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©Curriculum Associates, LLC Copying is not permitted. 39Lesson 5 Terminating and Repeating Decimals

Name: Terminating and Repeating Decimals

Lesson 5

Prerequisite: Understand Fractions as Division

Study the example showing fractions as division. Then solve problems 1–5.

1 The teacher has 5 cups of broth to share among the 4 students

a. How much broth will each student get? Draw a model to explain your answer

Example

A teacher has 3 cups of brown rice to share equally among 4 students in cooking class How much rice will each student get?

You can model this problem with a picture

Look at one of the cups One cup

divided among 4 students is 1 ·· 4 cup

You can think of this as 1 4 4 5 1 ·· 4

The teacher has 3 cups, so each student will receive 3 ·· 4 cup of rice

You can think of this as 1 ·· 4 3 3 5 3 ·· 4 or 3 4 4 5 3 ·· 4

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cup

12

14

cup

12

14

cup

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b. Will each student get more than or less than 1 cup of broth? Explain

2 What if the teacher divides the 5 cups of broth among 6 students? Will each student get more than or less than 1 cup of broth? Explain

3939

5 ·· 4 cups Possible model shown above.

more than 1 cup; Possible explanation: 5 ·· 4 is more than 4 ·· 4 , which is 1 whole.

less than 1 cup; Possible explanation: 5 4 6 5 5 ·· 6 , which is less than 6 ·· 6 , or 1.

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©Curriculum Associates, LLC Copying is not permitted.40 Lesson 5 Terminating and Repeating Decimals

Solve.

3 Five seventh-grade students are decorating 3 bulletin boards and want to divide the work equally

a. Will each student decorate more than or less than 1 bulletin board? Explain how you know

b. How many fifths of a bulletin board are there in 3 bulletin boards? Use a model to explain

c. How many fifths of a bulletin board will each student decorate?

fifths

d. Write this as a fraction of a bulletin board

of a bulletin board

4 A science teacher has 15 ounces of vinegar to divide

equally among 8 students for an experiment Matt said

that each student should get 8 ·· 15 ounce How did he get

that answer? Do you agree? Explain

5 One restaurant uses 5 quarts of soup to make 20 equal servings Another restaurant uses 7 quarts of soup to make 25 equal servings Which restaurant has larger servings of soup? Write division expressions to represent the problem and then solve

Show your work.

Solution:

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less than 1 bulletin board; Possible explanation: There are more students than there

are bulletin boards.

15 fifths

3

3 ·· 5

Possible answer: Matt found 8 4 15 instead of 15 4 8; no; The 15 ounces of vinegar are

to be divided among 8 students, so each student should get 15 ··· 8 ounces.

Possible work: 5 4 20 5 5 ··· 20 5 25 ···· 100 ; 7 4 25 5 7 ··· 25 5 28 ···· 100 ; 25 ···· 100 28 ···· 100

The second restaurant has larger servings.

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©Curriculum Associates, LLC Copying is not permitted. 41Lesson 5 Terminating and Repeating Decimals

Name: Lesson 5

Using Patterns to Write Fractions as Decimals

Study the example showing how to use patterns to write fractions as equivalent decimals. Then solve problems 1–7.

1 Describe the patterns in the table

a. What is the pattern in the first column?

b. What is the pattern in the second column?

c. How can you use the fraction in the first column to write the equivalent decimal in the second column?

2 Explain how you can use the patterns in the table to fi nd

the decimal form of 17 ·· 20

Example

Carlos knows that 1 ·· 20 5 0 05 He can use a table of values

to find a pattern that will help him find the decimal

forms of 2 ·· 20 , 3 ·· 20 , 4 ·· 20 , and 5 ·· 20

0 0520 q ······ 1 00

21 00

0

Because 2 ·· 20 is twice 1 ·· 20 ,

the decimal form of 2 ·· 20

is twice 0 05, or 0 10

Fraction Decimal

1 ·· 20 0 05

2 ·· 20 0 10

3 ·· 20 0 15

4 ·· 20 0 20

5 ·· 20 0 25

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The fractions increase by 1 ··· 20 .

The decimals increase by 0.05.

Possible answer: Multiply the numerator of the fraction by 0.05 to get the

decimal form.

Possible answer: Because 17 ··· 20 5 17 3 1 ··· 20 , I know that 17 ··· 20 5 17 3 0.05 5 0.85. So the

decimal form of 17 ··· 20 is 0.85.

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©Curriculum Associates, LLC Copying is not permitted.42 Lesson 5 Terminating and Repeating Decimals

Solve.

3 What kind of decimal is 0 05? How do you know?

4 Use the fact that 1 ·· 20 5 0 05 to write the decimal form

of the unit fractions 1 ·· 10 , 1 ·· 5 , and 1 ·· 4 in the table below

Unit Fraction Decimal

1 ·· 10

1 ·· 5

1 ·· 4

5 Explain how you can fi nd each decimal form in

problem 4 using the decimal form of 1 ·· 20

6 Marita knows that 1 ·· 5 written as a decimal is 0 20

a. How can she find the decimal form of 9 ·· 5 without dividing?

b. What is another way that Marita could figure out the

decimal form of 9 ·· 5 without dividing?

7 Chuck knows that 1 ·· 50 5 0 02 How can he use this

information to fi nd the fraction form for the decimal 0 86?

Vocabularyterminating decimal a decimal that ends, or

terminates

0 5; 4 08; 0 300

42

a terminating decimal; The decimal 0.05 ends and does not repeat.

0.10

0.20

0.25

Possible answer: She can multiply 0.20 by 9 to get 1.80.

Possible answer: I can double the decimal for 1 ··· 20 to get the decimal for 1 ··· 10 because 1 ··· 10 5

2 ··· 20 5 2 3 1 ··· 20 . Similarly, I can multiply the decimal for 1 ··· 20 by 4 to get the decimal for 1 ·· 5 because

1 ·· 5 5 4 ··· 20 . I can multiply the decimal for 1 ··· 20 by 5 to get the decimal for 1 ·· 4 because 1 ·· 4 5 5 ··· 20 .

Possible answer: Because 9 ·· 5 5 5 ·· 5 1 4 ·· 5 and 5 ·· 5 5 1, she could find the decimal form of 4 ·· 5 and

add 1 to get 1 1 0.80, or 1.80.

Possible answer: Divide 0.86 by 0.02 to get 43.

Because 0.86 is 43 3 0.02, the fraction is 43 ··· 50 .

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©Curriculum Associates, LLC Copying is not permitted.44 Lesson 5 Terminating and Repeating Decimals

Solve.

4 You saw on the previous page that 2 ·· 11 5 0 ··· 18 Show how

you can use what you know about the decimal for 2 ·· 11 to

fi nd the decimal for 1 ·· 11

5 A 9-member team plans to run a 4-mile relay race Distance markers are placed on the racecourse every 0 25 mile

a. Place an X on the number line at the approximate locations where the relay exchanges will take place

b. Will any of the relay exchanges take place at any of the 0 25-mile markers? If so, which one(s)? List the locations of all of the exchanges in decimal form

6 If the decimal form for a unit fraction is a repeating decimal, is it possible for a multiple of that fraction to have a decimal form that is not a repeating decimal? Use examples to explain your reasoning

7 Mario claims that if the denominator of a fraction is a prime number, then its decimal form is a repeating decimal Do you agree? Use an example to explain

21 3 4Distance (mi)

Start End

44

Possible answer: Divide 0. ··· 18 by 2; 1 ··· 11 5 0. ··· 09 .

no; 0. ··· 4 , 0. ··· 8 , 1. ··· 3 , 1. ··· 7 , 2. ··· 2 , 2. ··· 6 , 3. ··· 1 , 3. ··· 5

yes; Possible answer: 1 ··· 11 5 0. ··· 09 , but 11 ··· 11 5 1, which is not a repeating decimal.

no; Possible answer: 5 is a prime number, but 1 ·· 5 5 0.2, which is not a repeating decimal.

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©Curriculum Associates, LLC Copying is not permitted. 43Lesson 5 Terminating and Repeating Decimals

Name: Lesson 5

Writing Fractions as Decimals

Study the example problem showing how to write a fraction as a decimal. Then solve problems 1–7.

1 How do you know that the decimal for 2 ·· 11 repeats without end?

2 Between which two tenths of a mile will the second relay exchange occur? Explain

3 Will there be more than one relay exchange between 2 consecutive tenths of a mile? Explain

Example

A relay race is 2 miles long Eleven students will run the relay,

splitting the distance as equally as possible for a distance of

about 2 ·· 11 mile each Does the first relay exchange occur

before or after 0 2 mile?

Vocabularyrepeating decimal a

decimal that never ends

but instead repeats

the same digit or

digits over and over

0 6666… 5 0 ·· 6

0.17 0.18 0.19 0.20

0.18

The first relay exchange occurs before 0 2 mile because 0 ··· 18 , 0 2

2 ·· 11 5 0 1818 5 0 ··· 18 0 1818

11 q ······ 2 0000 2 1 1

902 88

202 11

902 88

2

2 ·· 11 5 2 4 11

43

Possible answer: The digits 1 and 8 will continue to repeat in the quotient because

differences in the long division will continue to alternate between 90 and 20.

no; Possible answer: Because 0. ··· 18 0.1, there cannot be

more than one relay exchange within a 0.1-mile interval.

between the markers at 0.3 and 0.4; Possible explanation:

The second relay exchange will occur 4 ··· 11 mile from the

beginning of the course, so I can double the decimal for

2 ··· 11 mile; 2 3 0. ··· 18 5 0. ··· 36 ; 0. ··· 36 0.3 and 0. ··· 36 0.4.

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©Curriculum Associates, LLC Copying is not permitted.46 Lesson 5 Terminating and Repeating Decimals

Solve.

4 Mark estimates that he spends 3 ·· 8 of his money on lunches

What percent of his money does he spend on lunches?

A 2 6% C 26%

B 3 75% D 37 5%

Sue chose B as the correct answer How did she get that answer?

5 Mrs Gelb is making costumes for a play The table shows how many yards of each color ribbon are needed and how many yards she has on hand

Color Yards She Needs Yards She Has

Red 2 5 ·· 8 2 5

Green 7 ·· 8 0 36

Black 4 3 ·· 8 4 5

Tell whether each statement is True or False

a. She does not need to buy any black ribbon u True u False

b. She needs to buy 3 ·· 8 yard of red ribbon u True u False

c. She needs to buy at least 3 ·· 5 yard of green ribbon u True u False

6 Write a fraction and its decimal equivalent for the following conditions

a. a fraction that is a repeating decimal between 0 25 and 0 5

b. a fraction that is a repeating decimal between 0 5 and 1

c. a fraction that is a terminating decimal between 1 25 and 1 5

How can you compare fractions and decimals?

What fractions can you think of that you know repeat? When will a fraction be greater than 1?

When you write a decimal as a percent, what do you need to do to the decimal point?

46

Possible answer: Sue moved the decimal point in 0.375 one decimal place to the

right, but she should have moved the decimal point two places to the right.

Possible answer: 1 ·· 3 = 0. ·· 3

Possible answer: 5 ·· 6 = 0.8 ·· 3

Possible answer: 11 ··· 8 = 1.375

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©Curriculum Associates, LLC Copying is not permitted. 45Lesson 5 Terminating and Repeating Decimals

Name:

1 Carla says that 5 ·· 6 is greater than 0 8 Is she correct?

Show your work.

Solution:

Terminating and Repeating Decimals

Solve the problems.

2 Which of the following are repeating decimals when written in decimal form? Select all that apply

A 11 ·· 16

B 8 ·· 11

C 3 ·· 9

D 9 ·· 18

3 Complete the table Then describe a pattern in thedecimal forms of the fractions

Unit Fraction 1 ·· 2 1 ·· 3 1 ·· 4 1 ·· 5 1 ·· 6

Decimal 0 5

How can you write a fraction as a division problem?

What happens to the value of the decimal as the denominator of the fraction increases?

Can you simplify any fractions to fractions that you know are not repeating decimals?

Lesson 5

45

Carla is correct because 5 ·· 6 5 0.8 ·· 3 , which is greater than 0.8.

Possible answer: As the denominator of the

fraction increases, the decimal form of the

fraction decreases.

0. ·· 3 0.25 0.2 0.1 ·· 6

Possible work: 5 ·· 6 5 5 4 60.8333...

6 q ······ 5.0000 2 48

202 18

202 18

202 18

2

0.8 ·· 3 0.8

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