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2: Addition and Subtraction of Rational Numbers

# 2: Addition and Subtraction of Rational Numbers

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2.2 • Addition and Subtraction of Rational Numbers

51

Deﬁnition 2.3

If a, b, and c are integers, and b is not zero, then

c

aϩc

a

ϩ ϭ

b

b

b

a

c

aϪc

Ϫ ϭ

b

b

b

Subtraction

We say that rational numbers with common denominators can be added or subtracted by adding

or subtracting the numerators and placing the results over the common denominator. Consider

the following examples:

3

2

3 ϩ 2

5

ϩ

ϭ

ϭ

7

7

7

7

7

2

7 Ϫ 2

5

Ϫ ϭ

ϭ

8

8

8

8

2

1

2 ϩ 1

3

1

ϩ

ϭ

ϭ

ϭ

6

6

6

6

2

We agree to reduce the final answer

3

5

3Ϫ5

Ϫ2

2

Ϫ

ϭ

ϭ

or    Ϫ

11

11

11

11

11

5

7

5 ϩ 7

12

ϩ

ϭ

ϭ

x

x

x

x

9

3

9 Ϫ 3

6

Ϫ ϭ

ϭ

y

y

y

y

In the last two examples, the variables x and y cannot be equal to zero in order to exclude

division by zero. It is always necessary to restrict denominators to nonzero values, although we

will not take the time or space to list such restrictions for every problem.

How do we add or subtract if the fractions do not have a common denominator?

We use the fundamental principle of fractions,

a

aиk

, and obtain equivalent fractions

ϭ

b

bиk

that have a common denominator. Equivalent fractions are fractions that name the same

number. Consider the following example, which shows the details.

Classroom Example

1

1

4

5

EXAMPLE 1

1

1

ϩ .

2

3

Solution

1

1и3

3

ϭ

ϭ

2

2и3

6

1и2

2

1

ϭ

ϭ

3

3и2

6

3

1

and are equivalent fractions naming the same number

2

6

1

2

and are equivalent fractions naming the same number

2

6

1

1

3

2

3 ϩ 2

5

ϩ

ϭ ϩ

ϭ

ϭ

2

3

6

6

6

6

Notice that we chose 6 as the common denominator, and 6 is the least common multiple

of the original denominators 2 and 3. (Recall that the least common multiple is the smallest

52

Chapter 2 • Real Numbers

nonzero whole number divisible by the given numbers.) In general, we use as a least common denominator (LCD) the least common multiple of the denominators of the fractions to

Recall from Section 1.2 that the least common multiple may be found either by inspection or by using prime factorization forms of the numbers. Let’s consider some examples

involving these procedures.

Classroom Example

1

4

3

7

EXAMPLE 2

1

2

ϩ .

4

5

Solution

By inspection we see that the LCD is 20. Thus both fractions can be changed to equivalent

fractions that have a denominator of 20.

1 2 1и5 2и4

5

8

13

ϩ ϭ

ϩ

ϭ

ϩ ϭ

4 5 4 и 5 5 и 4 20 20 20

Use of fundamental

principle of fractions

Classroom Example

7

4

Subtract Ϫ .

9

15

EXAMPLE 3

Subtract

5

7

.

Ϫ

8

12

Solution

By inspection it is clear that the LCD is 24.

5

7

5и3

7и2

15

14

1

Ϫ

ϭ

Ϫ

ϭ

Ϫ

ϭ

8

12

8и3

12 и 2

24

24

24

If the LCD is not obvious by inspection, then we can use the technique from Chapter 1

to find the least common multiple. We proceed as follows.

Step 1 Express each denominator as a product of prime factors.

Step 2 The LCD contains each different prime factor as many times as the most times it

appears in any one of the factorizations from step 1.

Classroom Example

7

6

ϩ

.

12

15

EXAMPLE 4

7

5

.

ϩ

18

24

Solution

If we cannot find the LCD by inspection, then we can use the prime factorization forms.

18 ϭ 2 и 3

и 3 f ¡ LCD ϭ 2 2 2 3 3 ϭ 72

и и и и

24 ϭ 2 и 2 и 2 и 3

5

7

5и4

7и3

20

21

41

ϩ

ϭ

ϩ

ϭ

ϩ

ϭ

18

24

18 и 4

24 и 3

72

72

72

2.2 • Addition and Subtraction of Rational Numbers

Classroom Example

3

11

Subtract

Ϫ

.

10

15

EXAMPLE 5

Subtract

53

8

3

.

Ϫ

14

35

Solution

и7

f

35 ϭ 5 и 7

14 ϭ 2

¡ LCD ϭ 2

и 5 и 7 ϭ 70

3

8

3и5

8и2

15

16

Ϫ1

1

Ϫ

ϭ

Ϫ

ϭ

Ϫ

ϭ

or      Ϫ

14

35

14 и 5

35 и 2

70

70

70

70

Classroom Example

7

Ϫ7

ϩ .

9

15

EXAMPLE 6

3

-5

ϩ

.

8

14

Solution

8 ϭ 2 и 2

и2

f ¡ LCD ϭ 2 и 2 и 2 и 7 ϭ 56

14 ϭ 2 и 7

Ϫ5

3

Ϫ5 и 7

3и4

Ϫ35

12

Ϫ23

ϩ

ϭ

ϩ

ϭ

ϩ

ϭ

or

8

14

8и7

14 и 4

56

56

56

Classroom Example

4

9

EXAMPLE 7

Ϫ

23

56

2

.

5

Solution

Ϫ3 ϩ

2

Ϫ3 и 5

2

Ϫ15

2

Ϫ15 ϩ 2

Ϫ13

13

ϭ

ϩ ϭ

ϩ ϭ

ϭ

or       Ϫ

5

1и5

5

5

5

5

5

5

Denominators that contain variables do not complicate the situation very much, as the

next examples illustrate.

Classroom Example

5

4

ϩ .

m

n

EXAMPLE 8

2

3

ϩ .

x

y

Solution

By inspection, the LCD is xy.

Commutative property

2иy

2y

2y ϩ 3x

2

3

3иx

3x

ϩ

ϭ

ϩ

ϭ

ϩ

ϭ

x

y

xиy

yиx

xy

xy

xy

Classroom Example

3

7

Subtract

.

Ϫ

4x

18y

EXAMPLE 9

Subtract

3

5

.

Ϫ

8x

12y

Solution

и 2 и 2 и x ¡ LCD ϭ 2 2 2 3 x y ϭ 24xy

f

и и и и и

12y ϭ 2 и 2 и 3 и y

3 и 3y

9y

9y Ϫ 10x

3

5

5 и 2x

10x

Ϫ

ϭ

Ϫ

ϭ

Ϫ

ϭ

8x

12y

8x и 3y

12y и 2x

24xy

24xy

24xy

8x ϭ 2

54

Chapter 2 • Real Numbers

EXAMPLE 10

Classroom Example

7

Ϫ4

.

ϩ

6x

9yz

Ϫ5

7

.

ϩ

4a

6bc

Solution

4a ϭ 2 и 2

и a f ¡ LCD ϭ 2 2 3 a b c ϭ 12abc

и и и и и

6bc ϭ 2 и 3 и b и c

7

Ϫ5

7 и 3bc

Ϫ5 и 2a

21bc

Ϫ10a

21bc Ϫ 10a

ϩ

ϭ

ϩ

ϭ

ϩ

ϭ

4a

6ac

4a и 3bc

6bc и 2a

12abc

12abc

12abc

Simplifying Numerical Expressions

Let’s now consider simplifying numerical expressions that contain rational numbers. As with

integers, multiplications and divisions are done first, and then the additions and subtractions

are performed. In these next examples only the major steps are shown, so be sure that you can

fill in all of the other details.

Classroom Example

3

1

3

1

Simplify Ϫ и

ϩ

5

3

4

4

EXAMPLE 11

и

1

.

2

3

2

ϩ

4

3

3

1

1

и 5 Ϫ 2 и 5.

Solution

3

2

ϩ

4

3

Classroom Example

Simplify

2

-3

1

5

1

Ϭ ϩa

ba b ϩ .

3

3

4

2

6

Simplify

3

1

1

3

2

1

и 5 Ϫ 2 и 5 ϭ 4 ϩ 5 Ϫ 10

EXAMPLE 12

Perform the multiplications

ϭ

15

8

2

ϩ

Ϫ

20

20

20

ϭ

21

15 ϩ 8 Ϫ 2

ϭ

20

20

Simplify

Change to equivalent fractions

and combine numerators

3

8

1 1

5

Ϭ ϩ aϪ ba b ϩ .

5

5

2 3

12

Solution

3

8

1 1

5

3

Ϭ ϩ aϪ ba b ϩ

ϭ

5

5

2 3

12

5

и 8 ϩ aϪ 2 ba 3 b ϩ 12

5

1

ϭ

3

Ϫ1

5

ϩ

ϩ

8

6

12

ϭ

Ϫ4

10

9

ϩ

ϩ

24

24

24

ϭ

9 ϩ (Ϫ4) ϩ 10

24

ϭ

15

5

ϭ

24

8

1

5

Change division to multiply

by the reciprocal

Reduce!

2.2 • Addition and Subtraction of Rational Numbers

55

The distributive property, a(b ϩ c) ϭ ab ϩ ac, holds true for rational numbers and, as

with integers, can be used to facilitate manipulation.

Classroom Example

1

1

Simplify 18 a ϩ b .

2

6

Simplify 12 a

EXAMPLE 13

1

1

ϩ b.

3

4

Solution

For help in this situation, let’s change the form by applying the distributive property.

12 a

Classroom Example

5 1

1

Simplify a ϩ b .

7 6

4

1

1

1

1

ϩ b ϭ 12 a b ϩ 12 a b

3

4

3

4

ϭ4ϩ3

ϭ7

EXAMPLE 14

Simplify

5 1

1

a ϩ b.

8 2

3

Solution

In this case it may be easier not to apply the distributive property but to work with the expression in its given form.

5 1

1

5 3

2

a ϩ bϭ a ϩ b

8 2

3

8 6

6

ϭ

5 5

a b

8 6

ϭ

25

48

Examples 13 and 14 emphasize a point we made in Chapter 1. Think first, and decide

whether or not the properties can be used to make the manipulations easier. Example 15

illustrates how to combine similar terms that have fractional coefficients.

Classroom Example

1

2

1

Simplify m Ϫ m ϩ m by

3

5

2

combining similar terms.

EXAMPLE 15

1

2

3

Simplify x ϩ x Ϫ x by combining similar terms.

2

3

4

Solution

We can use the distributive property and our knowledge of adding and subtracting rational

numbers to solve this type of problem.

2

3

1

2

3

1

x ϩ x Ϫ x ϭ a ϩ Ϫ bx

2

3

4

2

3

4

ϭ a

ϭ

6

8

9

ϩ

Ϫ bx

12

12

12

5

x

12

56

Chapter 2 • Real Numbers

EXAMPLE 16

Classroom Example

Matt bought 16 pounds of peanuts. If

2

pound of peanuts can be packaged

3

to sell, how many packages can be

Brian brought 5 cups of flour along on a camping trip. He wants to make biscuits and cake for

3

3

tonight’s supper. It takes of a cup of flour for the biscuits and 2 cups of flour for the cake.

4

4

How much flour will be left over for the rest of his camping trip?

Solution

Let’s do this problem in two steps. First add the amounts of flour needed for the biscuits and cake.

3

3

3

11

14

7

ϩ2 ϭ ϩ

ϭ

ϭ

4

4

4

4

4

2

Then to find the amount of flour left over, we will subtract

7

10

7

3

1

ϭ

Ϫ ϭ ϭ1

2

2

2

2

2

7

from 5.

2

1

So 1 cups of flour are left over.

2

Concept Quiz 2.2

For Problems 1–10, answer true or false.

1. To add rational numbers with common denominators, add the numerators and place the

result over the common denominator.

2

6

ϩ , c can be equal to zero.

c

c

3. Fractions that name the same number are called equivalent fractions.

4. The least common multiple of the denominators can always be used as a common

denominator when adding or subtracting fractions.

3

1

and , we need to find equivalent fractions with a common denominator.

8

5

5

2

6. To multiply and , we need to find equivalent fractions with a common denominator.

7

3

1

3

7. Either 20, 40 or 60 can be used as a common denominator when adding and , but 20

4

5

is the least common denominator.

5. To subtract

9. 36 a

10.

3y

2x

and , the least common denominator is ac.

ab

bc

1

4

Ϫ b simplifies to 2.

2

9

2

1

5

13

x Ϫ x ϩ x simplifies to x.

3

4

6

12

Problem Set 2.2

For Problems 1–64, add or subtract as indicated, and express

1.

3

2

ϩ

7

7

2.

5

3

ϩ

11

11

3.

7

2

Ϫ

9

9

4.

11

6

Ϫ

13

13

5.

3

9

ϩ

4

4

6.

5

7

ϩ

6

6

2.2 • Addition and Subtraction of Rational Numbers

7.

11

3

Ϫ

12

12

13

7

Ϫ

16

16

53.

5

7

ϩ

3x

3y

54.

3

7

ϩ

2x

2y

9.

1

5

Ϫ

8

8

10.

2

5

Ϫ

9

9

55.

8

3

ϩ

5x

4y

56.

1

5

ϩ

5x

6y

11.

5

11

ϩ

24

24

12.

7

13

ϩ

36

36

57.

5

7

Ϫ

4x

9y

58.

11

2

Ϫ

7x

14y

13.

8

7

ϩ

x

x

14.

17

12

ϩ

y

y

59. Ϫ

15.

5

1

ϩ

3y

3y

16.

3

1

ϩ

8x

8x

61. 3 ϩ

2

x

62.

17.

1

1

ϩ

3

5

18.

1

1

ϩ

6

8

63. 2 Ϫ

3

2x

64. Ϫ1 Ϫ

19.

15

3

Ϫ

16

8

20.

13

1

Ϫ

12

6

21.

7

8

ϩ

10

15

22.

7

5

ϩ

12

8

11

5

ϩ

24

32

24.

5

13

Ϫ

18

24

26.

5

2

Ϫ

8

3

28.

23.

25.

27.

29. Ϫ

31. Ϫ

5

8

ϩ

18

27

1

7

Ϫ

24

36

3

5

Ϫ

4

6

2

7

Ϫ

13

39

30. Ϫ

3

1

ϩ

14

21

32. Ϫ

33. Ϫ4 Ϫ

35.

8.

3

7

3

Ϫ6

4

3

4

37. ϩ

x

y

3

13

Ϫ

11

33

3

14

ϩ

20

25

34. Ϫ2 Ϫ

36.

5

6

5

Ϫ7

8

5

8

38. ϩ

x

y

3

5

Ϫ

2x

4y

60. Ϫ

57

13

11

Ϫ

8a

10b

5

ϩ4

x

1

3x

For Problems 65–80, simplify each numerical expression

65.

1

3

5

1

Ϫ ϩ

Ϫ

4

8

12

24

66.

3

2

1

5

ϩ Ϫ ϩ

4

3

6

12

67.

5

2

ϩ

6

3

68.

2

1

ϩ

3

2

#

2

1

Ϫ

5

3

69.

3

4

#

6

5

Ϫ

9

6

#

8

2

ϩ

10

3

#

70.

3

5

#

5

2

ϩ

7

3

#

3

1

Ϫ

5

7

2

5

1

2

#

1

Ϫ2

3

71. 4 Ϫ

2

3

3

1

Ϫ

4

4

#

#

#

2

5

#

1

5

6

8

3

Ϫ6

5

72. 3 ϩ

73.

4

10

5

14

10

Ϫ

Ϫ Ϭ

ϩ

5

12

6

8

21

74.

3

6

8

Ϭ ϩ

4

5

12

#

#

6

5

Ϫ

9

12

75. 24 a

3

1

Ϫ b

4

6

76. 18 a

2

1

ϩ b

3

9

77. 64 a

3

5

1

1

ϩ Ϫ ϩ b

16

8

4

2

78. 48 a

5

1

3

Ϫ ϩ b

12

6

8

Don’t forget the distributive property!

39.

7

2

Ϫ

a

b

40.

13

4

Ϫ

a

b

41.

2

7

ϩ

x

2x

42.

5

7

ϩ

x

2x

43.

10

2

Ϫ

x

3x

44.

13

3

Ϫ

x

4x

45.

7

1

Ϫ

x

5x

46.

17

2

Ϫ

x

6x

47.

3

5

ϩ

2y

3y

48.

7

9

ϩ

3y

4y

For Problems 81– 96, simplify each algebraic expression by

combining similar terms. (Objective 2)

49.

5

3

Ϫ

12y

8y

50.

9

5

Ϫ

4y

9y

81.

1

2

xϩ x

3

5

82.

1

2

xϩ x

4

3

51.

1

7

Ϫ

6n

8n

52.

3

11

Ϫ

10n

15n

83.

1

1

aϪ a

3

8

84.

2

2

aϪ a

5

7

79.

1

7 2

a Ϫ b

13 3

6

80.

1

5 1

a ϩ b

9 2

4

58

85.

87.

Chapter 2 • Real Numbers

1

2

1

xϩ xϩ x

2

3

6

86.

3

1

3

nϪ nϩ n

5

4

10

88.

89. n ϩ

4

1

nϪ n

3

9

7

5

91. Ϫn Ϫ n Ϫ

n

9

12

93.

1

2

5

xϩ xϩ x

3

5

6

2

7

8

nϪ nϩ n

5

10

15

90. 2n Ϫ

6

5

nϩ n

7

14

3

3

92. Ϫ n Ϫ n Ϫ

n

8

14

3

1

1

7

xϩ yϩ xϩ y

7

4

2

8

94.

5

3

4

7

xϩ yϩ xϩ y

6

4

9

10

95.

2

5

7

13

xϩ yϪ xϪ y

9

12

15

15

1

feet long. If he cuts off

2

3

feet long, how long is the remaining

4

piece of board?

1

miles. One day a

2

3

thunderstorm forced her to stop her walk after of

4

a mile. By how much was her walk shortened that day?

101. Mindy takes a daily walk of 2

rational numbers. (Objective 3)

1

of his estate to the Boy Scouts,

4

2

to the local cancer fund, and the rest to his

102. Blake Scott leaves

97. Beth wants to make three sofa pillows for her new sofa.

After consulting the chart provided by the fabric shop,

she decides to make a 12ЈЈ round pillow, an 18ЈЈ square

pillow, and a 12ЈЈ ϫ 16ЈЈ rectangular pillow. According

to the chart, how much fabric will Beth need to purchase?

Fabric shop chart

5

church. What fractional part of the estate does the

103. A triangular plot of ground measures 14

by 12

10ЈЈ round

3

yard

8

12ЈЈ round

1

yard

2

12ЈЈ ϫ 16ЈЈ rectangular

1

3

feet long, a piece 1 feet

2

4

long is cut off from one end. Find the length of the

remaining piece of board.

99. From a board that is 12

a piece 2

For Problems 97–104, solve using addition and subtraction of

18ЈЈ square

3

inches wide. He is

8

1

going to hang the prints side by side with 2 inches

4

between the prints. What width of wall space is

needed to display the three prints?

three prints that are each 13

100. Vinay has a board that is 6

9

3

2

5

96. Ϫ x Ϫ

yϩ xϩ y

10

14

25

21

12ЈЈ square

98. Marcus is decorating his room and plans on hanging

5

yard

8

3

yard

4

7

yard

8

1

yards

2

1

5

yards by 9 yards. How many yards of fenc3

6

ing are needed to enclose the plot?

1

miles,

2

3

then walks for

of a mile, and finally jogs for

4

1

another 1 miles. Find the total distance that Lian

4

104. For her exercise program, Lian jogs for 2

covers.

105. If your calculator handles rational numbers in

a

form,

b

Thoughts Into Words

106. Give a step-by-step description of the best way to add

3

5

the rational numbers and .

8

18

107. Give a step-by-step description of how to add the frac7

5

tions

and .

4x

6x

108. The will of a deceased collector of antique automobiles specified that his cars be left to his three

1

children. Half were to go to his elder son, to his

3

1

daughter, and

to his younger son. At the time of

9

his death, 17 cars were in the collection. The

2.3 • Real Numbers and Algebraic Expressions

administrator of his estate borrowed a car to make 18.

Then he distributed the cars as follows:

Elder son:

1

(18) ϭ 9

2

Daughter:

1

(18) ϭ 6

3

1

(18) ϭ 2

9

This takes care of the 17 cars, so the administrator then returned

the borrowed car. Where is the error in this solution?

1. True

2. False

3. True

4. True

9. True

10. False

2.3

Younger son:

59

5. True

6. False

7. True

8. False

Real Numbers and Algebraic Expressions

OBJECTIVES

1

Classify real numbers

2

Add, subtract, multiply, and divide rational numbers in decimal form

3

Combine similar terms whose coefﬁcients are rational numbers in decimal form

4

Evaluate algebraic expressions when the variables are rational numbers

5

Solve application problems that involve the operations of rational numbers in

decimal form

We classify decimals—also called decimal fractions— as terminating, repeating, or nonrepeating. Here are examples of these classifications:

Terminating

decimals

Repeating

decimals

Nonrepeating

decimals

0.3

0.26

0.347

0.9865

0.333333 . . .

0.5466666 . . .

0.14141414 . . .

0.237237237 . . .

0.5918654279 . . .

0.26224222722229 . . .

0.145117211193111148 . . .

0.645751311 . . .

Technically, a terminating decimal can be thought of as repeating zeros after the last digit.

For example, 0.3 ϭ 0.30 ϭ 0.300 ϭ 0.3000, and so on.

A repeating decimal has a block of digits that repeats indefinitely. This repeating block

of digits may contain any number of digits and may or may not begin repeating immediately

after the decimal point.

In Section 2.1 we defined a rational number to be any number that can be written in the

a

form , where a and b are integers and b is not zero. A rational number can also be defined

b

as any number that has a terminating or repeating decimal representation. Thus we can

express rational numbers in either common-fraction form or decimal-fraction form, as the

next examples illustrate. A repeating decimal can also be written by using a bar over the digits that repeat; for example, 0.14.

Terminating

decimals

Repeating

decimals

3

ϭ 0.75

4

1

ϭ 0.3333 . . .

3

60

Chapter 2 • Real Numbers

1

ϭ 0.125

8

5

ϭ 0.3125

16

7

ϭ 0.28

25

2

ϭ 0.66666 . . .

3

1

ϭ 0.166666 . . .

6

1

ϭ 0.08333 . . .

12

14

ϭ 0.14141414 . . .

99

2

ϭ 0.4

5

The nonrepeating decimals are called “irrational numbers” and do appear in forms other

than decimal form. For example, 12, 13, and p are irrational numbers; a partial representation for each of these follows.

22 ϭ 1.414213562373 . . .

23 ϭ 1.73205080756887 . . . t

Nonrepeating decimals

p ϭ 3.14159265358979 . . .

(We will do more work with the irrational numbers in Chapter 9.)

The rational numbers together with the irrational numbers form the set of real numbers. The

following tree diagram of the real number system is helpful for summarizing some basic ideas.

Real numbers

Rational

Irrational

Ϫ

Integers

Ϫ

0

ϩ

Nonintegers

ϩ

Ϫ

ϩ

Any real number can be traced down through the diagram as follows.

5 is real, rational, an integer, and positive

Ϫ4 is real, rational, an integer, and negative

3

is real, rational, a noninteger, and positive

4

0.23 is real, rational, a noninteger, and positive

Ϫ0.161616 . . . is real, rational, a noninteger, and negative

17 is real, irrational, and positive

Ϫ12 is real, irrational, and negative

In Section 1.3, we associated the set of integers with evenly spaced points on a line

as indicated in Figure 2.4. This idea of associating numbers with points on a

−4 −3 −2 −1

0

1

2

3

4

Figure 2.4

line can be extended so that there is a one-to-one correspondence between points on a line

and the entire set of real numbers (as shown in Figure 2.5). That is to say, to each real

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