees at a small company. Use the histogram to solve 8–11.

Question 8.
How many employees make less than $20 per hour?
_______ employees

Answer: 7 employees.

Explanation: 7 employees make less than $20 per hour.

Question 10.
Pose a Problem Write and solve a new problem that uses the histogram.

Answer: How many employees make more than $40 per hour?

Explanation: 3 employees.

Question 11.
Analyze Describe the overall shape of the histogram. What does this tell you about the salaries at the company?

Answer: The histogram shows that the employees are high at the interval of $25-$29 and it tells us that there is the same number of employees with salaries less than $25 as there are with salaries greater than $29.

Question 12.
The frequency table shows the TV ratings for the show American Singer. Complete the histogram for the data.

Answer:

Histograms – Page No. 671

For 1–4 use the data at right.

Question 1.
Complete the histogram for the data.

Answer:

Question 2.
What do the numbers on the y-axis represent?

Answer: The numbers on the y-axis represent the number of students.

Question 3.
How many students scored from 60 to 69?
_______ students

Answer: 3 students scored 60 to 69

Question 4.
Use your histogram to find the number of students who got a score of 80 or greater. Explain.
_______ students

Answer: 12 students.

Explanation: Students who scored 80-89 are 8 students and students who scored 90-99 are 4 students. So total students are
8+4= 12 students.

Problem Solving

For 5–6, use the histogram.

Question 5.
For which two age groups are there the same number of customers?

Answer: The same number of customers is 10-19 and 50-59.

Question 6.
How many customers are in the restaurant? How do you know?
_______ customers

Answer: 63 customers

Explanation: Total number of customers are 6+9+13+11+15+9= 63 customers. By adding all frequencies we can get a number of customers.

Question 7.
Write a letter to another student that explains how to make a histogram and what type of data a histogram displays.

Answer: A histogram represents a bar graph with a vertical axis and a horizontal axis. The histogram displays the vertical axis with frequencies and the horizontal axis with a certain amount of intervals. We must place the intervals from lower to higher, and the height of each bar should be equal to the frequency of its corresponding intervals.

Lesson Check – Page No. 672

Question 1.
The histogram shows the amount, to the nearest dollar, that customers spent at a museum gift shop. How many customers spent less than $20?

_______ customers

Answer: 14 customers.

Explanation: The number of customers who spent less than $20 is 8+6=14 customers.

Question 2.
Use the histogram in Problem 1. How many customers bought something at the gift shop?
_______ customers

Answer: 27 customers.

Explanation: The number of customers who bought something at the gift shop is 8+6+7+4+2= 27 customers.

Spiral Review

Question 3.
Marguerite drew a rectangle with vertices A(−2, −1), B(−2, −4), and C(1, −4). What are the coordinates of the fourth vertex?

Answer: As Marguerite draw a rectangle, so the fourth vertex is D(1,-1)

Explanation:

Question 5.
DeShawn is using this frequency table to make a relative frequency table. What percent should he write in the Relative Frequency column for 5 to 9 push-ups?

_______ %

Answer: 35%

Explanation: Data Values are 3+7+8+2= 20

As there are 3+7+8+2= 20 data values, so
5-9  7÷20= 0.35= 35% relative frequency.

Mid-Chapter Checkpoint – Vocabulary – Page No. 673

Choose the best term from the box to complete the sentence.

Question 1.
A _____ is a kind of bar graph that shows the frequency of data grouped into intervals.

Answer: A histogram is a kind of bar graph that shows the frequency of data grouped into intervals.

Question 2.
A question that asks about a set of data that varies is called a _____.

Answer: A question that asks about a set of data that varies is called a statistical question.

Concepts and Skills

Question 3.
A sports reporter records the number of touchdowns scored each week during the football season. What statistical question could the reporter ask about the data?

Answer: What was the greatest number of touchdowns scored in one week?

Question 4.
Flora records her pet hamster’s weight once every week for one year. How many observations does she make?
_______ observations

Answer: 52 observations.

Explanation: As there are 52 weeks in a year, so Flora makes 52 observations.

Question 5.
The number of runs scored by a baseball team in 20 games is given below. Draw a dot plot of the data and use it to find the most common number of runs scored in a game.

Answer:

Page No. 674

Question 6.
Write a statistical question you could ask about a set of data that shows the times visitors arrived at an amusement park.

Answer: How many visitors arrived at an amusement park each hour?

Question 7.
A school principal is trying to decide how long the breaks should be between periods. He plans to time how long it takes several students to get from one classroom to another. Name a tool he could use to collect the data.

Answer: He could use it to collect the data by stopwatch.

Question 8.
The U.S. Mint uses very strict standards when making coins. On a tour of the mint, Casey asks, “How much copper is in each penny?” Lenny asks, “What is the value of a nickel?” Who asked a statistical question?

Answer: Casey asked a statistical question.

Question 10.
The table shows the lengths of the songs played by a radio station during a 90-minute period. Alicia is making a histogram of the data. What frequency should she show for the interval 160–169 seconds?

Answer: 5.

Explanation: As there are 5 values between 160-169, so Alicia makes a frequency of 5 for the interval 160-169 seconds.

Share and Show – Page No. 677

Use counters to find the mean of the data set.

Make a dot plot for the data set and use it to check whether the given value is a balance point for the data set.

Question 2.
Rosanna’s friends have 0, 1, 1, 2, 2, and 12 pets at home. Rosanna says the mean of the data is 3. Is Rosanna correct?

Answer:

Explanation: Yes, Rosanna is correct. As the mean is
= \(\frac{0+1+1+2+2+12}{6}\)
= \(\frac{18}{6}\)
= 3
So Rosanna is correct.

Problem Solving + Applications

Question 3.
Four people go to lunch, and the costs of their orders are $6, $9, $10, and $11. They want to split the bill evenly. Find each person’s fair share. Explain your work.
Each person’s fair share is $ _______ .

Answer: $9.

Explanation: Each person’s fair share is
= \(\frac{ $6+$9+$10+$11}{4}
=\frac{$36}{4}\)
= $9.

Page No. 678

Use the table for 4–6.

Question 4.
A grocer is preparing fruit baskets to sell as holiday presents. If the grocer rearranges the apples in baskets A, B, and C so that each has the same number, how many apples will be in each basket? Use counters to find the fair share.
_______ apples

Answer: 3 apples.

Explanation: Mean for the apples are
= \(\frac{4+1+4}{3}
=\frac{9}{3}\)
= 3
So there will be 3 apples in each basket.

Question 5.
Make Arguments Can the pears be rearranged so that there is an equal whole number of pears in each basket? Explain why or why not.

Answer: No pears cannot rearrange, as three stacks of counters height are 2,1,5 so that there is an equal number in each stack, So we cannot rearrange.

Question 6.
Use counters to find the mean of the number of pears originally in baskets B and C. Draw a dot plot of the data set. Use your plot to explain why the mean you found is a balance point.

Answer: Mean= 3

Explanation: Mean= \(\frac{1+5}{2}
=\frac{6}{2}\)
= 3
As data point 1 is 2 times less than the mean and data point 5 is 2 times greater than the mean, so the points are the same distance from the mean and the mean is the balance point.

Question 7.
Four friends go to breakfast and the costs of their breakfasts are $5, $8, $9, and $10. Select True or False for each statement.
7a. The mean of the cost of the breakfasts can be found by adding each of the costs and dividing that total by 4.
7b. The mean cost of the four breakfasts is $10.
7c. The difference between the greatest cost and the mean is $2.
7d. The difference between the least cost and the mean is $2.

Answer:
7a. True

Explanation: As mean = (sum of the terms)/ (No.of terms)

7b. False

Explanation: The mean cost of the four breakfast is \(\frac{$5+$8+$9+$10}{4}
=\frac{$32}{4}\)
= $8.

7c. True

Explanation: The difference between the greatest cost and the mean is $10-$8= $2.

7d. False

Explanation: The difference between the least cost and the mean is $8-$5= $3

Mean as Fair Share and Balance Point – Page No. 679

Use counters to find the mean of the data set.

Make a dot plot for the data set and use it to check whether the given value is a balance point for the data set.

Question 3.
Sandy’s friends ate 0, 2, 3, 4, 6, 6, and 7 pretzels. Sandy says the mean of the data is 4. Is Sandy correct?

Answer: Yes, Sandy is correct.

Explanation: The mean of the data set is \(\frac{0+2+3+4+6+6+7}{7}
=\frac{28}{7}\)
= 4.

Problem Solving

Question 4.
Three baskets contain 8, 8, and 11 soaps. Can the soaps be rearranged so that there is an equal whole number of soaps in each basket? Explain why or why not.

Answer: Yes, the soaps can be rearranged.

Explanation: As the mean is 9, the soaps can be rearranged so that there is an equal whole number of soaps in each basket. And we can fit 9 in each group.

Question 5.
Five pages contain 6, 6, 9, 10, and 11 stickers. Can the stickers be rearranged so that there is an equal whole number of stickers on each page? Explain why or why not.

Answer: No

Explanation: No, we cannot rearrange the stickers. As there is a 5 stack counter which is unable to fit in for 6, 6, 9, 10, and 11 stickers

Question 6.
Describe how to use counters to find the mean of a set of data. Give a data set and list the steps to find the mean.

Answer: We will start with an unequal stack then we will move a counter from the tallest stack to the shortest stack and we will repeat it until the stacks have the same height.

Lesson Check – Page No. 680

Spiral Review

Question 3.
What figure does the net below represent?

Answer: As the net has 6 equal square surfaces, it represents a cube.

Question 4.
Sarah paints the box below. She paints the whole box except for the front face. What area of the box does she paint?

_______ cm²

Answer: 586 cm²

Explanation: The area of the box is
= 2×20×7 + 2 ×9×7 + 9×20
= 280+126+180
= 586 cm²

Question 5.
Chloe collected data and then displayed her results in the table to the right. What is the unit of measure of the data?

Answer: The unit of measure is Fahrenheit.

Share and Show – Page No. 683

Question 2.
Julie goes to several grocery stores and researches the price of a 12 oz bottle of juice. Find the mean, median, and mode of the prices shown.

Answer:
The mean is $1.21
The Median is $2.08
The mode is $0.99

Explanation:
The mean is \(\frac{$0.95+$1.09+$0.99+$1.25+$0.99+$1.99}{6}
=\frac{$7.26}{6}\)
= $1.21.
First, we must set the data from smallest to greatest
$0.95,$0.99,$0.99,$1.09,$1.25,$1.99 as the count is even number we will add both middle numbers and will divide with 2
so, the median is $0.99+$1.09= $2.08÷2
= $1.04.
As $0.99 appears twice the mode is $0.99.

On Your Own

Question 3.
T.J. is training for the 200-meter dash event for his school’s track team. Find the mean, median, and mode of the times shown in the table.

Answer:
The mean is $1.21
The Median is $2.08
The mode is $0.99

Explanation:
The mean is \(\frac{22.3+22.4+23.3+24.5+22.5}{5}
=\frac{115}{5}\)
= 23
First, we must set the data from smallest to greatest
22.3,22.4,22.5,23.3,24.5
so, the median is 22.5
As no value appear twice there is no mode.

Problem Solving + Applications – Page No. 684

Sense or Nonsense?

Question 6.
Jeremy scored 85, 90, 72, 88, and 92 on five math tests, for a mean of 85.4. On the sixth test he scored a 95. He calculates his mean score for all 6 tests as shown below, but Deronda says he is incorrect. Whose answer makes sense? Whose answer is nonsense? Explain your reasoning.

Jeremy’s Work:
The mean of my first 5 test scores was 85.4, so to find the mean of all 6 test scores, I just need to find the mean of 85.4 and 95.
Mean = \(\frac{85.4+95}{2}=\frac{180.4}{2}\) = 90.2.
So, my mean score for all 6 tests is 90.2.

Deronda’s Work:
To find the mean of all 6 test scores, you need to add up all 6 scores and divide by 6.
Mean = \(\frac{85+90+72+88+92+95}{6}=\frac{522}{6}\) = 87.
So, Jeremy’s mean score for all 6 tests is 87.

Answer: Jeremy’s answer is nonsense, Deronda’s answer makes sense.

Explanation: Jeremy should add all 6 test scores and then divide the sum by 6. And Deronda used the mean formula in the right way.

Measures of Center – Page No. 685

Use the table for 1–4.

Question 1.
What is the mean of the data?
The mean is _______ points.

Answer: 9.4 points.

Explanation: The mean is \(\frac{10+8+11+12+6}{5}
=\frac{47}{5}\)
= 9.4

Question 2.
What is the median of the data?
The median is _______ points.

Answer: 10 points.

Explanation: The median is 6,8,10,11,12
10.

Question 3.
What is the mode(s) of the data?

Answer: No mode.

Explanation: As there are no repeated values, so there is no mode.

Question 4.
Suppose Blaine played a sixth game and scored 10 points during the game. Find the new mean, median, and mode.

Answer:
Mean 9.5.
Median 10.
Mode 10.

Explanation:
The mean is \(\frac{10+8+11+12+6+10}{6}
= \frac{57}{6}\)
= 9.5.
The median is 6,8,10,10,11,12
= \(\frac{10+10}{2}
= \frac{20}{2}\)
= 10.
As 10 is repeated, so the mode is 10.

Problem Solving

Question 5.
An auto manufacturer wants their line of cars to have a median gas mileage of 25 miles per gallon or higher. The gas mileage for their five models are 23, 25, 26, 29, and 19. Do their cars meet their goal? Explain.

Answer: 25.

Explanation:
The median is 19,23,25,26,29
25.
So the car meets its goal.

Question 7.
Explain how to find the mean of a set of data.

Answer: By dividing the sum of data by the number of data we can find the mean.

Lesson Check – Page No. 686

Question 1.
The prices for a video game at 5 different stores are $39.99, $44.99, $29.99, $35.99, and $31.99. What is the mode(s) of the data?

Answer: As there are no repeated values, so there is no mode.

Question 2.
Manuel is keeping track of how long he practices the saxophone each day. The table gives his practice times for the past five days. What is the mean of his practice times?

Answer: 39.

Explanation: The mean is \(\frac{25+45+30+65+30}{5}
= \frac{195}{5}\)
= 39.

Spiral Review

Question 3.
What is the surface area of the triangular prism shown below?

_______ cm²

Answer: 1008 cm²

Explanation: The Surface area triangular prism= 25×9+25×12+25×15+2×12×9×12
= 225+300+375+108
= 1008 cm²

Question 4.
Kate records the number of miles that she bikes each day. She displayed the number of daily miles in the dot plot below. Each dot represents the number of miles she biked in one day. How many days did she bike 4–7 miles?

Answer: 7 days

Explanation: By counting dots from 4-7 we will get to know how many days she bikes. So it is for 7 days.

Share and Show – Page No. 689

Question 1.
Find the outlier by drawing a dot plot of the data.

Answer: The outlier is 15.

Explanation:

Question 3.
Identify the outlier in the data set of melon weights. Then describe the effect the outlier has on the mean and median.

Answer: The outlier is 14.

Explanation:
The mean with outlier is \(\frac{47+45+48+45+49+47+14+45+51+46+47}{11}
= \frac{$484}{11}\)
= 44 oz.
The mean without outlier is \(\frac{47+45+48+45+49+47+45+51+46+47}{10}
= \frac{$470}{10}\)
= 47 oz.
The outlier decreases the mean from 47 to 44 oz.
The median is 14,45,45,45,46,47,47,47,48,49,51.
= 47
There is no change in the median with the outlier.

Question 4.
Use Reasoning In a set of Joanne’s test scores, there is an outlier. On the day of one of those tests, Joanne had the flu. Do you think the outlier is greater or less than the rest of her scores? Explain

Answer: The outlier is less than the rest of her score because if Joanne had the flu her test score from the day is probably lower thn her score.

Problem Solving + Applications – Page No. 690

Use the table for 5–7.

Question 5.
Which player’s number of stolen bases is an outlier?

Answer: Rickey Henderson.

Explanation: The player is Rickey Henderson number 1,406 is an outlier.

Question 6.
What effect does the outlier have on the median of the data set?

Answer: The outlier increases from 905.5 to 914.

Explanation: The median with outlier is 914 and the median without outlier is \(\frac{897+914}{2}
= \frac{1811}{2}\)
=  905.5
The outlier increases from 905.5 to 914.

Question 7.
Miguel wrote that the mean of the data set is 992.6. Is this the mean with or without the outlier? Explain how you can tell without doing a calculation.

Answer: This is the mean is with outlier as the mean 992.6 is greater than the data values except for the outlier.

Question 8.
Does an outlier have any effect on the mode of a data set? Explain

Answer: The outlier will not effect the mode of a data set because an outlier must be greater or lesser than the data value, so it cannot be the same as any other data value.

Effects of Outliers – Page No. 691

Question 1.
Identify the outlier in the data set of students in each class. Then describe the effect the outlier has on the mean and median.

Answer: The outlier is 12. The outlier decreases both mean and median.

Explanation:
The mean with outlier is \(\frac{30+22+26+21+24+28+23+26+28+12}{10}
= \frac{240}{10}\)
= 24.
The mean without outlier is \(\frac{30+22+26+21+24+28+23+26+28}{9}
= \frac{228}{9}\)
= 25.3.
The outlier decreases mean from 24 to 25.3.
The median with outlier is 12,21,22,23,24,26,26,28,28,30.
= \(\frac{24+26}{2}
= \frac{50}{2}\)
= 25
The median without outlier is 21,22,23,24,26,26,28,28,30.
= 26
The outlier decreases both mean and median.

Question 2.
Identify the outlier in the data set of pledge amounts. Then describe the effect the outlier has on the mean and median.

Answer: The outlier is $100. The outlier increases mean from $22 to $31.75 and no effect on median.

Explanation:
The mean with outlier is \(\frac{$100+$10+$15+$20+$17+$20+$32+$40}{8}
= \frac{$254}{8}\)
= $31.75.
The mean without outlier is \(\frac{$10+$15+$20+$17+$20+$32+$40}{7}
= \frac{$154}{7}\)
= 22.
The outlier increases mean from $22 to $31.75.
The median with outlier is $10,$15,$17,$20,$20,$32,$40,$100.
= \(\frac{$20+$20}{2}
= \frac{$40}{2}\)
= $20
The median without outlier is $10,$15,$17,$20,$20,$32,$40.
= $20.
The outlier has no effect on median.

Problem Solving

Question 3.
Duke’s science quiz scores are 99, 91, 60, 94, and 95. Describe the effect of the outlier on the mean and median.

Answer: The outlier is 60. The outlier decreases the mean from 94.75 to 87.8 and decreases the median from 94.5 to 94.

Explanation:
The mean with outlier is \(\frac{99+91+60+94+95}{5}
= \frac{439}{5}\)
= 87.8.
The mean without outlier is \(\frac{99+91+94+95}{4}
= \frac{379}{4}\)
= 94.75.
The outlier decreases mean from 94.75 to 87.8.
The median with outlier is 60,91,94,95,99.
= 94.
The median without outlier is 91,94,95,99.
= \(\frac{94+95}{2}
= \frac{189}{2}\)
= 94.5
The outlier decreases the median from 94.5 to 94.

Question 4.
The number of people who attended an art conference for five days was 42, 27, 35, 39, and 96. Describe the effect of the outlier on the mean and median.

Answer: The outlier is 96. The outlier increases the mean from 35.75 to 47.8 and increases the median from 37 to 39.

Explanation:
The mean with outlier is \(\frac{42+27+35+39+96}{5}
= \frac{239}{5}\)
=47.8 .
The mean without outlier is \(\frac{42+27+35+39}{4}
= \frac{143}{4}\)
= 35.75.
The outlier increases mean from 35.75 to 47.8.
The median with the outlier is 27,35,39,42,96.
= 39.
The median without outlier is 27,35,39,42.
= \(\frac{35+39}{2}
= \frac{74}{2}\)
= 37.
The outlier increases the median from 37 to 39.

Question 5.
Find or create a set of data that has an outlier. Find the mean and median with and without the outlier. Describe the effect of the outlier on the measures of center.

Answer:

Lesson Check – Page No. 692

Question 1.
What is the outlier for the data set?
19, 19, 27, 21, 77, 18, 23, 29

Answer: The outlier is 77.

Explanation: As 77 is not fit in the data set, so 77 is an outlier.

Spiral Review

Question 3.
Hector covers each face of the pyramid below with construction paper. The area of the base of the pyramid is 28 square inches. What area will he cover with paper?

_______ in.²

Answer: 196 in.²

Explanation:
Area= 1/2 bh
= 1/2 × 8×14
= 4×14
= 56 in.²
The surface area is 28+3×56
= 28+168
= 196 in.²

Question 4.
Mr. Stevenson measured the heights of several students and recorded his findings in the chart below. How many observations did he complete?

Answer: No.of observations are 14.

Question 5.
Kendra is making a histogram for the data in the chart. She uses the intervals 0–4, 5–9, 10–14, and 15–19. What should be the height of the longest bar in her histogram?

Answer: So the height of the longest bar is 5.

Explanation:
The frequency of intervals from 0-4 is 2.
The frequency of intervals from 5-9 is 5.
The frequency of intervals from 10-14 is 4.
The frequency of intervals from 15-19 is 4.
So the height of the longest bar is 5.

Share and Show – Page No. 695

Question 1.
The table shows the number of goals scored by the Florida Panthers National Hockey League team in the last 20 games of the 2009 season. What was the most common number of goals the team scored?

Answer: The most common number of goals the team scored is 2.

Explanation: As 2 has appeared 6 times.

Question 2.
Draw a histogram of the hockey data. Use it to find the percent of the games in which the Panthers scored more than 3 goals.

Answer:

Explanation:

Question 3.
Use Appropriate Tools If you needed to find the mean of a data set, which data display—dot plot or histogram—would you choose? Explain your reasoning.

Answer: To find the mean data set the best tool is a dot plot because in a dot plot we can add all the data values to find the mean but in a histogram does not show individual values.

On Your Own – Page No. 696

Question 4.
Corey collected data on the ages of the parents of his classmates. Make a data display and use it to find the percent of parents at least 30 years old but under 50 years old.
42, 36, 35, 49, 52, 43, 41, 32, 45, 39, 50, 38, 27, 29, 37, 39

Answer: 75% of parents at least 30 years old but under 50 years old.

Explanation: Total parents are 16 and 12 parents who are at least 30 years old but under 50 years. So percent is 12/16= 0.75
= 75% of parents at least 30 years old but under 50 years old.

Question 5.
What is the mode of the data in Exercise 4?

Answer: 39

Explanation: As 39 appears two times, the mode is 39.

Question 7.
A recipe for punch calls for apple juice and cranberry juice. The ratio of apple juice to cranberry juice is 3:2. Tyrone wants to make at least 20 cups of punch, but no more than 30 cups of punch. Describe two different ways he can use apple juice and cranberry juice to make the punch.

Answer: Tyrone can use 60:40 and 90:60.

Explanation: For 20 cups Tyrone can use 60:40 and for 30 cups he can use 90:60

Question 8.
The data set shows the total points scored by the middle school basketball team in the last 14 games. What is the most common number of points scored in a game? Explain how to find the answer using a dot plot.

Answer: The most common number of points scored in a game is 39

Explanation:

Problem Solving Data Displays – Page No. 697

Read each problem and solve.

Question 1.
Josie collected data on the number of siblings her classmates have. Make a data display and determine the percent of Josie’s classmates that have more than 2 siblings.
5, 1, 2, 1, 2, 4, 3, 2, 2, 6
_______ %

Answer: 40%.

Explanation: Total number of classmates are 10 members and 4 of them have more than 2 siblings, so the percent of Josie’s classmates is 4÷10= 0.4= 40%.

Question 2.
The following data show the number of field goals a kicker attempted each game. Make a data display and tell which number of field goals is the mode.
4, 6, 2, 1, 3, 1, 2, 1, 5, 2, 2, 3

Answer: The mode of data is 2.

Explanation: As 2 is repeated 4 times, so the mode is 2.

Question 3.
The math exam scores for a class are shown below. Make a data display. What percent of the scores are 90 and greater?
91, 68, 83, 75, 81, 99, 97, 80, 85, 70, 89, 92, 77, 95, 100, 64, 88, 96, 76, 88

Answer: 35% of the scores are 90 and greater.

Explanation: Total data display is 20 scores and 7 of them are greater than 90. So the percent of scores is 7÷20= 0.35
= 35%.

Question 4.
The heights of students in a class are shown below in inches. Make a data display. What percent of the students are taller than 62 inches?
63, 57, 60, 64, 59, 62, 65, 58, 63, 65, 58, 61, 63, 64

Answer: 50% of the students are taller than 62 inches.

Explanation: Total data display is 14 scores and 7 of them are taller than 62 inches. So the percent of scores is 7÷14= 0.5
= 50%.

Question 5.
Write and solve a problem for which you would use a dot plot or histogram to answer questions about given data.

Answer:

Lesson Check – Page No. 698

Question 1.
The number of student absences is shown below. What is the mode of the absences?
2, 1, 3, 2, 1, 1, 3, 2, 2, 10, 4, 5, 1, 5, 1

Answer: 1

Explanation: The mode is the data value with the most dots, so the mode of absence is 1.

Question 2.
Kelly is making a histogram of the number of pets her classmates own. On the histogram, the intervals of the data are 0–1, 2–3, 4–5, 6–7. What is the range of the data?

Answer: 7

Explanation: The range of data is 7.

Spiral Review

Question 3.
The area of the base of the rectangular prism shown below is 45 square millimeters. The height is 5 \(\frac{1}{2}\) millimeters. What is the volume of the prism?

_______ \(\frac{□}{□}\) mm³

Answer: 247 1/2 mm³

Explanation:
As l×w= 45 mm²
Area of the base is l×w
V= l×w×h
=  45×h
= 45 × 5 1/2
= 45 × 11/2
= 495/2
= 247 1/2 mm³

Question 4.
The frequency table shows the number of runs scored by the Cougars in 20 of their baseball games. In what percent of the games did they score 5 or fewer runs?

_______ %

Answer: 85%.

Explanation: The Cougars scores 5 or fewer runs in 17 of their 20 games. So the percent of the games did they score 5 or fewer runs is 17/20= 0.85 = 85%.

Question 5.
There are 5 plates of bagels. The numbers of bagels on the plates are 8, 10, 9, 10, and 8. Shane rearranges the bagels so that each plate has the same amount. How many bagels are now on each plate?

Answer: 9

Explanation: There are 9 bagels on each plate.

Chapter 12 Review/Test – Page No. 699

Question 1.
The data set shows the total number of sandwiches sold each day for 28 days. What is the most common number of sandwiches sold in a day?

______ sandwiches

Answer: 13 sandwiches.

Explanation: The most common number of sandwiches sold are 13.

Question 2.
Michael’s teacher asks, “How many items were sold on the first day of the fund raiser?” Explain why this is not a statistical question.

Answer: As there is no variability in the number of items sold on the first day of the fundraiser, so it is not a statistical question.

Question 3.
Describe the data set by writing the attribute measured, the unit of measure, the likely means of measurement, and the number of observations in the correct location on the chart.

Answer:

Page No. 700

Question 4.
The numbers of points scored by a football team in 7 different games are 26, 38, 33, 20, 27, 3, and 28. For numbers 4a–4c, select True or False to indicate whether the statement is correct.
4a. The outlier in the data set is 3
4b. The difference between the outlier and the median is 24.
4c. The outlier in this set of data affects the mean by increasing it.

4a.
Answer:  True.

Explanation: The outlier is 3.

4b.
Answer: True.

Explanation: The median is 3,20,26,27,28,33,38
= 27
and outlier is 3, so difference between median and outlier is 27-3= 24.

4c.
Answer: False

Explanation: The mean with outlier is \(\frac{26+38+33+20+27+3+28}{7}
= \frac{175}{7}\)
= 25
The mean without outlier is \(\frac{26+38+33+20+27+28}{6}
= \frac{172}{6}\)
= 28.6
The mean increases without the outlier.

Question 5.
Mr. Jones gave a quiz to his math class. The students’ scores are listed in the table. Make a dot plot of the data.

Answer:

Question 6.
Melanie scored 10, 10, 11, and 13 points in her last 4 basketball games.
The mean of the test scores is _____.
The median of the test scores is _____.
The mode of the test scores is _____.

Answer:
The mean of the test scores is 11.
The median of the test scores is 10.5
The mode of the test scores is 10.

Explanation:
The mean of the test scores is \(\frac{10+10+11+13}{4}
= \frac{44}{4}\)
= 11.
The median of the test scores is 10,10,11,13
= \(\frac{10+11}{2}
= \frac{21}{2}\)
= 10.5
The mode of the test scores is 10. As 10 is repeated twice.

Page No. 701

Question 7.
The Martin family goes out for frozen yogurt to celebrate the last day of school. The costs of their frozen yogurts are $1, $1, $2, and $4. Select True or False for each statement.
7a. The mean cost for the frozen yogurts can be found by adding each cost and dividing that total by 4.
7b. The mean cost of the four frozen yogurts is $2.
7c. The difference between the greatest cost and the mean is $1.
7d. The difference between the least cost and the mean is $1.

7a.
Answer: True.

Explanation: To find the mean we will add each cost and divide that total by 4.

7b.
Answer: True.

Explnation: The mean is \(\frac{$1+$1+$2+$4}{4}
= \frac{$8}{4}\)
= $2.

Answer:
7c. False.

Explanation: The difference between the greatest cost and the mean is $4-$2= $2.

Answer:
7d. True.

Explanation: The difference between the least cost and the mean is $2-$1= $1.

Question 8.
The histogram shows the amount of time students spent on homework for the week. For numbers 8a–8d, choose True or False to indicate whether the statement is correct.

8a. The number of students that spent between 30 minutes and 59 minutes on homework is 2.
8b. The greatest number of students spent between 90 minutes and 119 minutes on homework.
8c. Five of the students spent less than 60 minutes on homework for the week.
8d. Six of the students spent 60 minutes or more on homework for the week.

8a.
Answer: True.

8b.
Answer: True.

8c.
Answer: False

Explanation: Three of the students spent less than 60 minutes.

8d.
Answer: True.

Page No. 702

Question 9.
The dot plot shows how many games of chess 8 different members of the chess club played in one month. If Jackson is a new member of the chess club, how many games of chess is he likely to play in one month? Explain how the dot plot helped you find the answer.

Answer: Jackson played 5 games of chess in one month.

Explanation: As the tallest stack in this dot plot is 5 games.

Question 10.
Larry is training for a bicycle race. He records how far he rides each day. Find the mode of the data.

Answer: 15

Explanation: As 15 is repeated 3 times, so the mode of the data is 15.

Question 12.
The frequency table shows the height, in inches, of 12 basketball players. What fraction of the players are 70 inches or taller?

\(\frac{□}{□}\)

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

Explanation: The total number of players who are 70 inches or taller are 6+3= 9, so fraction is \(\frac{9}{12}\)
= \(\frac{3}{4}\).

Page No. 703

Question 13.
A teacher surveys her students to find out how much time the students spent eating lunch on Monday.

She uses _____ as the unit of measure.
She uses ______ as the unit of measure.

Answer: She uses minutes as the unit of measure.

Question 14.
For numbers 14a–14d, choose Yes or No to indicate whether the question is a statistical question.
14a. What are the heights of the trees in the park?
14b. How old are the trees in the park?
14c. How tall is the cypress tree on the north side of the lake this morning?
14d. What are the diameters of the trees in the park?

14a.
Answer: Yes.

14b.
Answer: Yes.

14c.
Answer: No.

14d. Yes.

Question 16.
The data set shows the ages of the members of the cheerleading squad. What is the most common age of the members of the squad? Explain how to find the answer using a dot plot.

Answer: 11 is the most common age of the members of the squad.

Explanation:

Page No. 704

Question 17.
The band director kept a record of the number of concert tickets sold by 20 band members. Complete the frequency table by finding the frequency and the relative frequency.

Answer:

Explanation:

Question 18.
Gilbert is training for a marathon by running each week. The table shows the distances, in miles, that he ran each week during the first 7 weeks.

Part A
Gilbert set a goal that the mean number of miles he runs in 7 weeks is at least 14 miles. Did Gilbert reach his goal? Use words and numbers to support your answer.

Answer: No, Gilbert did not reach his goal as 13 is less than 14.

Explanation: The mean number of miles he runs in 7 weeks \(\frac{8+10+9+10+15+18+21}{7}
= \frac{91}{7}\)
= 13

Question 18.
Part B
Suppose Gilbert had run 18 miles during week 5 and 22 miles during week 6. Would he have reached his goal? Use words and numbers to support your answer

Answer: As the mean is 14, Gilbert reached his goal.

Explanation: The mean is \(\frac{8+10+9+10+22+18+21}{7}
= \frac{98}{7}\)
= 14.

Final Words

Tap the above links and start your preparation from now itself. You can understand the concepts of Data Displays and Measures of Center in-depth here. Bookmark our ccssanswers.com to get the answers with explanations for all grade 6 chapters.

Go Math Grade 6 Answer Key Chapter 13 deals with Variability and Data Distributions. The HMH Go Math 6th Grade Answer Key is a very helpful resource for students to prepare for the exams. The solutions are mentioned topic-wise to all the questions for chapter 13 Variability and Data Distributions. You can understand the problem-solving methods in a better way by using Go Math Grade 6 Answer Key Chapter 13 Variability and Data Distributions for free.

Go Math Grade 6 Answer Key Chapter 13 Variability and Data Distributions

The main concepts of chapter 13 Variability and Data Distributions are discussed in the below sections. So, tap the links and practice the problems. This Go Math Grade 6 Answer Key Ch 13 Variability and Data Distributions helps you to secure the highest marks in the exams. For better performance, you try to compare the problems with real time.

Chapter 13 – Lesson: 1

• Share and Show – Page No. 709
• Big Cats – Page No. 710
• Patterns in Data – Page No. 711
• Lesson Check – Page No. 712

Chapter 13 – Lesson: 2

• Share and Show – Page No. 715
• Problem Solving + Applications – Page No. 716
• Box Plots – Page No. 717
• Lesson Check – Page No. 718

Chapter 13 – Lesson: 3

• Share and Show – Page No. 721
• Problem Solving + Applications – Page No. 722
• Mean Absolute Deviation – Page No. 723
• Lesson Check – Page No. 724

Chapter 13 – Lesson: 4

• Share and Show – Page No. 727
• Problem Solving + Applications – Page No. 728
• Measures of Variability – Page No. 729
• Lesson Check – Page No. 730

Chapter – 13 – Mid-Chapter Checkpoint

• Mid-Chapter Checkpoint – Vocabulary – Page No. 731
• Mid-Chapter Checkpoint – Page No. 732

Chapter 13 – Lesson: 5

• Share and Show – Page No. 735
• Unlock the Problem – Page No. 736
• Choose Appropriate Measures of Center and Variability – Page No. 737
• Lesson Check – Page No. 738

Chapter 13 – Lesson: 5

• Share and Show – Page No. 741
• Problem Solving + Applications – Page No. 742
• Apply Measures of Center and Variability – Page No. 743
• Lesson Check – Page No. 744

Chapter 13 – Lesson: 6

• Share and Show – Page No. 747
• Problem Solving + Applications – Page No. 748
• Describe Distributions – Page No. 749
• Lesson Check – Page No. 750

Chapter 13 – Lesson: 7

• Share and Show – Page No. 753
• On Your Own – Page No. 754
• Problem Solving Misleading Statistics – Page No. 755
• Lesson Check – Page No. 756

Chapter 13 – Review/Test

• Chapter 13 Review/Test – Page No. 757
• Chapter 13 Review/Test – Page No. 758
• Chapter 13 Review/Test – Page No. 759
• Chapter 13 Review/Test – Page No. 760
• Chapter 13 Review/Test – Page No. 761
• Chapter 13 Review/Test – Page No. 762

Share and Show – Page No. 709

For 1–3, use the dot plot.

Question 1.
The dot plot shows the number of paintings students in the art club displayed at the art show. Does the dot plot contain any gaps?
If so, where?

Type below:
_________________

Answer: Between the intervals of 4 – 7 excluding 4 and 7

Explanation:

The dots are filled from 1 – 4 and 7 but the region between these two intervals is left unfilled so the region containing gaps is 5-6 including 5 and 6

Question 2.
Identify any clusters in the data.
Type below:
_________________

Answer: 1-4

Explanation:
A group of dots is called a cluster
The dots form a cluster at 1 – 4

On Your Own

Question 4.
What patterns do you see in the histogram data?

Type below:
_________________

Answer:

Explanation:
STEP 1 Identify any peaks in the data.
The histogram has 6 peaks.
The interval representing the greatest number of visitors is for ages between 60 and 69 age group.
STEP 2 The data changes across the intervals.
The number of visitors increases from 0 to 29 age group and from 40 to 69 age group.
So, the data values increase to one peak in the interval from 0 to 9 age group and then decrease.
The visitors of the age group 30 – 39 did not visit the zoo.

Question 5.
The dot plot shows the number of errors made by a baseball team in the first 16 games of the season. For numbers 5a-5e, choose Yes or No to indicate whether the statement is correct.

5a. There is a gap from 4 to 5.
5b. There is a peak at 0.
5c. The dot plot has symmetry.
5d. There are two modes.
5e. There is one cluster.
5a. __________
5b. __________
5c. __________
5d. __________
5e. __________

Answer:
5a. Yes
5b. Yes
5c. No
5d. No
5e. No

Explanation:
5a. There are dots between 4-5 so we can say that there is a gap from 4 to 5.
5b. The number of dots is more at the interval 0 So we can say that there is a peak at 0.
5c. The symmetrical view is nothing but having the same number of dots on both sides of the figure but we cannot observe it in the above figure. Therefore we can say that the dot plot has no symmetry.
5d. The most frequently occurring observation is known as a mode. One dot repeats in all the intervals so we can say that the mode is 1.
5e. A group of observations form a cluster, there are more than 1 group of dots in the figure given above.

Big Cats – Page No. 710

There are 41 species of cats living in the world today. Wild cats live in places as different as deserts and the cold forests of Siberia, and they come in many sizes. Siberian tigers may be as long as 9 feet and weigh over 2,000 pounds, while bobcats are often just 2 to 3 feet long and weigh between 15 and 30 pounds.

You can find bobcats in many zoos in the United States. The histogram below shows the weights of several bobcats. The weights are rounded to the nearest pound.

Use the histogram for 6 and 7.

Question 6.
Look for a Pattern Describe the overall shape of the histogram.
Type below:
_________________

Answer: The graph starts from a small interval and increases to the highest and then decreases to the smallest interval.
The histogram has rectangles which are closely packed.

Explanation:
STEP 1 Identify any peaks in the data.
The histogram has 1 peak(s).
The interval representing the greatest number of bobcats is for weights between 18 and 20 pounds.
STEP 2 Describe how the data changes across the intervals. The number of bobcats increases from 12 to 17 pounds and from 21 to 29 pounds.
STEP 3 Describe any symmetry the graph has. If I draw a vertical line through the interval for 18 to 20 pounds, the left and right parts of the histogram are very close to being mirror images. The histogram has line symmetry.

So, the data values increase to one peak in the interval for 18 to 20 pounds and then decrease. The data set has a vertical line
symmetry about the peak.

Patterns in Data – Page No. 711

For 1–2, use the dot plot.

Question 1.
The dot plot shows the number of omelets ordered at Paul’s Restaurant each day. Does the dot plot contain any gaps?
Type below:
_________________

Answer: Yes, the dot plot contains gaps

Explanation:
The dots are filled from 10 – 11, from 14 – 16, and from 18 – 19 but the region between these two intervals is left unfilled so the region containing gaps is 12-13 including 12 and 13, 17 is also left unfilled.

Question 2.
Identify any clusters in the data.
Type below:
_________________

Answer: 14 – 16 and 18 – 19

Explanation:
A group of dots is called a cluster. The dots which form a cluster are 14 – 16 and from 18 – 19.

For 3–4, use the histogram.

Question 3.
The histogram shows the number of people that visited a local shop each day in January. How many peaks does the histogram have?
Type below:
_________________

Answer: The histogram has only one peak.

Explanation:
The rectangle with a tall length represents the highest peak in the graph given above.
The number of people who visited a local shop each day in January was 0 – 9  visitors and this was the highest frequency having 14 days.
The highest peak is in the interval of 0 – 9.

Question 4.
Describe how the data values change across the intervals.
Type below:
_________________

Answer: They decrease from highest to low values in the given picture above.

Explanation:
The graph represents the number of visitors in the month of January the visitors of number 0 – 9 have the highest frequency,
10 – 19 are the second-highest among the visitors who went to the local shop in the month of January, followed by 20 – 29,
30 – 39

Problem Solving

Question 5.
Look at the dot plot at the right. Does the graph have symmetry? Explain.

Type below:
_________________

Answer: Yes, the graph has a symmetry

Explanation:
If I draw a vertical line through the interval for _ to_ pounds, the left and right parts of the histogram are very close to being mirror images. The histogram __ line symmetry.
A geometric figure has line symmetry if you can draw a line through it so that the two parts are mirror images of each other.
So, the data values increase to one peak in the interval for _ to _ pounds and then decrease. The data set __ line symmetry about the peak.

Question 6.
A histogram that shows the ages of students at a library has intervals 1–5, 6–10, 11–15, 16–20, and 21–25. There is a peak at 11–15 years and the graph is symmetric. Sketch what the histogram could look like and describe the patterns you see in the data.
Type below:
_________________

Answer:

The histogram shows the ages of students at a library has intervals 1–5, 6–10, 11–15, 16–20, and 21–25. There is a peak at 11–15 years and the graph is symmetric.

Explanation:

The histogram is a graph with continuous rectangles which are closely packed.
The asymmetric graph is a graph which has a mirror-like view with equal rectangles on each side.
The graph with the highest peak represents the highest number of students who visit the library in the age group 11 – 15

Lesson Check – Page No. 712

Question 1.
What interval in the histogram has the greatest frequency?

Type below:
_________________

Answer: 10 – 14 interval has highest frequency of 6

Explanation:
The rectangle with a peak can be said as it has the greatest frequency. The interval with a peak is 11 – 15 and the frequency of the peak is 6

Question 2.
Meg makes a dot plot for the data 9, 9, 4, 5, 5, 3, 4, 5, 3, 8, 8, 5. Where does a gap occur?
Type below:
_________________

Answer: 6 – 7 including 6 and 7

Explanation:
Let us consider an axis with 3 to 9 numbers on it and plot the dots as given in the question at points 3,4,5,8 and 9 the gap occurs between 6 and 7 including 6 and 7.

Spiral Review

Question 4.
Look at the histogram below. How many students scored 81 or higher on the math test?
________ students

Answer: 14

Explanation:
The interval 81 – 90 has 10 frequencies and the interval 91 – 100 has 4 frequencies. So the total number of students = 14

Question 5.
The Little League coach uses a radar gun to measure the speed of several of Kyle’s baseball pitches. The speeds, in miles per hour, are 52, 48, 63, 47, 47. What is the median of Kyle’s pitch speeds?
The median is ________ miles.

Answer: Median is 48

Explanation:
First, write the observations in ascending order or descending order.
The formula to calculate the median is (n+1/2)th observation, for odd number of observations
n = 5 (odd)
Median = (5 + 1 / 2) = (6/2) = 3rd observation = 48
Therefore the median is 48.

Share and Show – Page No. 715

Find the median, lower quartile, and upper quartile of the data.

Question 2.
the lengths, in seconds, of 9 videos posted online:
50, 46, 51, 60, 62, 50, 65, 48, 53
Type below:
_________________

Answer: Median: 51 Lower quartile: 46 Upper quartile: 65

Explanation:
First, write the observations in ascending order or descending order.
The formula to calculate the median is (n+1/2)th observation, for odd number of observations
n = 9 (odd)
Median = (9 + 1 / 2) = (10/2) = 5th observation = 51
Therefore the median is 51.

Lower quartile: 46 Upper quartile: 65

Question 3.
Make a box plot to display the data set in Exercise 2.
Type below:
_________________

Answer: The box plot is drawn on the topic: Lengths of the videos (in seconds) posted in online.

Explanation:
The box is drawn to understand the clear view of the raw data, in a precise manner.
This box gives us information about the lengths of the videos posted online. We can directly say the median, lower quartile, and upper quartile by seeing the box plot.

On Your Own

Find the median, lower quartile, and upper quartile of the data.

Question 5.
61, 23, 49, 60, 83, 56, 51, 64, 84, 27
Type below:
_________________

Answer: 58

Explanation:
First, write the observations in ascending order or descending order.
The formula to calculate the median is mean of (n/2) and (n/2+1)th observations, for even number of observations
n = 10 (even)
Median = Mean of (5)th and (6)th observations = 56 + 60 divided by 2 = 116/2 = 58
Therefore the median is 58.

Question 6.
The chart shows the height of trees in a park. Display the data in a box plot.

Type below:
_________________

Answer:

Explanation:
Lower limit: 8
Upper limit: 30
Median:
First, write the observations in ascending order or descending order.
The formula to calculate the median is mean of (n/2) and (n/2+1)th observations, for even number of observations
n = 12 (even)
Median = Mean of (6)th and (7)th observations = 18 + 20 divided by 2 = 38/2 = 19
Therefore the median is 19.

Question 7.
Analyze Eric made this box plot for the data set below. Explain his error.


Type below:
_________________

Answer: The lower and upper limits are marked wrong.

Explanation:
The box drew above the number line is wrong.
It does not show the correct upper and lower limits.
The lower limit is 5 and the upper limit is 35.

Problem Solving + Applications – Page No. 716

Pose a Problem

Question 8.
The box plots show the number of flights delayed per day for two different airlines. Which data set is more spread out?

Airline A: greatest value − least value = _____
Airline B: greatest value − least value = _____
So, the data for _____ is more spread out.
Write a new problem that can be solved using the data in the box plots.
Type below:
_________________

Answer:
Airline A: greatest value − least value = 8
Airline B: greatest value − least value = 10
The data for airline B is more spread out.

A problem which can be solved using the box plot can be:

Find the median, lower and upper limits.

Explanation:
The greatest value and lowest value can be identified by seeing the box drew above the number line. The ends represent the lower and upper limits in both the box plots.

The solution to the question framed:
The start and end of the rectangle represent the lower and upper limits. And the middle line represents the median.
The lower limit is 5
Upper limit is 35
Median:
First, write the observations in ascending order or descending order.
The formula to calculate the median is mean of (n/2) and (n/2+1)th observations, for even number of observations
n = 6 (even)
Median = Mean of (3)th and (4)th observations = 15 + 25 divided by 2 = 40/2 = 20
Therefore the median is 20.

Question 9.
The data set shows the cost of the dinner specials at a restaurant on Friday night.

The median is _____.
The lower quartile is _____.
The upper quartile is _____.

Answer:
Median: 24
The lower quartile is 16.
The upper quartile is 30.

Explanation:
Seeing the data in the box we can identify the lower and upper quartiles.
Median:
First, write the observations in ascending order or descending order.
The formula to calculate the median is (n+1/2)th observation, for odd number of observations
n = 11 (odd)
Median = (11 + 1 / 2) = (12/2) = 6th observation =24
Therefore the median is 24.

Box Plots – Page No. 717

Find the median, lower quartile, and upper quartile of the data.

Question 1.
the amounts of juice in 12 glasses, in fluid ounces:
11, 8, 4, 9, 12, 14, 9, 16, 15, 11, 10, 7
Type below:
_________________

Answer:
Median: 10.5
Lower quartile: 4
Upper quartile: 16

Explanation:
Median:
First, write the observations in ascending order or descending order.
The formula to calculate the median is mean of (n/2) and (n/2+1)th observations, for even number of observations
n = 12 (even)
Median = Mean of (6)th and (7)th observations = 10 + 11 divided by 2 = 21/2 = 10.5
Therefore the median is 10.5.
After writing in ascending or descending order the first and last terms justify the lower and upper limits respectively.
They are:
Lower quartile: 4
Upper quartile: 16

Question 2.
the lengths of 10 pencils, in centimeters:
18, 15, 4, 9, 14, 17, 16, 6, 8, 10
Type below:
_________________

Answer:
Median: 12
Lower quartile: 4
Upper quartile: 18

Explanation:
Median:
First, write the observations in ascending order or descending order.
The formula to calculate the median is mean of (n/2) and (n/2+1)th observations, for even number of observations
n = 10 (even)
Median = Mean of (5)th and (6)th observations = 10 + 14 divided by 2 = 24/2 = 12
Therefore the median is 12.
After writing in ascending or descending order the first and last terms justify the lower and upper limits respectively.
They are:
Lower quartile: 4
Upper quartile: 18

Question 3.
Make a box plot to display the data set in Exercise 2.
Type below:
_________________

Answer:

The above box plot represents the lower and upper quartiles, the median.

Explanation:
Box plot is drawn using the number line and the rectangle which is drawn above it.
The ends of the rectangles say about the lower and upper limits and the middle line indicates the median.

Question 4.
The numbers of students on several teams are 9, 4, 5, 10, 11, 9, 8, and 6. Make a box plot for the data.
Type below:
_________________

Answer:

Explanation:
Box plot is drawn using the number line and the rectangle which is drawn above it.
The ends of the rectangles say about the lower and upper limits and the middle line indicates the median.
Therefore the lower and upper quartiles are 4 and 11 respectively.
Median:
First, write the observations in ascending order or descending order.
The formula to calculate the median is mean of (n/2) and (n/2+1)th observations, for even number of observations
n = 8 (even)
Median = Mean of (4)th and (5)th observations = 8 + 9 divided by 2 = 17/2 = 8.5
Therefore the median is 8.5.

Problem Solving

Question 6.
The weights of the six puppies in ounces are 8, 5, 7, 5, 6, and 9. What is the upper quartile of the data?
Type below:
_________________

Answer: Upper quartile: 9

Explanation:
The highest value in the data is defined as the upper quartile.
The highest value in the raw data given is 9

Question 7.
Draw a box plot to display this data: 81, 22, 34, 55, 76, 20, 56.
Type below:
_________________

Answer:

Explanation:
A box plot gives information about the lower and upper quartiles and about the median.
The box plot is drawn using a rectangle and the number line.

Lesson Check – Page No. 718

Question 1.
The values in a data set are 15, 7, 11, 12, 6, 3, 10, and 6. Where would you draw the box in a box plot for the data?
Type below:
_________________

Answer: The box is drawn above the number line.

Explanation:
Example:

The rectangle which can be seen above the number line is the box plot which is drawn.
The box plot gives information about the lower and upper quartiles and about the median.

Question 2.
What is the lower quartile of the following data set?
22, 27, 14, 21, 22, 26, 18
Type below:
_________________

Answer: 14

Explanation:
The value which is lowest in the given data is called the lowest quartile.
Therefore the lowest quartile in the given data is 14.

Spiral Review

Question 3.
Jenn says that “What is the average number of school lunches bought per day?” is a statistical question. Lisa says that “How many lunches did Mark buy this week?” is a statistical question. Who is NOT correct?
Type below:
_________________

Answer: Lisa’s statement is wrong.

Explanation:

Question 5.
By how much does the mean of the following data set change if the outlier is removed?
13, 19, 16, 40, 12
Type below:
_________________

Answer: The mean shows a difference if the lower limit is removed the mean increases and if the upper limit is removed the mean decreases.

Explanation:
Outliers are nothing but both upper and lower limits.
The actual mean is 20
But when the lower limit is removed the mean increases to 22 while when the upper limit is removed the mean decreases to 15.
Therefore, we can say that the mean shows a difference when the outliers are removed.

Question 6.
Where in the dot plot does a cluster occur?

Type below:
_________________

Answer: 52 – 54

Explanation:
A cluster is nothing but a group of dots.
In the intervals 52 – 54 a cluster has occurred.

Share and Show – Page No. 721

Use counters, a dot plot, or iTools to find the mean absolute deviation of the data.

Question 1.
Find the mean absolute deviation for both data sets. Explain which data set is more spread out.
the number of laps Shawna swam on 5 different days:
5, 6, 6, 8, 10
mean = 7

the number of laps Lara swam on 5 different days:
1, 3, 7, 11, 13
mean = 7
Type below:
_________________

Answer: Case 2 is more spread out.

Explanation:
CASE1
The number of laps Shawna swam on 5 different days:
5,6,6,8,10
Mean = 7
Deviations:
7 – 5 = 2
7 – 6 = 1
7 – 6 = 1
7 -8 = -1
7 -10=-3
Mean of deviations = 2+1+1+1+3/5 = 8/5 = 1.6

CASE2
The number of laps Lara swam on 5 different days:
1, 3, 7, 11, 13
Mean = 7
Deviations:
7 – 1 = 6
7 – 3 = 4
7 – 7 = 0
7 -11= -4
7 -13= -6
Mean of deviations = 6+ 4 + 0 + 4 + 6 / 5 = 20/5 = 4

Use the dot plot to find the mean absolute deviation of the data.

Question 2.
mean = 7 books

______ books

Answer: Mean absolute deviation is 2.4

Explanation:
STEP 1 Label each dot with its distance from the mean.
Starting from left to right:
4: 7-4=3
5: 7-5=2
6: 7-6=1
9: 7-9=-2
10: 7-10=-3
11: 7-11=-4

STEP 2 Find the mean of the distances.
(3) + (2) +(2) +(2) +(2) +(1) + (2) +(3) +(3) +(4) / 10 = 24/10 = 2.4

So, the mean absolute deviation of the data is 2.4

Question 3.
mean = 29 pounds

_______ pounds

Answer: Mean Absolute deviation is 3.2

Explanation:
STEP 1 Label each dot with its distance from the mean.
Starting from left to right:
26: 29-26=03
27: 29-27=02
32: 29-32=-3
33: 29-33=-4
35: 29-35=-6

STEP 2 Find the mean of the distances.
(3) + (2) +(3) +(4) +(6) +(3) + (3) +(2) / 8 = 26/8 = 3.2

So, the mean absolute deviation of the data is 3.2

Question 4.
The mean absolute deviation of the number of daily visits to Scott’s website for February is 167.7. In March, the absolute mean deviation is 235.9. In which month did the number of visits to Scott’s website vary more? Explain how you know.
Type below:
_________________

Answer: As the mean absolute deviation is more in the month of February we can say that there are more visitors in this month.

Explanation:
As the mean of the month of February is less it means that the number of observations are more.
Similarly, as the mean of the month of March is more it means that the number of observations are less.
Therefore we can say that the number of visitors were more in the month of February compared to March.

Question 5.
Write an Inequality Algebra In April, the data for Scott’s website visits are less spread out than they were in February. Use a to represent the mean absolute deviation for April. Write an inequality to describe the possible values of a.
Type below:
_________________

Answer: a < February

Explanation:
Since the data is more spread out in the month of April than they were in February. Therefore the inequality represents “less than” sign.

Problem Solving + Applications – Page No. 722

Question 6.
Use the table.

The mean of the data is 11. What is the mean absolute deviation of the data?
_______ days

Answer: 3

Explanation:
STEP 1 Label each observation with its distance from the mean.
Starting from left to right:
10: 11-10= 1
12: 11-12=-1
13: 11-13=-2
18: 11-18=-7
10: 11-10= 1
08: 11-08= 3
07: 11-07= 4
06: 11-06= 5
16: 11-16=-5
14: 11-14=-3
08: 11-08= 3
10: 11-10= 1

STEP 2 Find the mean of the distances.
1+1 +2 +7 +1 +3 +4 +5+5+3+3+1/ 12
= 36/12 = 3

So, the mean absolute deviation of the data is 3.

Question 8.
Explain Tell how an outlier that is much greater than the mean would affect the mean absolute deviation of the data set. Explain your reasoning.
Type below:
_________________

Answer: An outlier increases the mean absolute deviation of the data set.

Explanation:
The difference between the outlier and the mean is a greater number when added in the sum of observations the mean absolute deviation increases.

Question 9.
The data set shows the number of soccer goals scored by players in 3 games.

For numbers 9a–9c, choose Yes or No to indicate whether the statement is correct.
9a. The mean absolute deviation of Player A is 1.
9b. The mean absolute deviation of Player B is 0.
9c. The mean absolute deviation of Player C is greater than the mean absolute deviation of Player A.
9a. __________
9b. __________
9c. __________

Answer:
9a. No
9b. Yes
9c. No

Explanation:
Player A
Mean = 1+2+3/3 = 6/3 = 2
Mean absolute deviation = 1+0+1/3 = 2/3 = 0.6

Player B
Mean =2+2+2/3 = 6?3 = 2
Mean absolute deviation = 0/3 = 0

Player C
Mean = 1+2+1/3 = 4/3 = 1.3
Mean absolute deviation = 0.3+0+0.3/3 = 0.2

Mean Absolute Deviation – Page No. 723

Use counters and a dot plot to find the mean absolute deviation of the data.

Question 1.
the number of hours Maggie spent practicing soccer for 4 different weeks:
9, 6, 6, 7
mean = 7 hours
_______ hour

Answer: The mean absolute deviation of the data is 1.

Explanation:
STEP 1 Label each observation with its distance from the mean.
Starting from left to right:
9: 7-9=-2
6: 7-6=-1
6: 7-6=-1
7: 7-7= 0

STEP 2 Find the mean of the distances.
2 +1 +1+0/ 4
= 4 /4 = 1

So, the mean absolute deviation of the data is 1.

Question 2.
the heights of 7 people in inches:
60, 64, 58, 60, 70, 71, 65
mean = 64 inches
_______ inches

Answer: The mean absolute deviation of the data is 4.

Explanation:
STEP 1 Label each observation with its distance from the mean.
Starting from left to right:
60: 64-60= 4
64: 64-64= 0
58: 64-58= 6
60: 64-60= 4
70: 64-70=-6
71: 64-71=-7
65: 64-65=-1

STEP 2 Find the mean of the distances.
4+0+6+ 4+6+7+1/7
= 28/7 = 4

So, the mean absolute deviation of the data is 4.

Use the dot plot to find the mean absolute deviation of the data.

Question 3.
mean = 10

_______ year

Answer: The mean absolute deviation of the data is 1

Explanation:
STEP 1 Label each dot with its distance from the mean.
Starting from left to right:
08: 10-08=02
09: 10-09=01
10: 10-10= 0
11: 10-11=-1
12: 10-12=-2

STEP 2 Find the mean of the distances.
(2) + (1) +(0) +(1) +(2) +(2)+(1)+(0)+(0)+(0)+(1)+(2) /12 = 12/12= 1

So, the mean absolute deviation of the data is 1

Question 4.
mean = 8

_______ hours

Answer: The mean absolute deviation of the data is 2.4

Explanation:
STEP 1 Label each dot with its distance from the mean.
Starting from left to right:
03: 8-03=05
04: 8-04=04
05: 8-05=03
07: 8-07=01
08: 8-08= 0
09: 8-09=-1
10: 8-10=-2
11: 8-11=-3
12: 8-12=-4

STEP 2 Find the mean of the distances.
(5) + (4) +(3) +(1) +(0) +(1) + (2) +(3) +(4)+(5)+(0)+(1)+(1)+(2)+(4)/ 15 = 36/15 = 2.4

So, the mean absolute deviation of the data is 2.4

Problem Solving

Question 5.
In science class, Troy found the mass, in grams, of 6 samples to be 10, 12, 7, 8, 5, and 6. What is the mean absolute deviation?
_______ grams

Answer: The mean absolute deviation of the data is 2.

Explanation:
Mean = 10+12+7+8+5+6/6 = 48/6 = 8

Mean absolute deviation:

STEP 1 Label each observation with its distance from the mean.
Starting from left to right:
10: 8-10= -2
12: 8-12= -4
07: 8-07= 01
08: 8-08= 0
05: 8-05=03
06: 8-06=02

STEP 2 Find the mean of the distances.
2+4+1+0+3+2/6
= 12/6 = 2

So, the mean absolute deviation of the data is 2.

Question 6.
Five recorded temperatures are 71°F, 64°F, 72°F, 81°F, and 67°F. What is the mean absolute deviation?
_______ °F

Answer: The mean absolute deviation of the data is 4.4.

Explanation:
Mean = 71+64+72+81+67/5 = 355/5 = 71

Mean absolute deviation:

STEP 1 Label each observation with its distance from the mean.
Starting from left to right:
71: 71-71= 0
64: 71-64= 07
72: 71-72= -1
81: 71-81=-10
67: 71-67= 04

STEP 2 Find the mean of the distances.
0+7+1+10+4/5
= 22/5 = 4.4

So, the mean absolute deviation of the data is 4.4.

Question 7.
Make a dot plot of the following data: 10, 10, 11, 12, 12, 13, 13, 15. Use the dot plot to find the mean absolute deviation.
Type below:
_________________

Answer: The mean absolute deviation of the data is 1.25

Explanation:
Mean = 10+10+11+12+12+13+13+15/8 = 96/8 = 12

Mean absolute deviation:

Box plot:

STEP 1 Label each dot with its distance from the mean.
Starting from left to right:
10: 12-10=02
11: 12-11=01
12: 12-12=0
13: 12-13=-1
15: 12-15=-3

STEP 2 Find the mean of the distances.
(2) + (2) +(1) +(0) +(0) +(1) + (1) +(3) / 8 = 10/8 = 1.25

So, the mean absolute deviation of the data is 1.25

Lesson Check – Page No. 724

Question 1.
The six test grades are 86, 88, 92, 90, 82, and 84. The mean of the data is 87. What is the mean absolute deviation?
_______

Answer: The mean absolute deviation of the data is 3.5

Explanation:

Mean absolute deviation:

STEP 1 Label each observation with its distance from the mean.
Starting from left to right:
86: 87-86= 01
88: 87-88= -1
92: 87-92= -5
90: 87-81= 06
82: 87-82= 05
84: 87-84= 03

STEP 2 Find the mean of the distances.
1+5+1+6+5+3/6
= 21/6 = 3.5

So, the mean absolute deviation of the data is 3.5

Question 2.
Eight heights in inches are 42, 36, 44, 46, 48, 42, 48, and 46. The mean of the data is 44. What is the mean absolute deviation?
_______ inches

Answer:

Explanation: The mean absolute deviation of the data is 3

Mean absolute deviation:

STEP 1 Label each observation with its distance from the mean.
Starting from left to right:
42: 44-42= 02
36: 44-36= 08
44: 44-44= 0
46: 44-46= -2
48: 44-48= -4
42: 44-42= 02
48: 44-48= -4
46: 44-46= -2

STEP 2 Find the mean of the distances.
2+8+2+4+0+2+4+2/8
= 24/8 = 3

So, the mean absolute deviation of the data is 3

Spiral Review

Question 4.
Carrie is making a frequency table showing the number of miles she walked each day during the 30 days of September. What value should she write in the Frequency column for 9 to 11 miles?

_______

Answer: 1

Explanation:
Total number of days in the month of September = 30
Number of days given in the frequency table = 17+8+4 = 29
Frequency in the interval 9 – 11 = 30 – 29 = 1 day

Question 6.
What is the upper quartile of the following data?
43, 48, 55, 50, 58, 49, 38, 42, 50
The upper quartile is _______

Answer: The upper quartile is 58

Explanation:
The highest observation in the data given is known as upper quartile. The upper quartile is 58

Share and Show – Page No. 727

Question 1.
Find the range and interquartile range of the data in the box plot.

The range is $ __________ .
The interquartile range is $ __________ .

Answer: $12, $3

Explanation:
The difference between the highest observation and the lowest observation is called a range.
Range = 19 – 7 = $12
The difference and the highest and the lowest dots of the dot plot is called as interquartile range.
Interquartile range = 15 – 12 = $3

Practice: Copy and Solve Find the mean absolute deviation for the data set.

Question 2.
heights in inches of several tomato plants:
16, 18, 18, 20, 17, 20, 18, 17
_______ inch

Answer: The mean absolute deviation of the data is 1

Explanation:
Mean:
Mean = 16+18+18 +20+17+20+18+17/8 = 144/8 = 18

Mean absolute deviation:

STEP 1 Label each observation with its distance from the mean.
Starting from left to right:
16: 18-16= 02
18: 18-18= 0
18: 18-18= 0
20: 18-20= -2
17: 18-17= 01
20: 18-20= -2
18: 18-18= 0
17: 18-17= 01

STEP 2 Find the mean of the distances.
2+0+0+2+1+2+0+1/8
= 8/8 = 1

So, the mean absolute deviation of the data is 1

Question 3.
times in seconds for students to run one lap:
68, 60, 52, 40, 64, 40
_______ seconds

Answer: The mean absolute deviation of the data is 10

Explanation:
Mean:
Mean = 68+60+52+40+64+40/6 = 54

Mean absolute deviation:

STEP 1 Label each observation with its distance from the mean.
Starting from left to right:
68: 54-68= -14
60: 54-60= -6
52: 54-52= 02
40: 54-40= 14
64: 54-64= -10
40: 54-40= 14

STEP 2 Find the mean of the distances.
14+6+2+14+10+14/6
= 60/6 = 10

So, the mean absolute deviation of the data is 10

On Your Own

Use the box plot for 4 and 5.

Question 4.
What is the range of the data?
$ _______

Answer: $24

Explanation:
The difference between the highest observation and the lowest observation is called a range.
Range = $56 – $32 = $24

Question 5.
What is the interquartile range of the data?
$ _______

Answer: $16

Explanation:
The difference and the highest and the lowest dots of the dot plot is called as interquartile range.
Interquartile range = $52 – $36 = $16

Practice: Copy and Solve Find the mean absolute deviation for the data set.

Question 6.
times in minutes spent on a history quiz:
35, 35, 32, 34, 34, 32, 34, 36
_______ minute

Answer: The mean absolute deviation of the data is 1

Explanation:
Mean:
Mean = 35+ 35+ 32+ 34+34+ 32+ 34+36/8 = 272/8 = 34

Mean absolute deviation:

STEP 1 Label each observation with its distance from the mean.
Starting from left to right:
35: 34-35= -1
35: 34-35= -1
32: 34-32= 02
34: 34-34= 0
34: 34-34= 0
32: 34-32=02
34: 34-34=0
36: 34-36=-2

STEP 2 Find the mean of the distances.
1+1+2+0+0+2+0+2/8
= 8/8 = 1

So, the mean absolute deviation of the data is 1

Question 7.
the number of excused absences for one semester:
1, 2, 1, 10, 9, 9, 10, 6, 1, 1
_______

Answer: The mean absolute deviation of the data is 3.8

Explanation:

Mean:
Mean =1+2+1+10+9+9+10+6+1+1 /10 = 50/10 = 5

Mean absolute deviation:

STEP 1 Label each observation with its distance from the mean.
Starting from left to right:
1: 5-1= 4
2: 5-2= 3
1: 5-1= 4
10: 5-10= -5
9: 5-9= -4
9: 5-9=-4
10: 5-10=-5
6: 5-6=-1
1: 5-1=4
1: 5-1=4

STEP 2 Find the mean of the distances.
4+3+4+5+4+4+5+1+4+4/10
=38/10 = 3.8

So, the mean absolute deviation of the data is 3.8

Question 8.
The chart shows the price of different varieties of dog food at a pet store. Find the range, interquartile range, and the mean absolute deviation of the data set.

Type below:
_________________

Answer:

The mean absolute deviation of the data is 3.6
Range = 32-16 = 16
Interquartile range = 24 – 20 = 4

Explanation:

Mean:
Mean =18+24+20+26+24+20+32+20+16+20 /10 = 220/10 = 22

Mean absolute deviation:

STEP 1 Label each observation with its distance from the mean.
Starting from left to right:
18: 22-18= 4
24: 22-24= -2
20: 22-20= 2
26: 22-26= -4
24: 22-24= -2
20: 22-20= 2
32: 22-32=-10
20: 22-20= 2
16: 22-16= 6
20: 22-20= 2

STEP 2 Find the mean of the distances.
4+2+2+4+2+2+10+2+6+2/10
=36/10 = 3.6

So, the mean absolute deviation of the data is 3.6
The difference between the highest observation and the lowest observation is called a range.
Range = 32-16 = 16
The difference and the highest and the lowest dots of the dot plot is called the interquartile range.
Interquartile range = 24 – 20 = 4

Problem Solving + Applications – Page No. 728

Question 10.
Compare Jack recorded the number of minutes his family walked each day for a month. The range of the data is 15. How does this compare to the data for Hyato’s family?
Type below:
_________________

Answer: Jack’s family walked less number of minutes each day compared to Hyato’s family.

Explanation:
The range of Hyato’s family is 20 while the range of Jack’s family is 15. Therefore we can say that Hyato’s family walked in more minutes compared to Jack’s family in a day.
The range can define the data with large observations and the data with the least observations.

Question 11.
Sense or Nonsense? Nathan claims that the interquartile range of a data set can never be greater than its range. Is Nathan’s claim sense or nonsense? Explain.
Type below:
_________________

Answer: Nonsense, Interquartile range of a data set can be less than or greater than the range.

Explanation:
The interquartile range is the difference between the medians of the observations.
Nathan’s claim is nonsense as he said, ” The interquartile range can never be greater than its range.”
The range is the difference between the highest observation and the lowest observation.
The interquartile range can be less or greater than the range.

Example:
Ascending order: 25,30,30,35,40,40,45
n=7 (odd)
Median= Number of (n+1/2) = 8/2 = 4th observation = 35
Median:(four terms of the data)
Median = 30+30/2 = 60/2 = 30
Median:(last 3 terms of the data)
Median = (n+1/2) = 2nd observation = 40
Interquartile range = 30+40/2 = 70/2 = 35

Range= 45-25 = 20

In the above case the interquartile range is more than the range proving that the given statement is nonsense.

Question 12.
The box plot shows the heights of corn stalks from two different farms.

The range of Farm A’s heights is _____ the range of Farm B’s heights.

Answer: greater than

Explanation:
The range is the difference between the highest and the lowest observations.
Range of Farm A: 72-58 = 14
Range of Farm B: 70-55 = 15

Therefore, The range of Farm A’s heights is greater than the range of Farm B’s heights.

Measures of Variability – Page No. 729

Question 1.
Find the range and interquartile range of the data in the box plot.

The range is __________ miles.
The interquartile range is __________ miles.

Answer: 16, 8

Explanation:
The difference between the highest and the lowest observations is range.
Range = 17 – 1 = 16
The difference between the highest and lowest observations of the box is the interquartile range.
Interquartile range = 12 – 4 = 8

Use the box plot for 2 and 3.

Question 2.
What is the range of the data?
_____

Answer: 35

Explanation:
The difference between the highest and the lowest observations is range.
Range = 95 – 60 = 35

Question 3.
What is the interquartile range of the data?
_____

Answer: 20

Explanation:
The difference between the highest and lowest observations of the box is the interquartile range.
Interquartile range = 90 – 70 = 20

Find the mean absolute deviation for the set.

Question 4.
heights in centimeters of several flowers:
14, 7, 6, 5, 13
_____ cm

Answer: The mean absolute deviation of the data is 3.6

Explanation:

Mean:
Mean =14+7+ 6+5+13/5= 45/5 = 9

Mean absolute deviation:

STEP 1 Label each observation with its distance from the mean.
Starting from left to right:
14: 9-14= -5
07: 9-07= 02
06: 9-06= 03
05: 9-05= 04
13: 9-13= -4

STEP 2 Find the mean of the distances.
5+2+3+4+4/5
= 18/5 = 3.6

So, the mean absolute deviation of the data is 3.6

Question 5.
ages of several children:
5, 7, 4, 6, 3, 5, 3, 7
_____ years

Answer: The mean absolute deviation of the data is 1.25

Explanation:
Mean:
Mean = 5+7+4+6+ 3+5+3+7/8 = 40/8 = 5

Mean absolute deviation:

STEP 1 Label each observation with its distance from the mean.
Starting from left to right:
5: 5-5= 0
7: 5-7= -2
4: 5-4= 01
6: 5-6= -1
3: 5-3= 02
5: 5-5= 0
3: 5-3= 02
7: 5-7=-2

STEP 2 Find the mean of the distances.
0+2+1+1+2+0+2+2/8
= 10/8 = 1.25

So, the mean absolute deviation of the data is 1.25

Problem Solving

Question 6.
The following data set gives the amount of time, in minutes, it took five people to cook a recipe. What is the mean absolute deviation for the data?
33, 38, 31, 36, 37
_____ minutes

Answer: The mean absolute deviation of the data is 2.4

Explanation:
Mean:
Mean = 33+38+31+36+37/5 = 175/5 = 35

Mean absolute deviation:

STEP 1 Label each observation with its distance from the mean.
Starting from left to right:
33: 35-33= 02
38: 35-38= -3
31: 35-31= 04
36: 35-36= -1
37: 35-37= -2

STEP 2 Find the mean of the distances.
2+3+4+1+2/5
= 12/5 = 2.4

So, the mean absolute deviation of the data is 2.4

Question 7.
The prices of six food processors are $63, $59, $72, $68, $61, and $67. What are the range, interquartile range, and mean absolute deviation for the data?
Type below:
_________________

Answer: Range = $9 The mean absolute deviation of the data is 4

Explanation:
The difference between the highest and the lowest observations is range.
Range = $68 – $59 = $9
The difference between the highest and lowest observations of the box is the interquartile range.
Interquartile range = 12 – 4 = 8

Mean:
Mean = $63+$59+$72+$68+$61+$67/6 = 390/6 = 65

Mean absolute deviation:

STEP 1 Label each observation with its distance from the mean.
Starting from left to right:
63: 65-63= 02
59: 65-59= 06
72: 65-72= -7
68: 65-68= -3
61: 65-61= -4
67: 65-67= -2

STEP 2 Find the mean of the distances.
2+6+7+3+4+2/6
= 24/6 = 4

So, the mean absolute deviation of the data is 4

Question 8.
Find the range, interquartile range, and mean absolute deviation for this data set: 41, 45, 60, 61, 61, 72, 80.
Type below:
_________________

Answer: The mean absolute deviation of the data is 9.7

Explanation:

Mean:
Mean = 41+45+60+61+61+72+80 /7 = 420/7 = 60

Mean absolute deviation:

STEP 1 Label each observation with its distance from the mean.
Starting from left to right:
41: 60-41= 19
45: 60-45= 15
60: 60-60= 0
61: 60-61= -1
61: 60-61= -1
72: 60-72= -12
80: 60-80= -20

STEP 2 Find the mean of the distances.
19+15+0+1+1+12+20/7
= 68/7 = 9.7

So, the mean absolute deviation of the data is 9.7

Lesson Check – Page No. 730

Question 1.
Daily high temperatures recorded in a certain city are 65°F, 66°F, 70°F, 58°F, and 61°F. What is the mean absolute deviation for the data?
_____ °F

Answer: The mean absolute deviation of the data is 3.6

Explanation:

Mean:
Mean = 65+66+70+58+61 /5 = 320/5 = 64

Mean absolute deviation:

STEP 1 Label each observation with its distance from the mean.
Starting from left to right:
65: 64-65=-1
66: 64-66=-2
70: 64-70=-6
58: 64-58=06
61: 64-61=03

STEP 2 Find the mean of the distances.
1+2+6+6+3/5
= 18/5 = 3.6

So, the mean absolute deviation of the data is 3.6

Spiral Review

Question 3.
Look at the histogram. How many days did the restaurant sell more than 59 pizzas?

________

Answer: 20

Explanation:
After 59 there is 1 interval 60-79
Number of days the restaurant sell more than 59 pizzas = 20

Question 4.
Look at the histogram. Where does a peak in the data occur?

Type below:
_________________

Answer: 20 – 39

Explanation:
Number of days the restaurant sold the maximum pizzas = 30
Number of pizzas sold each day = 20 – 39

Question 6.
The data set below lists the ages of people on a soccer team. The mean of the data is 23. What is the mean absolute deviation?
24, 22, 19, 19, 23, 23, 26, 27, 24
________

Answer: The mean absolute deviation of the data is 2

Explanation:

Mean absolute deviation:

STEP 1 Label each observation with its distance from the mean.
Starting from left to right:
24: 23-24=-1
22: 23-22= 1
19: 23-19= 4
19: 23-19= 4
23: 23-23=0
23: 23-23=0
26: 23-26=-3
27: 23-27=-4
24: 23-24=-1

STEP 2 Find the mean of the distances.
1+1+4+4+3+4+1+0+0/9
= 18/9 = 2

So, the mean absolute deviation of the data is 2

Mid-Chapter Checkpoint – Vocabulary – Page No. 731

Choose the best term from the box to complete the sentence.

Question 1.
The _____ is the difference between the upper quartile and the lower quartile of a data set.
Type below:
_________________

Answer: Range

Explanation:
The difference between the upper and lower quartiles of the data is known as range.

Question 2.
A graph that shows the median, quartiles and least and greatest values of a data set is called a(n) _____.
Type below:
_________________

Answer: Box plot

Explanation:
The figure which shows the median, quartiles and least and greatest values of a data set is called a box plot, a box plot is a figure which represents median with a horizontal line and the starting and ending line represents the upper and lower quartiles and the end dots represent the upper limit and the lower limit.

Question 3.
The difference between the greatest value and the least value in a data set is the _____.
Type below:
_________________

Answer: Range

Explanation:
Each data set consists of upper and lower limits the difference between these limits is called as range.

Question 4.
The _____ is the mean of the distances between the values of a data set and the mean of the data set.
Type below:
_________________

Answer: Mean absolute

Explanation:
Mean absolute deviation is calculated by subtracting each observation from the mean and then the mean is calculated for these observations.
Therefore we can say that the mean absolute is the mean of the distances between the values of a data set and the mean of the data set.

Concepts and Skills

Find the mean absolute deviation of the data.

Question 6.
43, 46, 48, 40, 38
________

Answer: The mean absolute deviation of the data is 3.2

Explanation:
Mean:

Mean = 43+46+48+40+38/5 = 215/5 = 43

Mean absolute deviation:

STEP 1 Label each observation with its distance from the mean.
Starting from left to right:
43: 43-43=0
43: 43-46=-3
43: 43-48=-5
43: 43-40= 3
43: 43-38= 5

STEP 2 Find the mean of the distances.
0+3+5+3+5/5 = 16/5 = 3.2

So, the mean absolute deviation of the data is 3.2

Question 7.
26, 20, 25, 21, 24, 27, 26, 23
________

Answer: The mean absolute deviation of the data is 2.125

Explanation:
Mean:

Mean = 26+20+25+21+24+27+26+23/8 = 192/8 = 24

Mean absolute deviation:

STEP 1 Label each observation with its distance from the mean.
Starting from left to right:
26: 24-26=-3
20: 24-20= 4
25: 24-25= -1
21: 24-21= 3
24: 24-24=0
27: 24-27=-3
26: 24-26=-2
23: 24-23=1

STEP 2 Find the mean of the distances.
3+4+1+3+0+3+2+1/8
= 17/8 = 2.125

So, the mean absolute deviation of the data is 2.125

Question 8.
99, 70, 78, 85, 76, 81
________

Answer: The mean absolute deviation of the data is 2

Explanation:
Mean:

Mean = 99+70+78+85+76+81/6 = 489/6 = 81.5

Mean absolute deviation:

STEP 1 Label each observation with its distance from the mean.
Starting from left to right:
24: 23-24=-1
22: 23-22= 1
19: 23-19= 4
19: 23-19= 4
23: 23-23=0
23: 23-23=0
26: 23-26=-3
27: 23-27=-4
24: 23-24=-1

STEP 2 Find the mean of the distances.
1+1+4+4+3+4+1+0+0/9
= 18/9 = 2

So, the mean absolute deviation of the data is 2

Find the range and interquartile range of the data.

Question 11.
39, 22, 33, 45, 42, 40, 28
The range is _________.
The interquartile range is _________.

Answer: 23

Explanation:
The difference between the upper quartile and lower quartile.
Range = 45 – 22 = 23

Page No. 732

Question 12.
Yasmine keeps track of the number of hockey goals scored by her school’s team at each game. The dot plot shows her data.

Where is there a gap in the data?
Type below:
_________________

Answer: There is a gap in the data in the intervals: between 1 and 2

Explanation:
As shown in the dot plot there is a gap between 1 and 2. This means that Yasmine’s team did not score only one goal when they played the game.

Question 13.
What is the interquartile range of the data shown in the dot plot with Question 12?
The interquartile range is _________ .

Answer: 2

Explanation:
The interquartile range is the difference between the lower and upper quartiles.
Interquartile range = 3-1 = 2

Question 14.
Randall’s teacher added up the class scores for the quarter and used a histogram to display the data. How many peaks does the histogram have? Explain how you know.

Type below:
_________________

Answer: The graph has only one peak

Explanation:
In the given graph there is are rectangles out of which one has a tall rectangle which can be addressed as peak.

Question 15.
In a box plot of the data below, where would the box be drawn?
55, 37, 41, 62, 50, 49, 64
Type below:
_________________

Answer: The box is drawn above the number line and this represents the median and the lower and upper limits.

Explanation:
A box is drawn to represent the median and the upper and lower limits in a box plot.

Share and Show – Page No. 735

Question 1.
The distances in miles students travel to get to school are 7, 1, 5, 9, 9, and 8. Decide which measure(s) of center best describes the data set. Explain your reasoning.
Type below:
_________________

Answer: Mean: 6.5
Median: 7.5
Mode: 9

Explanation:
Mean:
7+1+5+9+9+8/6 = 39/6 = 6.5
Median:
Ascending order: 1,5,7,8,9,9
Median = Mean of 7 and 8 = 7+8/2 = 15/2 = 7.5
Mode:
The most frequently occurring observation is known as the mode.
The mode is 9.

Question 2.
Use Graphs The numbers of different brands of orange juice carried in several stores are 2, 1, 3, 1, 12, 1, 2, 2, and 5. Make a box plot of the data and find the range and interquartile range. Decide which measure better describes the data set and explain your reasoning.
Type below:
_________________

Answer: Range: 11      Interquartile: 3 Interquartile range is the best way to represent the data.

Explanation:
Range = 12 – 1 = 11
Interquartile range :
Median(of first 4 terms):
Median = 1+1/2 = 2/2 = 1
Median (of last 4 terms):
Median = 3+5/2 = 8/2 = 4
Interquartile range = 4 – 1 = 3

On Your Own

Unlock the Problem – Page No. 736

Question 5.
Jaime is on the community swim team. The table shows the team’s results in the last 8 swim meets. Jaime believes they can place in the top 3 at the next swim meet. Which measure of center should Jaime use to persuade her team that she is correct? Explain.

a. What do you need to find?
Type below:
_________________

Answer: Mean, median,mode

Explanation:
Mean= 1+2+2+3+3+1+18+2/8 = 32/8 = 4
Median:
Ascending order: 1,1,2,2,2,3,3,18
Median = 2+2/2 = 2
Mode:
The most frequently occurring observation is called a mode.
Mode=2

Question 5.
b. What information do you need to solve the problem?
Type below:
_________________

Answer: We need to have the data to find the centre of tendencies.

Explanation:
The given data can be used to find the mean, median and mode.

Question 5.
c. What are the measures of center?
Type below:
_________________

Answer: Mean = 4 Median = 2 Mode = 2

Explanation:
There are three measures to calculate their approximate values.

Question 5.
d. Which measure of center should Jaime use? Explain.
Type below:
_________________

Answer: Median or mode

Explanation:
Median or mode are nearer to the solution, therefore, they can be used.

Question 6.
The numbers of sit-ups students completed in one minute are 10, 42, 46, 50, 43, and 49. The mean of the data values is 40 and the median is 44.5. Which measure of center better describes the data, the mean or median? Use words and numbers to support your answer.
Type below:
_________________

Answer: Median is the better way to represent the data.

Explanation:
44.5 is closer and represents the more number of observations compared to the mean.

Choose Appropriate Measures of Center and Variability – Page No. 737

Question 1.
The distances, in miles, that 6 people travel to get to work are 14, 12, 2, 16, 16, and 18. Decide which measure(s) of center best describes the data set. Explain your reasoning.
Type below:
_________________

Answer: Mean= 13 miles Median= 15 miles Mode= 16 miles

Explanation:
Mean is less than the data points.
Median describes the data in the best way compared to mean and mode.

Problem Solving

Question 3.
Brett’s history quiz scores are 84, 78, 92, 90, 85, 91, and 0. Decide which measure(s) of center best describes the data set. Explain your reasoning.
Type below:
_________________

Answer: Mean is the best measure of centre to describe the data set.

Explanation:
Mean:
Mean= 84+78+92+90+85+91+0/7 = 74.2
Median:
Ascending order: 0,78,84,85,90,91,92
Median = 4th observation = 85

Question 4.
Eight students were absent the following number of days in a year: 4, 8, 0, 1, 7, 2, 6, and 3. Decide if the range or interquartile range better describes the data set, and explain your reasoning.
Type below:
_________________

Answer: 8 represents all the terms range is more preferable compared to the interquartile range.

Explanation:
Ascending order: 0,1,2,3,4,6,7,8
Range = 8-0 = 8
Median = Mean of 3 and 4 = 3+4/2 = 7/2 = 3.5
Median of first 3 terms = 1
Median of last 3 terms = 7
Interquartile range = 7-1 = 6
Since 8 represents all the terms range is more preferable compared to the interquartile range.

Question 5.
Create two sets of data that would be best described by two different measures of centre.
Type below:
_________________

Answer: The given below are the examples of two sets of data that would be best described by two different measures of centre.

Explanation:
Example 1:

The numbers of pets that several children have are 2, 1, 2, 3, 4, 3, 10, 0, 1, and 0. Make a box plot of the data and find the range and interquartile range. Decide which measure better describes the data set and explain your reasoning.
Type below:
_________________

Answer: Range = 10-0 = 10
Interquartile range = 3.5 – 0.5 = 3
The interquartile range is the best way to represent the data.

Explanation:
Ascending order: 0,0,1,1,2,2,3,3,4,10
Median = 2+2/2 = 2
Lower quartile = 0.5
Upper quartile= 7/2 = 3.5
Highest observation= 10
Lowest observation = 0
Range = 10-0 = 10
Interquartile range = 3.5 – 0.5 = 3
The interquartile range is the best way to represent the data.

Example 2:

Brett’s history quiz scores are 5,6,7,8,9,10. Decide which measure(s) of centre best describes the data set. Explain your reasoning.
Type below:
_________________

Answer: Mean and median are the best measure of centre to describe the data set.

Explanation:
Mean:
Mean= 5+6+7+8+9+10/6 = 7.5
Median:
Ascending order: 5,6,7,8,9,10
Median = Mean of 7 and 8 = 15/2 = 7.5

Lesson Check – Page No. 738

Question 1.
Chloe used two box plots to display some data. The box in the plot for the first data set is wider than the box for the second data set. What does this say about the data?
Type below:
_________________

Answer: The graphs say that the interquartile range is more for the second graph compared to the first.

Explanation:
The interquartile range is the difference between the lower and upper quartiles.
It is more for wider data compared to the compact data.

Question 2.
Hector recorded the temperature at noon for 7 days in a row. The temperatures are 20°F, 20°F, 20°F, 23°F, 23°F, 23°F, and 55°F. Which measure of center would best describe the data?
Type below:
_________________

Answer: Mode

Explanation:
The most frequently occurring observation is known as a mode.
The mode of the above data describes the data well, the mode of the data is 20°F

Spiral Review

Question 4.
What percent of the people surveyed spent at least an hour watching television?

_______ %

Answer: 8 people

Explanation:
Total number of people= 40
Percentage = 8/40 x 100 = 20%

Question 5.
What is the lower quartile of the following data?
12, 9, 10, 8, 7, 12
The lower quartile is _______ .

Answer:

Explanation:
Ascending order: 7,8,9,10,12,12
Median = 9+10/2 = 9.5
Lower quartile = 8

Question 6.
What is the interquartile range of the data shown in the box plot?

The interquartile range is _______ .

Answer: 5

Explanation:
The difference between the upper and lower quartiles is called as interquartile range.
Interquartile range = 14 – 9 = 5

Share and Show – Page No. 741

Question 1.
Zoe collected data on the number of points her favourite basketball players scored in several games. Use the information in the table to compare the data.

The mean of Player 1’s points is __ the mean of Player 2’s
points.
The interquartile range of Player 1’s points is __ the
interquartile range of Player 2’s points.
So, Player 2 typically scores __ points than Player 1, but
Player 2’s scores typically vary __ Player 1’s scores
Type below:
_________________

Answer: less than ; less than ; more ; more

Explanation:
The mean of Player 1’s points is less than the mean of Player 2’s points.
The interquartile range of Player 1’s points is less than the interquartile range of Player 2’s points.
So, Player 2 typically scores more points than Player 1, but Player 2’s scores typically vary in more Player 1’s scores

Question 2.
Mark collected data on the weights of puppies at two animal shelters. Find the median and range of each data set, and use these measures to compare the data.

Type below:
_________________

Answer: They differ slightly but on average we can say that shelter B is more as compared to shelter A

Explanation:
Shelter A
Median:
Ascending order: 5,7,7,7,10,12,15
Median = 4th observation = 7
Range = 15-5 = 10
Shelter B
Median:
Ascending order: 4,5,5,11,11,13,15
Median = 4th observation = 11
Range = 15-4 = 11

On Your Own

Kwan analyzed data about the number of hours musicians in her band practice each week. The table shows her results. Use the table for Exercises 3–5.

Question 3.
Which two students typically practised the same amount each week, with about the same variation in practice times?
Type below:
_________________

Answer: Sally and Jennifer

Explanation:
They are slightly different but on a whole, the average shows no difference and we can say that Sally and Jennifer practised for the same amount each week

Question 4.
Which two students typically practised the same number of hours, but had very different variations in their practice times?
Type below:
_________________

Answer: Tim and Sally

Explanation:
They are different in range but on a whole, the average shows no difference and we can say that Sally and Tim practised for the same number of hours, but had very different variations in their practice times.

Question 5.
Which two students had the same variation in practice times, but typically practised a different number of hours per week?
Type below:
_________________

Answer: Matthew and Tim

Explanation:
Matthew and Tim practised for the same number of hours but they had a high variation in the range.

Problem Solving + Applications – Page No. 742

Question 6.
Compare The table shows the number of miles Johnny ran each day for two weeks. Find the median and the interquartile range of each data set, and use these measures to compare the data sets.

Type below:
_________________

Answer: Interquartile range is the best way to compare the data in the week 1
While the median is the best way to compare the data in the week 2

Explanation:
Week 1
Median:
Ascending order: 1,2,2,3,3,4,5
Median = 4th observation = 3
Lower quartile range= 2
Upper quartile range= 4
Interquartile range = 4-2 = 2

Week 2
Median:
Ascending order: 1,1,1,3,3,8,8
Median = 4th observation = 3
Lower quartile range= 1
Upper quartile range= 8
Interquartile range = 8-1 = 7

Question 7.
Sense or Nonsense? Yashi made the box plots at right to show the data he collected on plant growth. He thinks that the variation in bean plant growth was about the same as the variation in tomato plant growth. Does Yashi’s conclusion make sense? Why or why not?

Type below:
_________________

Answer: Sense

Explanation:
Yashi said that thinks that the variation in bean plant growth was about the same as the variation in tomato plant growth.
It is a true statement because the range of both bean and tomato plants growth is the same and they have the same medians.

Question 8.
Kylie’s teacher collected data on the heights of boys and girls in a sixth-grade class. Use the information in the table to compare the data.

The mean of the boys’ heights is _____ the mean of the girls’ heights.
The range of the boys’ heights is _____ the range of the girls’ heights.

Answer: more than ; more than

Explanation:
Mean of boys height:
Mean = 72+68+70+56+58+62+64/7 = 64.2
Range= 72-56 = 16

Mean of girls height:
Mean = 55+60+56+51+60+63+65/7 = 58.5
Range= 65-51 = 14
The mean of the boys’ heights is _more than____ the mean of the girls’ heights.
The range of the boys’ heights is _more than____ the range of the girls’ heights.

Apply Measures of Center and Variability – Page No. 743

Solve.

Question 1.
The table shows temperature data for two cities. Use the information in the table to compare the data.

The mean of City 1’s temperatures is the ———————– mean of City 2’s temperatures.
The ———————- of City 1’s temperatures is————— the —————–of City 2’s temperatures.
So, City 2 is typically —————-City 1, but City 2’s temperatures
vary ——————-City 1’s temperatures.
Type below:
_________________

Answer: less than; interquartile range; less than; interquartile range; warmer than; more than

Explanation:
The mean of City 1’s temperatures is the —-less than———- mean of City 2’s temperatures.
The –interquartile range——— of City 1’s temperatures is—less than—– the —-interquartile range—–of City 2’s temperatures.
So, City 2 is typically —warmer than—–City 1, but City 2’s temperatures
vary —–more than——-City 1’s temperatures.

Question 2.
The table shows weights of fish that were caught in two different lakes. Find the median and range of each data set, and use these measures to compare the data.

Type below:
_________________

Answer: Lake A’s average is greater but varies more.

Explanation:
Lake A
Median:
Ascending order: 4,6,7,9,10,12
Median = Mean 3rd and 4th observation = 7+9/2 = 8
Range = 12 – 4 = 8

Lake B
Median:
Ascending order: 4,4,5,6,6,7
Median = Mean 3rd and 4th observation = 5+6/2 = 5.5
Range = 7 – 3 = 4

Problem Solving

Question 3.
Mrs. Mack measured the heights of her students in two classes. Class 1 has a median height of 130 cm and an interquartile range of 5 cm. Class 2 has a median height of 134 cm and an interquartile range of 8 cm. Write a statement that compares the data.
Type below:
_________________

Answer: Class 2 is greater but varies more.

Explanation:
The interquartile range is the difference between the lower and upper quartiles. Since the interquartile range is more for class 2 we can say that the extremes are greater while the interquartile range is less for class 1 which means that the data is compact.

Question 4.
Richard’s science test scores are 76, 80, 78, 84, and 80. His math test scores are 100, 80, 73, 94, and 71. Compare the medians and interquartile ranges.
Type below:
_________________

Answer: Medians are equal but the interquartile range varies a large, math test scores are more spread out compared to science test scores.

Explanation:
Science test scores:
Median:
Ascending order: 76,78,80,80,84
Median = 3rd observation = 80
Interquartile range = 84 – 76 = 10

Math test scores:
Median:
Ascending order: 71,73,80,94,100
Median = 3rd observation = 80
Interquartile range = 100 – 71 = 29

Medians are equal but the interquartile range varies a large, math test scores are more spread out compared to science test scores.

Question 5.
Write a short paragraph to a new student that explains how you can compare data sets by examining the mean and the interquartile range.
Type below:
_________________

Answer: average and consistency

Explanation:
If the mean is more it means that the data has more observations or observations with more value.
Interquartile range and median range say about the consistency.

Lesson Check – Page No. 744

Question 1.
Team A has a mean of 35 points and a range of 8 points. Team B has a mean of 30 points and a range of 7 points. Write a statement that compares the data.
Type below:
_________________

Answer: Similar variation but team A average is more than team B

Explanation:
The range has only a difference of 1 point which can be said as a slight variation but while the average/ mean has a large variation.

Question 2.
Jean’s test scores have a mean of 83 and an interquartile range of 4. Ben’s test scores have a mean of 87 and an interquartile range of 9. Compare the students’ scores.
Type below:
_________________

Answer: Ben’s average is more than Jean’s but Ben is less consistent compared to Jean.

Explanation:
Ben’s average score is more than the average scores of Jean while the interquartile range

Spiral Review

Question 3.
Look at the box plots below. What is the difference between the medians for the two groups of data?

_______ students

Answer: 2 students

Explanation:
Median of students in a class of school A = 24
Median of students in a class of school B = 26
Difference between the medians of the schools = 26-24 = 2 students

Question 5.
Which two teams typically practice the same number of hours, but have very different variations in their practice times?

Type below:
_________________

Answer: Team A and C

Explanation:
Team A and C have medians which are only slightly different but the consistency varies a lot that is range.

Share and Show – Page No. 747

Connie asked people their ages as they entered the food court at the mall. Use the histogram of the data she collected for 1–5.

Question 1.
What statistical question could Connie ask about her data?
Type below:
_________________

Answer: Mean, Median, Mode

Explanation:
The graph shows the age and number of people. The questions which can be asked can be of mean, median, mode.

Question 2.
Describe any peak or gap in the data.
Type below:
_________________

Answer: Peak : 21-30   Gap : 61-70

Explanation:
There is a peak in the graph at the interval 21-30
There is a gap in between the bars of the histogram the bar which had a gap before it was 61-70

Question 3.
Does the graph have symmetry? Explain your reasoning.
Type below:
_________________

Answer: No the graph doesn’t have symmetry

Explanation:
The symmetry of the graph means there must be equal parts of the graph on both the sides of the line of the graph.
This is not possible in the above situation.

On Your Own

Question 4.
The lower quartile of the data set is 16.5 years, and the upper quartile is 51.5 years. Find the interquartile range. Is it a better description of the data than the range? Explain your reasoning.
Type below:
_________________

Answer: Interquartile range = 35; The interquartile range is better than the range.

Explanation:
The interquartile range is the difference between the upper quartile and the lower quartile.
Interquartile range = 51.5-16.5 = 35
The interquartile range is better than the range because if we take the example of the above graph we can see thatthe most of the data fall in the range of the interquartile range ie. 35.
Therefore we can say that the interquartile range is better than the range.

Question 5.
Make Arguments The mode of the data is 16 years old. Is the mode a good description of the center of the data? Explain
Type below:
_________________

Answer: No mode is not a good description of the data.

Explanation:
The mode is just a frequently occurring observation.
It cannot be the best way to describe the data.

Problem Solving + Applications – Page No. 748

Use the dot plot for 6–8.

Question 6.
Make Arguments Jason collected data about the number of songs his classmates bought online over the past 3 weeks. Does the data set have symmetry? Why or why not?

Type below:
_________________

Answer: No, the data has no symmetry.

Explanation:
No, the data has no symmetry. Because there are gaps between the dots drawn.

Question 7.
Jason claims that the median is a good description of his data set, but the mode is not. Does his statement make sense? Explain.
Type below:
_________________

Answer: Median can be a better centre of description. Therefore his statement makes a sense.

Explanation:
Median = 7+8/2 = 15/2 = 7.5
The number 7.5 represents more number of observations.

Question 8.
Trinni surveyed her classmates about how many siblings they have. A dot plot of her data increases from 0 siblings to a peak at 1 sibling and then decreases steadily as the graph goes to 6 siblings. How is Trinni’s dot plot similar to Jason’s? How is it different?
Type below:
_________________

Answer: Trinni graph represents a part of Jason’s graph

Explanation:
In Jason’s graph, there is a peak in the middle and then it decreases on both sides.
But according to Trinni graph, there is no peak in the middle.

Question 9.
Diego collected data on the number of movies seen last month by a random group of students.

Draw a box plot of the data and use it to find the interquartile range and range.
Type below:
_________________

Answer: Range = 12  Interquartile range = 2

Explanation:
The range is the difference between the highest and lowest observations.
Range = 12-0 = 12
The interquartile range is the difference between the upper and lower quartiles.
Ascending order: 0,0,1,1,2,2,2,2,3,3,3,5,12
Lower quartile = 1+1/2 = 2/2 = 1
Upper quartile = 3+3/2 = 6/2 = 3
Interquartile range = 3-1 = 2

Describe Distributions – Page No. 749

Chase asked people how many songs they have bought online in the past month. Use the histogram of the data he collected for 1–4.

Question 1.
What statistical question could Chase ask about the data?
Type below:
_________________

Answer: What is the median number of songs purchased?

Explanation:
Many questions can be formed from the data given to us
We can ask about the mean, median, mode.

Question 2.
Describe any peaks in the data.
Type below:
_________________

Answer: Peak : 0-4

Explanation:
The peak is a bar in the histogram which has the highest value. The peak of the given graph is 0-4

Question 3.
Describe any gaps in the data.
Type below:
_________________

Answer: There are no gaps in the graph.

Explanation:
The gap is something between a bar of the histogram and all the other adjacent bars.
There no such case in the graph. Therefore there are no gaps in the graph.

Question 4.
Does the graph have symmetry? Explain your reasoning.
Type below:
_________________

Answer: No the graph doesn’t have symmetry

Explanation:
The symmetry of the graph means there must be equal parts of the graph on both the sides of the line of the graph.
This is not possible in the above situation.

Problem Solving

Question 5.
Mr. Carpenter teaches five classes each day. For several days in a row, he kept track of the number of students who were late to class and displayed the results in a dot plot. Describe the data.

Type below:
_________________

Answer: Peaks: At 6 and 8
Gaps: Between 3 and 5 , 6 and 8
Clusters: Between 0-3 ; 9-11

Explanation:
The highest points in the graph are known as peaks.
They are at 6 and 8 in this graph.
The space between the dots in the dot plot graph is known as gaps.
Gaps are between 3 and 5; 6 and 8
The group of the dots in the dot plot are known as clusters.
The clusters are at the 0-3; 9-11

Question 6.
Describe how a graph of a data set can be used to understand the distribution of the data.
Type below:
_________________

Answer: Mean, median, mode

Explanation:
There are three measures of centre which can be used to describe the data given in the form of a graph.
The three measures of centre are mean, median, mode.

Lesson Check – Page No. 750

Question 2.
What is the median in the dot plot?

$ ________

Answer: 11

Explanation:
Median is the middlemost value and it is 11 in the above graph.
We need to consider the middle value by neglecting the same number on both the sides.

Spiral Review

Question 3.
Look at the dot plot. Where does a gap occur in the data?

Type below:
_________________

Answer: 30-33

Explanation:
The gap is a space between the intervals.
The intervals are 30-33.

Question 4.
Look at the dot plot. Where does a peak occur in the data?

Type below:
_________________

Answer: 37

Explanation:
The highest point in the graph is known as the peak.
The peak in the dot plot is 37.

Question 5.
Which two teams had similar variations in points earned, but typically earned a different number of points per game?

Type below:
_________________

Answer: Red and blue

Explanation:
The difference between the upper and lowest observations is called a range.
The range (consistency) in the data given is the same but they vary in the mean.
But we can say that the Red and Blue teams typically earned a different number of points per game.

Share and Show – Page No. 753

Question 1.
Josh is playing a game at the carnival. If his arrow lands on a section marked 25 or higher, he gets a prize. Josh will only play if most of the players win a prize. The carnival worker says that the average (mean) score is 28. The box plot shows other statistics about the game. Should Josh play the game? Explain your reasoning.

First, look at the median. The median is _ points.
Next, work backwards from the statistics.
The median is the __ value of the data.
So, at least __ of the values are scores
less than or equal to _.
Finally, use the statistics to draw a conclusion.
Type below:
_________________

Answer: middlemost value; 20; 15

Explanation:
First, look at the median. The median is 20 points.
Next, work backwards from the statistics.
The median is the middlemost value of the data.
So, at least 20 of the values are scores
less than or equal to 15.
Finally, use the statistics to draw a conclusion.

Question 2.
What if a score of 15 or greater resulted in a prize? How would that affect Josh’s decision? Explain.
Type below:
_________________

Answer: It doesn’t affect his decision.

Explanation:
Josh wanted to play only when most of the prizes were awarded.
Therefore if the minimum score was 15 or greater than it then they would get the prize. So there will be no difference in his decision.

Question 3.
A store collects data on the sales of DVD players each week for 3 months. The manager determines that the data has a range of 62 players and decides that the weekly sales were very inconsistent. Use the statistics in the table to decide if the manager is correct. Explain your answer.

Type below:
_________________

Answer: No the manager is not correct.

Explanation:
The range is given to be 62.
The range is correct when we check it with the help of the given data, therefore, the data is not inconsistent.

On Your Own – Page No. 754

Question 5.
Susanna wants to buy a fish that grows to be about 4 in. long. Mark suggests she buys the same type of fish he has. He has five of these fish with lengths of 1 in., 1 in., 6 in., 6 in., and 6 in., with a mean length of 4 in. Should Susanna buy the type of fish that Mark suggests? Explain.
Type below:
_________________

Answer: Yes Susanna buy the type of fish that Mark suggests

Explanation:
The length of fish suggested by Mark is 4 in. long.
Mark has 5 fish with a mean length of 4 in.
To buy a fish that grows to be about 4 in. long. Susanna should buy the fish suggested by Mark.

Question 6.
Look for a Pattern The graph shows the number of stamps that Luciano collected over several weeks. If the pattern continues, how many stamps will Luciano collect in Week 8? Explain.

_______ stamps

Answer: 7 stamps

Explanation:
In week 4 and 6, the number of stamps are 4, 5. Therefore in week 6 and 8, the number of stamps are 5, 7

Question 7.
The data set shows the number of hours Luke plays the piano each week. Luke says he usually plays the piano 3 hours per week. Why is Luke’s statement misleading?

Type below:
_________________

Answer: According to the question he should spend 3 hours per week. His statement is correct.

Explanation:
Sum of the data = 1+2+1+3+2+10+2 = 21
Number of days in a week = 7
Mean = 21/7 = 3 hours

Problem Solving Misleading Statistics – Page No. 755

Mr Jackson wants to make dinner reservations at a restaurant that has most meals costing less than $16. The Waterside Inn advertises that they have meals that average $15. The table shows the menu items.

Question 1.
What is the minimum price and the maximum price?
minimum: $ _________
maximum: $ _________

Answer: minimum: $6
maximum: $19

Explanation:
The minimum value is the most minimum price in the given data.
The maximum value is the most maximum price in the given data.

Question 2.
What is the mean of the prices?
$ ________

Answer: $15

Explanation:
Mean = sum of all the observations/ total number of observations = 6+16+18+16+18+19/6 = 93/6 = 15.2
Approximately therefore the mean of the observations is $15

Question 3.
Construct a box plot for the data.
Type below:
_________________

Answer: The box plot is a diagram which signifies the information about the data.

Explanation:

The box plot represents the range, lower and upper quartiles.

Question 4.
What is the range of the prices?
$ ________

Answer: the range is $13

Explanation:
The difference between the upper and lower observations is known as the range.
Range = 19-6 = $13

Question 5.
What is the interquartile range of the prices?
$ ________

Answer: $7.5

Explanation:
Ascending order: $6, $16, $16, $18, $18, $19
Median = 16+18/2 = 34/2 = 17
Lower quartile = 6+16/2 = 11
Upper quartile = 18+19/2 = 18.5
Interquartile range = 18.5-11 = 7.5

Question 6.
Does the menu match Mr. Jackson’s requirements? Explain your reasoning.
Type below:
_________________

Answer: Yes the menu matches Mr Jackson’s requirements.

Explanation:
Mr Jackson wants to make dinner arrangements with cost less than $16.
The mean of the items in the menu:
Mean = $6+$16+$16+$18+$18+$19/6 = 93/6 = $15.5
Therefore the requirements of Mr Jackson is satisfied.

Question 7.
Give an example of a misleading statistic. Explain why it is misleading.
Type below:
_________________

Answer: The Waterside Inn advertises a misleading statement.

Explanation:
According to the information given in the question, The Waterside Inn advertises that they have meals that average $15. But it is more than that, so this is a misleading statement.
The mean of the items on the menu:
Mean = $6+$16+$16+$18+$18+$19/6 = 93/6 = $15.5
Therefore the requirements of Mr. Jackson are satisfied.

Lesson Check – Page No. 756

Question 1.
Mary’s science test scores are 66, 94, 73, 81, 70, 84, and 88. What is the range of Mary’s science test scores?
________

Answer: 28

Explanation:
The difference between the highest and the lowest observations is called a range.
Range = 94 – 66 = 28

Spiral Review

Question 4.
Look at the box plot. What is the interquartile range of the data?

________

Answer: Interquartile range = 6

Explanation:
The difference between the lower and upper quartiles is known as the interquartile range.
Interquartile range = 50 – 44 = 6

Question 5.
Erin is on the school trivia team. The table shows the team’s scores in the last 8 games. Erin wants to build confidence in her team so that they will do well in the last game. If a score of 20 is considered a good score, what measure of center would be best for Erin to use to motivate her teammates?

Type below:
_________________

Answer: Mean and median are the best centre of tendencies to compare the data.

Explanation:
Mean = 20+20+18+19+23+40+22+19/8 = 181/8 = 22.6
Median:
Ascending order: 18,19,19,20,20,22,23,40
Median = 40/2 = 20

Chapter 13 Review/Test – Page No. 757

Question 1.
The dot plot shows the number of chin-ups done by a gym class.

For numbers 1a–1e, choose Yes or No to indicate whether the statement is correct.
1a. There are two peaks.
1b. There are no clusters.
1c. There is a gap from 6 to 8.
1d. The most chin-ups anyone did was 15.
1e. The modes are 3, 4, and 9.
1a. ____________
1b. ____________
1c. ____________
1d. ____________
1e. ____________

Answer: 1a. Yes
1b. No
1c. Yes
1d. No
1e. Yes

Explanation:
1a. The highest point in the dot plot is called the peak. The peak in the given dot plot is at 5 and 11 the value of the peak is 3
1b. The group of dots form a cluster with 3 or more intervals.
1c. There is a gap between the intervals 6-8
1d. The maximum number of people did 11 chin-ups while only a single person did 15 chin-ups.
1e. The most frequently occurring observation is known as mode.
The mode of the given data is at the intervals 3,4 and 9.

Question 2.
The histogram shows the high temperatures in degrees Fahrenheit of various cities for one day in March.

Select the best word to complete each sentence.
The histogram has _____ peak(s).
The histogram _____ symmetry.

Answer: The histogram has 1 peak(s).
The histogram is symmetry.

Explanation:
The is one and only one peak at the interval 41 – 50
We can say that the graph is symmetrical because if we draw a line between the graph we can observe that the graph has two parts symmetric to each other.

Chapter 13 Review/Test – Page No. 758

Question 3.
The data set shows the scores of the players on the winning team of a basketball game.

The median is _____.
The lower quartile is _____.
The upper quartile is _____.

Answer:Median = 6
Lower quartile = 1
Upper quartile =19.5

Explanation:
Ascending order: 0,0,1,1,4,5,6,9,13,17,22,30,47
Median = 6
Lower quartile = Mean of 1 and 1 = 1+1/2 = 2/2 = 1
Upper quartile = Mean of 17 and 22 = 17+22/2 = 39/2 = 19.5

Question 4.
The data set shows the number of desks in 12 different classrooms.

Find the values of the points on the box plot.

Type below:
_________________

Answer: A= 16 B=17 C= 20 D= 21 E=24

Explanation:
Ascending order: 16,17,17,18,19,20,20,21,21,21,22,24
Median = 20+20/2 = 20
Lower quartile = 17
Upper quartile = 21

Question 5.
The box plot shows the number of boxes sold at an office supply store each day for a week.

For numbers 5a–5d, select True or False for each statement.
5a. The median is 18.
5b. The range is 15.
5c. The interquartile range is 9.
5d. The upper quartile is 18.
5a. ____________
5b. ____________
5c. ____________
5d. ____________

Answer: 5a.  false
5b. true
5c. true
5d. true

Explanation:
Median is the middlemost value of the given data.
Median of the data is 14
The range is the difference between the upper and lower observations.
Range = 21-6 = 15
The interquartile range is the difference between the upper and lower observations.
Upper quartile range: 18
Interquartile range = 18-9 = 9

Chapter 13 Review/Test – Page No. 759

Question 6.
The data set shows the number of glasses of water Dalia drinks each day for a week.

Part A
What is the mean number of glasses of water Dalia drinks each day?
_______ glasses

Answer: The mean number of glasses of water Dalia drinks each day is 8 glasses.

Explanation:
Mean = sum of all the observations/ total number of observations= 6+7+9+9+8+7+10/7 = 8

Question 6.
Part B
What is the mean absolute deviation of the number of glasses of water Dalia drinks each day? Round your answer to the nearest tenth. Use words and numbers to support your answer.
_______

Answer: Mean absolute deviation is 1.14

Explanation:

Mean:
Mean = sum of all the observations/ total number of observations= 6+7+9+9+8+7+10/7 = 8

Mean absolute deviation:

STEP 1 Label each observation with its distance from the mean.
Starting from left to right:
06: 8-6  = 02
07: 8-7  = 01
09: 8-9  = -1
09: 8-9  = -1
08: 8-8  =  0
07: 8-7  = 01
10: 8-10= -2

STEP 2 Find the mean of the distances.
2+1+1+1+0+1+2/7
= 8/7 = 1.14

So, the mean absolute deviation of the data is 1.14

Question 7.
The numbers of emails Megan received each hour are 9, 10, 9, 8, 7, and 2. The mean of the data values is 7.5 and the median is 8.5. Which measure of center better describes the data, the mean or median? Use words and numbers to support your answer.
Type below:
_________________

Answer: Mean is the best center of the tendency to represent the data given in the question

Explanation:
Ascending order of the data: 2,7,8,9,9,10
Mean = 7.5
Mean represents the observations 8,9,9,10 which come after 7.5
Therefore mean is the best way to represent the data.

Question 8.
The number of miles Madelyn drove between stops was 182, 180, 181, 184, 228, and 185. Which measure of center best describes the data?
Options:
a. mean
b. median
c. mode

Answer: b. Median

Explanation:
Ascending order: 180,181,182,184,185,228
Median = 182+184/2 = 183
183 represents all the observations after 182
So the median is the best way to represent the data.

Chapter 13 Review/Test – Page No. 760

Question 9.
The histogram shows the weekly earnings of part-time workers. What interval(s) represents the most common weekly earnings?

Type below:
_________________

Answer: 321-330 ; 341-350

Explanation:
The histogram has 2 intervals which show equal heights which means that the monthly earnings of these intervals is the same.

Question 10.
Jordan surveyed a group of randomly selected smartphone users and asked them how many applications they have downloaded onto their phones. The dot plot shows the results of Jordan’s survey. Select the statements that describe patterns in the data. Mark all that apply.

Options:
a. The modes are 37 and 42.
b. There is a gap from 38 to 40.
c. There is a cluster from 41 to 44.
d. There is a cluster from 35 to 36.

Answer: b. There is a gap from 38 to 40.

Explanation:
The dot plot represents a gap between 38-40. So we can say that there is a gap between intervals 38 to 40.

Chapter 13 Review/Test – Page No. 761

Question 11.
Mrs. Gutierrez made a histogram of the birth month of the students in her class. Describe the patterns in the histogram by completing the chart.


Type below:
_________________

Answer: There are 2 peaks, Yes there is an increase across the intervals, Yes there is a decrease across the intervals

Explanation:
The highest point in the histogram is called is as a peak.
There is a peak near the month’s May and August.

There is an increase between the bars in the bar graph.
At the months February, March, November there is an increase in the graph.

There is a decrease between the bars in the bar graph.
At the months September, October, December.

Question 12.
Ian collected data on the number of children in 13 different families.

Draw a box plot of the data and use it to find the interquartile range and range.
Type below:
_________________

Answer: Range = 8-0 = 8 Interquartile range = 3-1 = 2

Explanation:

Ascending order:
0,0,1,1,1,1,2,2,2,3,3,4,8
Median = 2
Lower quartile = 1+1/2 = 1
Upper quartile = 3+3/2 = 3
Range = 8-0 = 8
Interquartile range = 3-1 = 2

Chapter 13 Review/Test – Page No. 762

Question 13.
Gavin wants to move to a county where it rains about 5 inches every month. The data set shows the monthly rainfall in inches for a county. The mean of the data is 5 and the median is 4.35. After analyzing the data, Gavin says that this county would be a good place to move. Do you agree or disagree with Gavin? Use words and numbers to support your answer.

Type below:
_________________

Answer: Yes I agree that it is a good place to move.

Explanation:
After analyzing the data we can say that this country would be a good place to move for Gavin.
Gavin wants to move to a place which has an average of 5 cm rainfall. So this country is the best choice because it has a mean equal to 5 and median equal to 4.35

Question 14.
The data set shows the number of books Peyton reads each month. Peyton says she usually reads 4 books per month. Why is Peyton’s statement misleading?

Type below:
_________________

Answer: No Peyton’s statement is not misleading because the mean of the data is 4.
Therefore Peyton says she usually reads 4 books per month.

Explanation:
Mean = 2+3+2+4+3+11+3/7 = 28/7 = 4

Question 15.
The data set shows the scores of three players for a board game.

For numbers 15a–15d, choose Yes or No to indicate whether the statement is correct.
15a. The mean absolute deviation of Player B’s scores is 0.
15b. The mean absolute deviation of Player A’s scores is 0.
15c. The mean absolute deviation of Player B’s scores is greater than the mean absolute deviation of Player C’s scores.
15a. ___________
15b. ___________
15c. ___________

Answer:15a. No
15b. Yes
15c.  Yes

Explanation:
Mean of player A = 90+90+90/3 = 90

Mean absolute deviation:

STEP 1 Label each observation with its distance from the mean.
Starting from left to right:
90: 90-90  = 0
90: 90-90  = 0
90: 90-90  = 0

STEP 2 Find the mean of the distances.
0+0+0/3 = 0
So, the mean absolute deviation of player A is 0

Mean of player B = 110+100+90/3 = 100

Mean absolute deviation:

STEP 1 Label each observation with its distance from the mean.
Starting from left to right:
100: 100-110  = -10
100: 100-100  = 0
100: 100-90    = 10

STEP 2 Find the mean of the distances.
10+0+10/3 = 20/3 = 6.67
So, the mean absolute deviation of player B is 6.67

Mean of player C = 95+100+95/3 = 96.67

Mean absolute deviation:

STEP 1 Label each observation with its distance from the mean.
Starting from left to right:
96.67: 96.67-95    = 1.67
96.67: 96.67-100  = -3.33
96.67: 96.67-95    = 1.67

STEP 2 Find the mean of the distances.
1.67+3.33+1.67/3 = 6.67/3 = 2.22
So, the mean absolute deviation of player C is 2.22

Conclusion:

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