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
Addition
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
Add ϩ .
4
5
EXAMPLE 1
Add
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
be added or subtracted.
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
Add ϩ .
3
7
EXAMPLE 2
Add
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
Add
ϩ
.
12
15
EXAMPLE 4
Add
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 .
Add
9
15
EXAMPLE 6
Add
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
Add Ϫ2 ϩ .
9
EXAMPLE 7
Add - 3 ϩ
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 .
Add
m
n
EXAMPLE 8
Add
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
Add
.
ϩ
6x
9yz
Add
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
made?
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
5Ϫ
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. When adding
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
8. When adding
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
your answers in lowest terms. (Objective 1)
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
and express your answers in reduced form. (Objective 1)
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
church receive?
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
check your answers for Problems 65–80.
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?
Answers to the Concept Quiz
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