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0.2
Fractions
OBJECTIVES
0.2
1. Simplify a fraction
2. Multiply or divide two fractions
3. Add or subtract two fractions
This section provides a review of the basic operations, addition, subtraction, division, and
multiplication, on fractions.
As mentioned in Section 0.1, the numbers used for counting are called the natural numbers. If we include zero in this group of numbers, we then call them the whole numbers.
The numbers of ordinary arithmetic consist of all the whole numbers and all fractions,
1
2
7
19
whether they are proper fractions such as and or improper fractions such as or .
2
3
2
5
a
Every number of ordinary arithmetic can be written in fraction form .
b
The number 1 has many different fractional forms. Any fraction in which the numerator
and denominator are the same (and not zero) is another name for the number one.
1
2
2
1
12
12
1
257
257
Because these fractions are just different names for the same quantity, they are called
equivalent fractions.
To write equivalent fractions, we use the Fundamental Principle of Fractions (FPF).
Rules and Properties: The Fundamental Principle of Fractions
a
ac
ac
a
or
, in which neither b nor c can equal zero.
b
bc
bc
b
Example 1
Rewriting Fractions
Write three fractional representations for each number.
NOTE Each representation is a
numeral, or name for the
number. Each number has many
names.
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NOTE In each case, we have
used the Fundamental Principle
of Fractions with c equal to a
different number.
(a)
2
3
We use the fundamental principle to multiply the numerator and denominator by the same
number.
2
22
4
3
32
6
4
6
2
23
6
3
33
9
6
9
2 10
20
2
3
3 10
30
20
30
17
CHAPTER 0
AN ARITHMETIC REVIEW
(b) 5
5
5 2
10
1 2
2
5
5 3
15
13
3
5
5 100
500
1 100
100
CHECK YOURSELF 1
Write three fractional representations for each number.
(a)
5
8
(b)
4
3
(c) 3
The simplest fractional representation for a number has the smallest numerator and
denominator. Fractions written in this form are said to be simplified.
Example 2
Simplifying Fractions
Simplify each fraction.
(a)
22
55
(b)
35
45
(c)
24
36
In each case, we first find the prime factors for the numerator and for the denominator.
(a)
22
2 11
55
5 11
We then use the fundamental principle.
22
2 11
2
55
5 11
5
(b)
35
7 5
75
45
335
95
Using the fundamental principle to remove the common factor of 5 yields
35
7
45
9
(c)
24
2223
36
2233
Removing the common factor 2 2 3 yields
2
3
CHECK YOURSELF 2
Simplify each fraction.
(a)
21
33
(b)
15
30
(c)
12
54
© 2001 McGraw-Hill Companies
18
FRACTIONS
SECTION 0.2
19
Rules and Properties: Multiplication of Fractions
ac
a c
b d
bd
NOTE This is how two
fractions, under the operation
of multiplication, become one
fraction.
When multiplying two fractions, rewrite them in factored form, and then simplify before
multiplying.
Example 3
Multiplying Fractions
Find the product of the two fractions.
NOTE A product is the result
from multiplication.
9 4
2 3
9 4
94
2 3
23
3322
23
32
1
6
The denominator of one is not necessary.
1
6
CHECK YOURSELF 3
Multiply and simplify each pair of fractions.
(a)
3 10
5 7
(b)
12 10
5
6
Rules and Properties: Division of Fractions
NOTE This is how two
fractions, under the operation
of division, become one
fraction.
a
c
a d
ad
b
d
b c
bc
To divide two fractions, the divisor is inverted, then the fractions are multiplied.
Example 4
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Dividing Fractions
Find the quotient of the two fractions.
NOTE A quotient is the result
from division.
7
5
3
6
7
5
7 6
76
3
6
3 5
35
723
72
35
5
14
5
20
CHAPTER 0
AN ARITHMETIC REVIEW
CHECK YOURSELF 4
Find the quotient of the two fractions
9
3
2
5
Rules and Properties: Addition of Fractions
ac
a
c
b
b
b
NOTE This is how two fractions
with the same denominator,
under the operation of
addition, become one fraction.
When adding two fractions, find the least common denominator (LCD) first. The least
common denominator is the smallest number that both denominators evenly divide. If you
have forgotten how to find the LCD, you might want to review the process from your arithmetic book. After rewriting the fractions with this denominator, add the numerators, then
simplify the result.
Example 5
Adding Fractions
Find the sum of the two fractions.
addition.
5
7
8
12
The LCD of 8 and 12 is 24. Each fraction should be rewritten as a fraction with that
denominator.
5
15
8
24
Multiply the numerator and denominator by 3.
7
14
12
24
Multiply the numerator and denominator by 2.
5
7
15
14
29
8
12
24
24
24
This fraction cannot be simplified.
CHECK YOURSELF 5
Find the sum for each pair of fractions.
(a)
4
7
5
9
(b)
5
4
6
15
© 2001 McGraw-Hill Companies
NOTE A sum is the result from
FRACTIONS
SECTION 0.2
21
Rules and Properties: Subtraction of Fractions
NOTE This is how two fractions
with like denominators become
one fraction under the
operation of subtraction.
a
ac
c
b
b
b
Subtracting fractions is treated exactly like adding them, except the numerator
becomes the difference of the two numerators.
Example 6
Subtracting Fractions
Find the difference.
NOTE The difference is the
result from subtraction.
1
7
9
6
The LCD is 18. We rewrite the fractions with that denominator.
7
14
9
18
1
3
6
18
7
1
14
3
11
9
6
18
18
18
This fraction cannot be simplified.
CHECK YOURSELF 6
Find the difference
11
5
.
12
8
Fractions with denominator 10 (or 100, 1000, etc.) can be written in decimal form. Example 7 demonstrates the addition or subtraction of decimal fractions.
Example 7
Adding or Subtracting Two Decimals
Perform the indicated operation.
(a) Add 2.356 and 15.6
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Aligning the decimal points, we get
2.356
15.600
17.956
Although the zeros are not necessary, they ensure proper alignment.
(b) Subtract 3.84 from 8.1
Again, we align the decimal points
8.10
3.84
4.26
When subtracting, always add zeros so that the right columns line up.
CHAPTER 0
AN ARITHMETIC REVIEW
CHECK YOURSELF 7
Perform the indicated operation.
(a) 34.76 2.419
(b) 71.82 8.197
Example 8 illustrates the multiplication of two decimal fractions.
Example 8
Multiplying Decimal Fractions
Multiply 4.6 and 3.27
4.6
3.27
322
920
13800
15.042
It is not necessary to align decimals being multiplied. Note that the two factors
have a total of three digits to the right of the decimal point.
The decimal point of the product is moved three digits to the left.
CHECK YOURSELF 8
Multiply 5.8 and 9.62.
CHECK YOURSELF ANSWERS
1. Answers will vary.
4.
15
2
5. (a)
7
1
2
6
; (b) ; (c)
3. (a) ; (b) 4
11
2
9
7
7
6.
7. (a) 37.179; (b) 63.623
24
2. (a)
71
11
; (b)
45
10
8. 55.796
© 2001 McGraw-Hill Companies
22
Name
0.2
Exercises
Section
Date
In Exercises 1 to 12, write three fractional representations for each number.
ANSWERS
3
1.
7
2
2.
5
4
3.
9
1.
4.
7
8
5.
5
6
6.
11
13
2.
3.
7.
10
17
8.
3
7
9.
9
16
10.
6
11
11.
7
9
12.
15
16
4.
5.
6.
7.
Write each fraction in simplest form.
13.
8
12
14.
12
15
15.
10
14
8.
9.
16.
15
50
17.
12
18
18.
28
35
10.
11.
19.
35
40
20.
21
24
21.
11
44
22.
10
25
23.
12
36
24.
18
48
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25.
24
27
26.
30
50
27.
32
40
28.
17
51
29.
75
105
30.
62
93
31.
48
60
32.
48
66
33.
105
135
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
23
ANSWERS
34.
35.
36.
37.
34.
54
126
35.
15
44
36.
10
63
Multiply. Be sure to simplify each product.
38.
39.
37.
3 7
4 5
38.
2 8
3 5
39.
3 5
5 7
40.
6 8
11 6
41.
6 4
13 9
42.
5 6
9 11
43.
3 7
11 9
44.
7 3
9 5
45.
3 5
10 9
40.
41.
42.
43.
44.
Divide. Write each result in simplest form.
45.
46.
47.
48.
49.
50.
51.
52.
53.
54.
55.
56.
46.
5
25
21
14
47.
1
3
5
4
48.
2
1
5
3
49.
2
3
5
4
50.
5
3
8
4
51.
8
4
9
3
52.
5
8
9
11
53.
7
5
10
9
54.
8
11
9
15
55.
8
2
15
5
56.
5
15
27
54
57.
5
25
27
36
58.
9
27
28
35
57.
58.
59.
60.
61.
62.
63.
64.
24
59.
2
1
5
4
60.
2
3
3
10
61.
2
7
5
15
62.
3
7
10
12
63.
3
5
8
12
64.
5
7
36
24
© 2001 McGraw-Hill Companies
Add.
ANSWERS
65.
2
9
15
20
66.
9
10
14
21
7
13
15
18
67.
65.
66.
68.
12
19
25
30
69.
1
1
1
2
4
8
1
1
1
3
5
10
70.
67.
68.
Subtract.
69.
71.
8
3
9
9
72.
1
5
73.
8
8
75.
77.
79.
6
9
10
10
70.
11
7
74.
12
12
7
2
8
3
76.
11
2
18
9
78.
5
1
8
6
80.
5
3
6
5
5
1
6
4
13
5
18
12
71.
72.
73.
74.
75.
76.
77.
78.
79.
80.
81.
82.
83.
84.
81.
8
1
21
14
82.
7
13
18
15
85.
86.
87.
Perform the indicated operations.
88.
83. 7.1562 14.78
84. 6.2358 3.14
85. 11.12 8.3792
86. 6.924 5.2
89.
90.
© 2001 McGraw-Hill Companies
91.
87. 9.20 2.85
89. 18.234 13.64
88. 17.345 11.12
90. 21.983 9.395
91. 3.21 2.1
92.
93.
94.
92. 15.6 7.123
93. 6.29 9.13
94. 8.245 3.1
25
ANSWERS
1
Roseann is making shirts for her three children. One shirt requires yard
2
1
of material, a second shirt requires yard of material, and the third shirt
3
1
requires yard of material. How much material is required for all three shirts?
4
95.
95. Sewing.
96.
97.
Jose rode his trail bike for 10 miles. Two-thirds of the distance was over a
mountain trail. How long is the mountain trail?
96. Hiking.
98.
97. Salary.
You make $240 a day on a job. What will you receive for working
99.
2
of a day?
3
3
A survey has found that of the people in a city own pets. Of those who
4
2
own pets, have cats. What fraction of those surveyed own cats?
3
98. Surveys.
100.
101.
102.
103.
Solve the following applications.
104.
99. Map scales. The scale on a map is 1 inch (in.) 200 miles (mi). What actual
distance, in miles, does
3
in. represent?
8
3
of a day?
4
3
101. Size. A lumberyard has a stack of 80 sheets of plywood. If each sheet is in. thick,
4
how high will the stack be?
100. Salary. You make $90 a day on a job. What will you receive for working
2
of its monthly income for housing and utilities
5
on average. If the family’s monthly income is $1750, what is spent for housing and
utilities? What amount remains?
102. Family budget. A family uses
3
were registered. Of those
4
5
registered, actually voted. What fraction of those people who were eligible voted?
9
7
of the people in a city own pets. Of those
10
2
who own pets, have dogs. What fraction of those surveyed own dogs?
3
104. Surveys. A survey has found that
26
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103. Elections. Of the eligible voters in an election,
ANSWERS
1
3
3
4
of linoleum must be bought to cover the floor?
105. Area. A kitchen has dimensions 3 by 3 yards (yd). How many square yards (yd2)
106.
3
4
106. Distance. If you drive at an average speed of 52 miles per hour (mi/h) for 1 h,
how far will you travel?
2
3
107. Distance. A jet flew at an average speed of 540 mi/h on a 4 -h flight. What was the
distance flown?
105.
2
3
107.
108.
109.
108. Area. A piece of land that has 11 acres is being subdivided for home lots. It is
estimated that
2
of the area will be used for roads. What amount remains to be used
7
110.
for lots?
109. Circumference. To find the approximate circumference or distance around a
22
circle, we multiply its diameter by . What is the circumference of a circle with a
7
diameter of 21 in.?
110. Area. The length of a rectangle is
6
21
yd, and its width is
yd. What is its area in
7
26
square yards?
1
7
5
4
8
6
112. Topsoil. Nico wishes to purchase mulch to cover his garden. The garden measures
7
1
1
7 feet (ft) by 10 ft. He wants the mulch to be ft deep. How much mulch should
8
8
3
Nico order if he must order a whole number of cubic feet?
111. Volume. Find the volume of a box that measures 2 in. by 3 in. by 4 in.
111.
112.
113.
114.
113. Every fraction (rational number) has a corresponding decimal form that either
5
terminates or repeats. For example,
0.3125 (the decimal form terminates), and
16
4
0.363636........ (the decimal form repeats). Investigate a number of fractions to
11
determine which ones terminate and which ones repeat. (Hint: you can focus on the
denominator; study the prime factorizations of several denominators.)
© 2001 McGraw-Hill Companies
114. Complete the following sums:
1
1
2
4
1
1
1
2
4
8
1
1
1
1
2
4
8
16
Based on these, predict the sum:
1
1
1
1
1
1
1
2
4
8
16
32
64
128
27
Answers
1.
6 9 12
, ,
14 21 28
9.
18 27 90
, ,
32 48 160
3.
8 16 40
, ,
18 36 90
14 35 140
, ,
18 45 180
11.
21.
1
4
35.
15
44
37.
21
20
49.
8
15
51.
2
3
63.
19
24
65.
7
12
67.
107
90
79.
11
24
81.
13
42
7
18
89. 4.594
77.
1
3
23.
25.
8
9
39.
53.
91. 6.741
99. 75 mi
101. 60 in.
109. 66 in.
111. 42
10 15 50
, ,
12 18 60
5.
13.
27.
4
5
15.
5
7
29.
3
7
41.
8
39
63
50
55.
4
3
69.
57.
7
8
5
12
4
15
17.
4
5
33.
45.
59.
5
9
2
3
73.
19.
7
9
1
6
47.
4
15
13
20
61.
13
15
1
2
75.
5
24
85. 19.4992
13
95.
yd
12
1
105. 12 yd2
2
7
8
87. 6.35
97. $160
107. 2520 mi
113.
© 2001 McGraw-Hill Companies
103.
7
33
71.
20 30 100
, ,
34 51 170
5
7
31.
43.
83. 21.9362
93. 57.4277
9
in.3
64
2
3
7.
28