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Lesson 3-6

Adding and Subtracting Unlike FractionsAdding and Subtracting Unlike Fractions

Vocab

Essential

Question

What happens when you add, subtract, multiply, and divide rational numbers?

Common Core

State Standards

Content Standards 7.NS.1, 7.NS.1d, 7.NS.3, 7.EE.3

Mathematical Practices 1, 3, 4

Vocabulary

unlike fractions

What You’ll Learn

• Add unlike fractions.• Subtract unlike fractions.

Real-World Link

Climate The Sahara Desert covers almost eleven countries, fully or partially. It is one of the driest places in the world, receiving only a couple of inches of rainfall each year.

Interactive

Study Guide

See pages 63–64 for:• Getting Started• Real-World Link• Notes

Unlike fractions are fractions with different denominators. Use the least common multiple of the denominators to rename the fractions before adding them.

Find 3 _ 5 + 1 _ 3 . Write in simplest form. Estimate 1 + 0 = 1

3 _ 5 + 1 _ 3 = 3 _ 5 · 3 _ 3 + 1 _ 3 · 5 _ 5 Use 3 � 5 or 15 as the common denominator.

= 9 _ 15 + 5 _ 15 Rename each fraction with the common denominator.

= 14 _ 15 Add the numerators.

Check for Reasonableness 14 __ 15 ≈ 1 �

Find each sum. Write in simplest form. 1a. 1 _ 6 + 3 _ 4

11

_ 12

1b. 2 _ 7 + 3 _ 14 1

_ 2

1c. 2 _ 5 + 3 _ 10 7

_ 10

1d. 3 _ 8 + 5 _ 24 7

_ 12

Tutor

Example 1

Do these problems to find out. Got It ?Got It ?

Words To add fractions with unlike denominators, rename the fractions with a common denominator. Then add and simplify as with like fractions.

Symbols 2 _ 3 + 1 _ 2 = 2 _ 3 · 2 _ 2 + 1 _ 2 · 3 _ 3

= 4 _ 6 + 3 _ 6

= 7 _ 6 or 1 1 _ 6

Add Unlike FractionsAAKey Concept

ers

aces uple

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Objectives add and subtract unlike fractions

Building on the Essential Question

At the end of the lesson, students should be able to answer “If you use 24 instead of 12 as a common denominator when finding 2 3 _ 4 - 5 5 _ 6 , will you get the correct answer?”

Example 1

What’s the Math? adding unlike fractions• Why was 15 chosen as the new denominator for both

fractions? The least common multiple of 5 and 3 is 15, so each fraction must be rewritten as an equivalent fraction with a denominator of 15 before they can be added.

Need Another Example?

Find 3 _ 4 + 1 _ 7 . Write in simplest form. 25 _ 28

2 Teach the Concept

126 Chapter 3 Operations with Rational Numbers

Rational Numbers and Exponents

Find each sum. Write in simplest form. a. − 5 _

6 + 1 _

8 Estimate -1 + 0 = -1

- 5 _ 6 + 1 _ 8 = -5 _ 6 · 4 _ 4 + 1 _ 8 · 3 _ 3 The LCD of 6 and 8 is 24.

= -20 _ 24 + 3 _ 24 Rename each fraction using the LCD, 24.

= -17 _ 24 or - 17 _ 24 Simplify.

b. − 3 _ 10

+ (−5 3 _ 4

) Estimate - 1 _

2 + -6 = -6 1 _

2

- 3 _ 10 + (-5 3 _ 4 ) = -3 _ 10 + -23 _ 4 Write -5 3 _ 4

as an improper fraction.

= -3 _ 10 · 2 _ 2 + -23 _ 4 · 5 _ 5 The LCD of 10 and 4 is 20.

= -6 _ 20 + -115 _ 20 Rename each fraction using the LCD, 20.

= -121 _ 20 or -6 1 _ 20 Simplify.

2a. 3 3 _ 4 + 5 _ 14 4 3

_ 28

2b. -6 8 _ 9 + 7 5 _ 12 19

_ 36

2c. 3 3 _ 5 + (-4 5 _ 6 )

Tutor

Example 2

Do these problems to find out. Got It ?Got It ?

-1 7

_ 30

To subtract fractions with unlike denominators, rename the fractions with a common denominator. Then subtract and simplify as with like fractions.

Subtract Unlike FractionsSSKey Concept

The rule for subtracting fractions with unlike denominators is similar to the rule for addition.

Find each difference. Write in simplest form. a. 3 _

8 - 3 _ 4

3 _ 8

- 3 _ 4

= 3 _ 8

- 3 _ 4

· 2 _ 2 The LCD is 8.

= 3 _ 8

- 6 _ 8

= -3 _ 8

or - 3

_ 8 Simplify.

b. 9 3 _ 5 - 7 2 _ 3

9 3 _ 5

- 7 2 _ 3

= 48 _ 5

- 23 _ 3

Rename 9 3 _ 5

.

= 48 _ 5

· 3 _ 3 - 23 _ 3

· 5 _ 5

= 144 _ 15

- 115 _ 15

= 29 _ 15

or 1 14 _ 15

Simplify.

3a. 3 _ 4 - 8 _ 9 - 5

_ 36

3b. 7 1 _ 6 - 6 5 _ 8 13

_ 24

3c. 5 1 _ 3 - (-4 5 __ 9

)

Tutor

Example 3

Rename using the LCD.

The LCD is 15.

Rename using the LCD.

Do these problems to find out. Got It ?Got It ?

89

_ 9

or 9 8

_ 9

LCD

You can rename fractions using any common denominator. However, using the least common denominator will make simplifying the solution easier.

Reasonableness

Use estimation to check whether your answer is reasonable.

3 _ 8 − 3 _ 4 ≈ 1 _ 2 − 1

≈ − 1 _ 2

− 3 _ 8 is close to − 1 _ 2 .

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Example 2

What’s the Math? add fractions and mixed numbers• In Example 2a, how do you get the numerator -17 in

the answer? -20 + 3 = -17

• In Example 2b, how does estimating the answer help you? An estimation helps you to know if your answer is reasonable.

• In Example 2b, how do you get the fraction -23_4 ? The

mixed number -5 3_4

is written as an improper fraction with a denominator of 4 by first multiplying and then adding to find the numerator, 5 × 4 + 3 = 23 and the

original mixed number is negative, so -5 3_4

=-23_

4 .

Need Another Example?

Find each sum. Write in simplest form.

a. 5_6 +

(

-3_

10 )8_

15

b. 2 1_8 +(

-3 2_3 ) -1 13_24

Example 3

What’s the Math? subtract fractions and mixed numbers• In Example 3a, why is the answer a negative number?

Since 6_8

is greater than 3_8

, subtracting a greater number,6_8

, from a lesser number, 3_8

, results in a negative

number, -3_8

.

• In Example 3b, why not just subtract the whole number parts, then subtract the fraction parts? This is not a good idea since the fraction parts may need to be renamed.

Need Another Example?

Find each difference. Write in simplest form.a.

9_16 -

5_8 -

1_16

b. 4 2_3 - 3 6_717_21

Lesson 3-6 Adding and Subtracting Unlike Fractions 127

To set up a computer network in an office, a 100-foot cable is cut and used to connect three computers to the server as shown. How much cable is left to connect the third computer?

?Server

Server

19 ft18

40 ft34

Add the measures of the cables that were already used and subtract that sum from 100, the length of the cable.

19 1 _ 8 + 40 3 _ 4 = 19 1 _ 8 + 40 6 _ 8 Rename 40 3 _ 4

using the LCD, 8.

= 59 7 _ 8 Simplify.

100 - 59 7 _ 8 = 99 8 _ 8 - 59 7 _ 8 Rename 100 as 99 8 _ 8

.

= 40 1 _ 8 Simplify.

There is 40 1 _ 8 feet of cable left to connect the third computer.

Check for Reasonableness 40 1 _ 8

≈ 40 �

4. At a recent frog-jumping contest, the winning frog jumped 21 1 _ 3 feet.

The second-place frog jumped 20 1 _ 2 feet. How much farther did the first-place frog jump?

Tutor

Example 4

Estimate 100 - (19 + 41) ≈ 100 - 60 or 40 feet

Do this problem to find out. Got It ?Got It ?

5

_ 6

ft

Find each sum. Write in simplest form. (Examples 1 and 2)

1. 1 _ 15 + 3 _ 5

2

_ 3

2. − 5 _ 9 + 1 _ 6 − 7

_ 18

3. 7 _ 8 + (− 2 _ 7 )

4. 8 5 _ 12 + 11 1 _ 4 19 2

_ 3

5. −2 1 _ 3 + (−7 1 _ 2 ) −9 5

_ 6

6. 4 3 _ 8 + 10 5 _ 12

Find each difference. Write in simplest form. (Example 3)

7. − 1 _ 4 - 7 _ 9 −1 1

_ 36

8. 3 _ 5 - 9 _ 10 −

3

_ 10

9. 5 _ 8 - 7 _ 12

1

_ 24

10. −1 1 _ 3 - 4 2 _ 7 −5 13

_ 21

11. 5 5 _ 6 - (−2 1 _ 4 ) 8 1

_ 12

12. 12 1 _ 2 - 6 3 _ 8

13. Dwayne needs 1 2 _ 3 cups of shredded cheese to put in his enchilada casserole and 3 _ 4 cup of cheese for the top. If he has 3 cups of cheese in all, how much cheese will

he have left? (Example 4)

Guided PracticeGuided PracticeCheck

33

_ 56

14 19

_ 24

6 1

_ 8

7_12

c

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Example 4

What’s the Math? add and subtract mixed numbers• Could you have used 16 as the common denominator?

Explain your answer. Yes; using the least common denominator eliminates steps in simplifying later, but any common denominator, such as 16, can be used.

• Why must 100 be written as a mixed number? The number being subtracted has a fraction part and so 100 must be renamed with a fraction part so the fraction parts can be subtracted.

Need Another Example?

Juyong jogged three days this week for a total of 11 17_

20 miles. If she jogged 3 1_2 miles one day and 4 1_4miles another, how many miles did she jog on the third day? 4 1_

10 mi

Formative AssessmentGuided Practice Use these exercises to assess students’ understanding of the concept of the lesson. If they need more help, use the Personal Tutors available online.

Have students explain how yesterday’s lesson on adding and subtracting like fractions helped them with today’s lesson on adding and subtracting unlike fractions. See students’ work.

Have students exp

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128 Chapter 3 Operations with Rational Numbers

Rational Numbers and Exponents

3 Practice and ApplyHomework

The Independent Practice pages are meant to be used as the homework assignment. If you do not wish to assign the entire exercise set, you can use the table below to select appropriate exercises for your students’ needs.

Differentiated Homework Options

ALApproaching Level

14–36, 47–50, 52–69

OL On Level15–35 odd, 37–39, 41–45 odd, 46–50, 52–69

BL Beyond Level 37–69

Create Your Own Homework Online

can be used to create worksheets for the suggested assignments above, or create your own worksheets for differentiated homework or review.

Additional Answer

37b. (8, 2), (7, 3), (6, 4), (5, 5)

Wid

th

23

1

0

45678

Length

321 5 74 6 8

y

x

Find each sum or difference. Write in simplest form. (Examples 1 and 3)

14. − 8 _ 35 + 2 _ 5 6

_ 35

15. 5 _ 7 + (− 10 _ 21 )

5

_ 21

16. − 5 _ 8 + (− 8 _ 9 ) −1 37

_ 72

17. − 1 _ 3 + (− 10 _ 11 ) −1 8

_ 33

18. 7 _ 8 − 2 _ 5 19

_ 40

19. 1 _ 6 − 5 _ 7 − 23

_ 42

20. − 2 _ 5 − 1 _ 3 − 11

_ 15

21. − 3 _ 10 − 1 _ 15 − 11

_ 30

22. − 3 _ 4 − 5 _ 6 −1 7

_ 12

During spring training, the Detroit Tigers won about 2 _ 3 of the games they

played while the Cleveland Indians won 7 _ 15 of the games they played. What

fraction more of the games did Detroit win than Cleveland? 1

_ 5

24. In a college dormitory, 1 _ 10 of the residents are juniors and 2 _ 5 of the residents are

sophomores. What fraction of the students at the dormitory are juniors and sophomores?

1

_ 2

Find each sum or difference. Write in simplest form. (Examples 2 and 3)

25. 8 1 _ 2 + 3 4 _ 5 12 3

_ 10

26. −10 2 _ 3 + 9 7 _ 12 −1 1

_ 12

27. 16 5 _ 6 − 12 1 _ 3 4 1

_ 2

28. 6 6 _ 7 − 11 7 _ 8 −5 1

_ 56

29. −4 1 _ 9 − 7 2 _ 3 −11 7

_ 9

30. − 5 _ 6 + 8 1 _ 4 7 5

_ 12

31. 9 _ 16 + 3 5 _ 6 4 19

_ 48

32. −5 3 _ 5 + (−7 1 _ 6 ) −12 23

_ 30

33. −10 1 _ 2 − 6 5 _ 7 −17 3

_ 14

34. Sybrina wants to make a 17-inch necklace with a 3 _ 4 -inch bead, a 1 1 _ 2 -inch bead,

and another 3 _ 4 -inch bead on it. What is the length of the remaining part of the

necklace? (Example 4) 14 in.

35. Kenzie is making three desserts for a party. The recipes call for 2 _ 3

cup sugar,

1 5 _ 6 cups sugar, and 2 3 _ 4 cups sugar. If she has 6 cups of sugar, how much sugar

will she have left over? 3

_ 4

c

36. The length of a page in a yearbook is 10 inches. The top margin is 1 _ 2 inch, and the

bottom margin is 3 _ 4 inch. What is the length of the page inside the margins? 8 3

_ 4

in.

37. Multiple Representations The perimeter of a geometric figure is the distance around the figure. You can find the perimeter of a rectangle by adding the measures of all four sides.

a. Table Copy and complete the table at the right by listing the lengths and widths of three additional rectangles that have a perimeter of 20.

b. Graph Write the values from the table as ordered pairs (�, w). Graph the ordered pairs on the coordinate plane. See margin.

c. Numbers Use the graph to predict the length of a rectangle

with a perimeter of 20 inches and a width of 5 1 _ 2 inches. Check the

prediction by finding the actual length of the rectangle. 4 1

_ 2

in.

Independent PracticeIndependent Practice Go online for Step-by-Step SolutionseHelp

23

B

Perimeter of 20

Length (�) Width (w)

8 2

� �

� �

� �

7 3

6 4

5 5

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Lesson 3-6 Adding and Subtracting Unlike Fractions 129