# Solution Manual for Beginning and Intermediate Algebra, 7th Edition

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R.1 Fractions
Chapter R
Prealgebra Review
R.1 Fractions
Classroom Examples, Now Try Exercises
1. 90 is composite and can be written as
N3. The fraction bar represents division. Divide the
numerator of the improper fraction by the
denominator.
18
5 92
5
42
40
2
Thus,
Writing 90 as the product of primes gives us
90 = 2 โ
3 โ
3 โ
5 .
N1. 60 is composite and can be written as
Writing 60 as the product of primes gives us
60 = 2 โ
2 โ
3 โ
5.
2. (a)
12 3 โ
4 3 4 3
3
=
= โ
= โ
1 =
20 5 โ
4 5 4 5
5
(b)
8
8
1
1
=
=
=
48 6 โ
8 6 โ
1 6
(c)
90 5 โ
18 5
5
=
= โ
1 =
162 9 โ
18 9
9
N2. (a)
30 5 โ
6 5 6 5
5
=
= โ
= โ
1 =
42 7 โ
6 7 6 7
7
(b)
10
10
1
1
=
=
โ
=
70 7 โ
10 7 โ
1 7
(c)
72 3 โ
24 3
3
=
= โ
1 =
120 5 โ
24 5
5
1
92
2
= 18 .
5
5
4. Multiply the denominator of the fraction by the
natural number and then add the numerator to
obtain the numerator of the improper fraction.
5 โ
3 = 15 and 15 + 4 = 19
The denominator of the improper fraction is the
same as the denominator in the mixed number.
4 19
Thus, 3 = .
5 5
N4. Multiply the denominator of the fraction by the
natural number and then add the numerator to
obtain the numerator of the improper fraction.
3 โ
11 = 33 and 33 + 2 = 35
The denominator of the improper fraction is the
same as the denominator in the mixed number.
2 35
Thus, 11 = .
3 3
5. (a) To multiply two fractions, multiply their
numerators and then multiply their
denominators. Then simplify and write the
answer in lowest terms.
5 18 5 โ
18
โ
=
9 25 9 โ
25
90
=
225
2 โ
45
=
5 โ
45
2
=
5
3. The fraction bar represents division. Divide the
numerator of the improper fraction by the
denominator.
3
10 37
30
7
37
7
Thus,
=3 .
10
10
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Chapter R Prealgebra Review
(b) To multiply two mixed numbers, first write
them as improper fractions. Multiply their
numerators and then multiply their
denominators. Then simplify and write the
answer as a mixed number in lowest terms.
1 3 10 7
3 โ
1 = โ
3 4 3 4
10 โ
7
=
3โ
4
2โ
5โ
7
=
3โ
2 โ
2
5
35
= , or 5
6
6
N5. (a) To multiply two fractions, multiply their
numerators and then multiply their
denominators. Then simplify and write the
answer in lowest terms.
4 5 4โ
5
โ
=
7 8 7 โ
8
20
=
56
5โ
4
=
14 โ
4
5
=
14
(b) To multiply two mixed numbers, first write
them as improper fractions. Multiply their
numerators and then multiply their
denominators. Then simplify and write the
answer as a mixed number in lowest terms.
2 2 17 20
3 โ
6 = โ
5 3 5 3
17 โ
20
=
5โ
3
17 โ
5 โ
4
=
5โ
3
68
2
= , or 22
3
3
6. (a) To divide fractions, multiply by the
reciprocal of the divisor.
9 3 9 5
รท = โ
10 5 10 3
3โ
3โ
5
=
2โ
5โ
3
3
1
= , or 1
2
2
(b) Change both mixed numbers to improper
fractions. Then multiply by the reciprocal of
the second fraction.
3
1 11 10
2 รท3 = รท
4
3 4 3
11 3
= โ
4 10
33
=
40
N6. (a) To divide fractions, multiply by the
reciprocal of the divisor.
2 8 2 9
รท = โ
7 9 7 8
2 โ
3โ
3
=
7โ
2โ
4
9
=
28
(b) To divide fractions, multiply by the
reciprocal of the divisor.
3
2 15 30
3 รท4 = รท
4
7 4 7
15 7
= โ
4 30
15 โ
7
=
4 โ
2 โ
15
7
=
8
7. To find the sum of two fractions having the
same denominator, add the numerators and
keep the same denominator.
1 5 1+ 5
+ =
9
9 9
6
=
9
2โ
3
=
3โ
3
2
=
3
Copyright ยฉ 2020 Pearson Education, Inc.
R.1 Fractions
N7.
To find the sum of two fractions having the
same denominator, add the numerators and
keep the same denominator.
1 3 1+ 3
+ =
8 8
8
4
=
8
1โ
4
=
2โ
4
1
=
2
8. (a) Since 30 = 2 โ
3 โ
5 and 45 = 3 โ
3 โ
5, the least
common denominator must have one factor
of 2 (from 30), two factors of 3 (from 45),
and one factor of 5 (from either 30 or 45),
so it is 2 โ
3 โ
3 โ
5 = 90.
Write each fraction with a denominator
of 90.
7
7 3 21
2
2 2 4
= โ
=
and
= โ
=
30 30 3 90
45 45 2 90
Now add.
7
2 21 4 21 + 4 25
+
=
+
=
=
30 45 90 90
90
90
25
Write
in lowest terms.
90
25 5 โ
5
5
=
=
90 18 โ
5 18
(b) Write each mixed number as an improper
fraction.
5
1 29 7
+
4 +2 =
6
3 6 3
The least common denominator is 6, so
write each fraction with a denominator of 6.
29
7 7 2 14
and = โ
=
6
3 3 2 6
Now add.
29 7 29 14 29 + 14
+ =
+ =
6 3 6
6
6
43
1
= , or 7
6
6
N8. (a) Since 12 = 2 โ
2 โ
3 and 8 = 2 โ
2 โ
2, the least
common denominator must have three
factors of 2 (from 8) and one factor of 3
(from 12), so it is 2 โ
2 โ
2 โ
3 = 24.
Write each fraction with a denominator
of 24.
3 3 3 9
5
5 2 10
and = โ
=
= โ
=
8 8 3 24
12 12 2 24
3
Now add.
5 3 10 9 10 + 9 19
+ =
+
=
=
12 8 24 24
24
24
(b) Write each mixed number as an improper
fraction.
1
5 13 45
3 +5 = +
4
8 4 8
The least common denominator is 8, so
write each fraction with a denominator of 8.
45
13 13 2 26
and
= โ
=
8
4
4 2 8
Now add.
13 45 26 45 26 + 45
+
=
+
=
4 8
8
8
8
71
7
= , or 8
8
8
9. (a) Since 10 = 2 โ
5 and 4 = 2 โ
2, the least
common denominator is 2 โ
2 โ
5 = 20. Write
each fraction with a denominator of 20.
3
3 2 6
1 1 5 5
= โ
=
and = โ
=
10 10 2 20
4 4 5 20
Now subtract.
3 1 6
5
1
โ =
โ
=
10 4 20 20 20
(b) Write each mixed number as an improper
fraction.
3 1 27 3
3 โ1 =
โ
8 2 8 2
The least common denominator is 8. Write
each fraction with a denominator of 8. 27
8
3 3 4 12
remains unchanged, and = โ
= .
2 2 4 8
Now subtract.
27 3 27 12 27 โ 12 15
7
โ =
โ =
= , or 1
8 2 8 8
8
8
8
N9. (a) Since 11 = 11 and 9 = 3 โ
3, the least
common denominator is 3 โ
3 โ
11 = 99. Write
each fraction with a denominator of 99.
5
5 9 45
2 2 11 22
and = โ
=
= โ
=
11 11 9 99
9 9 11 99
Now subtract.
5 2 45 22 23
โ =
โ
=
11 9 99 99 99
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4
Chapter R Prealgebra Review
(b) Write each mixed number as an improper
fraction.
1
5 13 17
4 โ2 = โ
3
6 3 6
The least common denominator is 6. Write
17
each fraction with a denominator of 6.
6
13 13 2 26
remains unchanged, and
= โ
= .
3
3 2 6
Now subtract.
13 17 26 17 26 โ 17 9
โ =
โ =
=
3 6
6 6
6
6
Now reduce.
9 3โ
3 3
1
=
= , or 1
6 2โ
3 2
2
10. To find out how many yards of fabric Jen
should buy, add the lengths needed for each
piece to obtain the total length. The common
denominator is 12.
1
2
1
3
8
6
17
1 +1 + 2 = 1 +1 + 2 = 4
4
3
2
12 12
12
12
17
5
Because
= 1 , we have
12
12
17
5
5
5
yd
4 = 4 + 1 = 5 . Jen should buy 5
12
12
12
12
of fabric.
N10. To find out how long each piece must be,
divide the total length by the number of pieces.
1
21 4 21 1 21
5
10 รท 4 = รท = โ
= , or 2
2
2 1 2 4 8
8
5
Each piece should be 2 feet long.
8
11. (a) In the circle graph, the sector for Other is
the second largest, so Other had the second
23
.
largest share of Internet users,
100
(b) The total number of Internet users, 3900
million, can be rounded to 4000 million (or
1
by 4000.
4 billion). Multiply
10
1
โ
4000 = 400 million
10
N11. (a) In the circle graph, the sector for Africa is
the smallest, so Africa had the least number
of Internet users.
(b) The total number of Internet users, 3900
million, can be rounded to 4000 million (or
1
4 billion). Multiply by 4000.
2
1
โ
4000 = 2000 million, or 2 billion
2
(c) Multiply the fraction from the graph for
Asia by the actual number of users.
1
โ
3900 = 1950 million, or 1.95 billion
2
Exercises
1. True; the number above the fraction bar is
called the numerator and the number below the
fraction bar is called the denominator.
2. True; 5 divides the 31 six times with a
31
1
remainder of one, so = 6 .
5
5
3. False; this is an improper fraction. Its value
is 1.
4. False; the number 1 is neither prime nor
composite.
13
can be written in lowest
39
1
13 13 โ
1 1
=
= .
terms as since
3
39 13 โ
3 3
5. False; the fraction
6. False; the reciprocal of
6
2 1
= 3 is = .
2
6 3
7. False; product refers to multiplication, so the
product of 10 and 2 is 20. The sum of 10 and 2
is 12.
8. False; difference refers to subtraction, so the
difference between 10 and 2 is 8. The quotient
of 10 and 2 is 5.
9.
16 2 โ
8 2
=
=
24 3 โ
8 3
Therefore, C is correct.
(c) Multiply the fraction from the graph for
Africa by the actual number of users.
1
โ
3900 = 390 million
10
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R.1 Fractions
10. Simplify each fraction to find which are equal
5
to .
9
15 3 โ
5 5
=
=
27 3 โ
9 9
30 6 โ
5 5
=
=
54 6 โ
9 9
40 2 โ
20 20
=
=
74 2 โ
37 37
55 11 โ
5 5
=
=
99 11 โ
9 9
Therefore, C is correct.
p
and r must be
q
s
a multiple of both denominators, q and s. Such
a number is q โ
s. Therefore, A is correct.
11. A common denominator for
12. We need to multiply 8 by 3 to get 24 in the
denominator, so we must multiply 5 by 3
as well.
5 5 โ
3 15
=
=
8 8 โ
3 24
Therefore, B is correct.
13. Since 19 has only itself and 1 as factors, it is a
prime number.
14. Since 31 has only itself and 1 as factors, it is a
prime number.
15. 30 = 2 โ
15
= 2 โ
3โ
5
Since 30 has factors other than itself and 1, it is
a composite number.
16. 50 = 2 โ
25
= 2 โ
5 โ
5,
so 50 is a composite number.
5
19. As stated in the text, the number 1 is neither
prime nor composite, by agreement.
20. The number 0 is not a natural number, so it is
neither prime nor composite.
21. 57 = 3 โ
19, so 57 is a composite number.
22. 51 = 3 โ
17, so 51 is a composite number.
23. Since 79 has only itself and 1 as factors, it is a
prime number.
24. Since 83 has only itself and 1 as factors, it is a
prime number.
25. 124 = 2 โ
62
= 2 โ
2 โ
31,
so 124 is a composite number.
26. 138 = 2 โ
69
= 2 โ
3 โ
23,
so 138 is a composite number.
27. 500 = 2 โ
250
= 2 โ
2 โ
125
= 2 โ
2 โ
5 โ
25
= 2 โ
2 โ
5 โ
5 โ
5,
so 500 is a composite number.
28. 700 = 2 โ
350
= 2 โ
2 โ
175
= 2 โ
2 โ
5 โ
35
= 2 โ
2 โ
5 โ
5 โ
7,
so 700 is a composite number.
29. 3458 = 2 โ
1729
= 2 โ
7 โ
247
= 2 โ
7 โ
13 โ
19
Since 3458 has factors other than itself and 1, it
is a composite number.
17. 64 = 2 โ
32
= 2 โ
2 โ
16
= 2 โ
2 โ
2 โ
8
= 2โ
2โ
2โ
2โ
4
= 2โ
2โ
2โ
2โ
2โ
2
Since 64 has factors other than itself and 1, it is
a composite number.
30. 1025 = 5โ
205
= 5 โ
5 โ
41
Since 1025 has factors other than itself and 1, it
is a composite number.
31.
8 1โ
8 1 8 1
1
=
= โ
= โ
1 =
16 2 โ
8 2 8 2
2
18. 81 = 3 โ
27
= 3โ
3โ
9
= 3โ
3โ
3โ
3
Since 81 has factors other than itself and 1, it is
a composite number.
32.
4 1โ
4 1 4 1
1
=
= โ
= โ
1 =
12 3 โ
4 3 4 3
3
33.
15 3 โ
5 3 5
5 5
=
= โ
= 1โ
=
18 3 โ
6 3 6
6 6
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6
Chapter R Prealgebra Review
34.
16 4 โ
4 4 4 4
4
=
= โ
= โ
1 =
20 5 โ
4 5 4 5
5
35.
90 3 โ
30 3 30 3
3
=
= โ
= โ
1 =
150 5 โ
30 5 30 5
5
100 5 โ
20 5 20 5
5
36.
=
= โ
= โ
1 =
140 7 โ
20 7 20 7
7
18 1 โ
18 1 18 1
1
37.
=
= โ
= โ
1 =
90 5 โ
18 5 18 5
5
38.
16 1 โ
16 1 16 1
1
=
= โ
= โ
1 =
64 4 โ
16 4 16 4
4
39.
144 6 โ
24 6 24 6
6
=
= โ
= โ
1 =
120 5 โ
24 5 24 5
5
40.
132 12 โ
11 12 11 12
12
=
= โ
= โ
1 =
77
7 โ
11 7 11 7
7
1
41. 7 12
7
5
12
5
=1 .
Therefore,
7
7
1
42. 9 16
9
7
Therefore,
16
7
=1 .
9
9
6
43. 12 77
72
5
Therefore,
77
5
=6 .
12
12
6
44. 15 101
90
11
Therefore,
101
11
=6 .
15
15
7
45. 11 83
77
6
Therefore,
83
6
=7 .
11
11
5
46. 13 67
65
2
Therefore,
67
2
=5 .
13
13
47. Multiply the denominator of the fraction by the
natural number and then add the numerator to
obtain the numerator of the improper fraction.
5 โ
2 = 10 and 10 + 3 = 13
The denominator of the improper fraction is the
same as the denominator in the mixed number.
3 13
Thus, 2 = .
5 5
48. Multiply the denominator of the fraction by the
natural number and then add the numerator to
obtain the numerator of the improper fraction.
7 โ
5 = 35 and 35 + 6 = 41
The denominator of the improper fraction is the
same as the denominator in the mixed number.
6 41
Thus, 5 = .
7 7
49. Multiply the denominator of the fraction by the
natural number and then add the numerator to
obtain the numerator of the improper fraction.
8 โ
10 = 80 and 80 + 3 = 83
The denominator of the improper fraction is the
same as the denominator in the mixed number.
3 83
Thus, 10 = .
8 8
50. Multiply the denominator of the fraction by the
natural number and then add the numerator to
obtain the numerator of the improper fraction.
3 โ
12 = 36 and 36 + 2 = 38
The denominator of the improper fraction is the
same as the denominator in the mixed number.
2 38
Thus, 12 = .
3 3
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R.1 Fractions
51. Multiply the denominator of the fraction by the
natural number and then add the numerator to
obtain the numerator of the improper fraction.
5 โ
10 = 50 and 50 + 1 = 51
The denominator of the improper fraction is the
same as the denominator in the mixed number.
1 51
Thus, 10 = .
5 5
52. Multiply the denominator of the fraction by the
natural number and then add the numerator to
obtain the numerator of the improper fraction.
6 โ
18 = 108 and 108 + 1 = 109
The denominator of the improper fraction is the
same as the denominator in the mixed number.
1 109
Thus, 18 =
.
6
6
53.
4 6 4 โ
6 24
โ
=
=
5 7 5 โ
7 35
54.
5 2 5 โ
2 10
โ
=
=
9 7 9 โ
7 63
55.
2 3 2โ
3
6
1โ
6
1
โ
=
=
=
=
15 8 15 โ
8 120 20 โ
6 20
56.
3 5
3โ
5
15
1 โ
15
1
โ
=
=
=
=
20 21 20 โ
21 420 28 โ
15 28
57.
1 12 1 โ
12 1 โ
2 โ
6 6
โ
=
=
=
10 5 10 โ
5 2 โ
5 โ
5 25
58.
1 10 1 โ
10 1 โ
2 โ
5
5
โ
=
=
=
8 7 8 โ
7 2 โ
4 โ
7 28
59.
60.
15 8 15 โ
8
โ
=
4 25 4 โ
25
3โ
5โ
4โ
2
=
4โ
5โ
5
3โ
2
=
5
6
1
= , or 1
5
5
21 4 21 โ
4
โ
=
8 7 8โ
7
3โ
7 โ
4
=
4โ
2โ
7
3
1
= , or 1
2
2
61. 21 โ
62. 36 โ
3 21 3
= โ
7 1 7
21 โ
3
=
1โ
7
3โ
7 โ
3
=
1โ
7
3โ
3
=
=9
1
4 36 4
= โ
9 1 9
36 โ
4
=
1โ
9
4โ
9โ
4
=
1โ
9
4โ
4
=
= 16
1
63. Change both mixed numbers to improper
fractions.
1 2 13 5
3 โ
1 = โ
4 3 4 3
13 โ
5
=
4โ
3
65
5
= , or 5
12
12
64. Change both mixed numbers to improper
fractions.
2 3 8 8
2 โ
1 = โ
3 5 3 5
8โ
8
=
3โ
5
64
4
= , or 4
15
15
65. Change both mixed numbers to improper
fractions.
3 1 19 16
2 โ
3 = โ
8 5 8 5
19 โ
16
=
8โ
5
19 โ
2 โ
8
=
8โ
5
38
3
= , or 7
5
5
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7
8
Chapter R Prealgebra Review
66. Change both mixed numbers to improper
fractions.
3 1 18 43
3 โ
7 = โ
5 6 5 6
18 โ
43
=
5โ
6
3 โ
6 โ
43
=
5โ
6
3 โ
43
=
5
129
4
=
, or 25
5
5
67. Change both numbers to improper fractions.
1 5 21
5โ
2 = โ
10 1 10
5 โ
21
=
1 โ
10
5 โ
21
=
1โ
2 โ
5
21
=
1โ
2
1
21
= , or 10
2
2
68. Change both numbers to improper fractions.
2 3 38
3โ
4 = โ
9 1 9
3 โ
38
=
1โ
9
3 โ
38
=
1โ
3 โ
3
38
=
1โ
3
38
2
= , or 12
3
3
69. To divide fractions, multiply by the reciprocal
of the divisor.
7 3 7 2
รท = โ
9 2 9 3
7โ
2
=
9โ
3
14
=
27
70. To divide fractions, multiply by the reciprocal
of the divisor.
6 5 6 4
รท = โ
11 4 11 5
6โ
4
=
11 โ
5
24
=
55
71. To divide fractions, multiply by the reciprocal
of the divisor.
5 3 5 8
รท = โ
4 8 4 3
5โ
8
=
4โ
3
5โ
4โ
2
=
4โ
3
5โ
2
=
3
10
1
= , or 3
3
3
72. To divide fractions, multiply by the reciprocal
of the divisor.
7 3 7 10
รท = โ
5 10 5 3
7 โ
10
=
5โ
3
7 โ
2โ
5
=
5โ
3
14
2
= , or 4
3
3
73. To divide fractions, multiply by the reciprocal
of the divisor.
32 8 32 15
รท = โ
5 15 5 8
32 โ
15
=
5โ
8
8โ
4 โ
3โ
5
=
1โ
5 โ
8
4โ
3
=
= 12
1
Copyright ยฉ 2020 Pearson Education, Inc.
R.1 Fractions
74. To divide fractions, multiply by the reciprocal
of the divisor.
24 6 24 21
รท =
โ
7 21 7 6
24 โ
21
=
7โ
6
4 โ
6 โ
3โ
7
=
1โ
7 โ
6
4โ
3
=
= 12
1
78. To divide fractions, multiply by the reciprocal
of the divisor.
4 8 9
8รท = โ
9 1 4
8โ
9
=
1โ
4
2โ
4โ
9
=
1โ
4
2โ
9
=
= 18
1
75. To divide fractions, multiply by the reciprocal
of the divisor.
3
3 1
รท 12 = โ
4
4 12
3 โ
1
=
4 โ
12
3 โ
1
=
4 โ
3โ
4
1
1
=
=
4 โ
4 16
79. Change the first number to an improper
fraction, and then multiply by the reciprocal of
the divisor.
3 3 27 3
รท
6 รท =
4 8 4 8
27 8
=
โ
4 3
27 โ
8
=
4โ
3
3โ
9 โ
2 โ
4
=
4โ
3
9โ
2
=
= 18
1
76. To divide fractions, multiply by the reciprocal
of the divisor.
2
2 1
รท 30 = โ
5
5 30
2 โ
1
=
5 โ
30
2 โ
1
=
5 โ
2 โ
15
1
1
=
=
5 โ
15 75
77. To divide fractions, multiply by the reciprocal
of the divisor.
3 6 5
6รท = โ
5 1 3
6โ
5
=
1โ
3
2 โ
3โ
5
=
1โ
3
2โ
5
=
= 10
1
9
80. Change the first number to an improper
fraction, and then multiply by the reciprocal of
the divisor.
3 7 28 7
รท
5 รท =
5 10 5 10
28 10
=
โ
5 7
28 โ
10
=
5โ
7
4โ
7โ
2โ
5
=
5โ
7
4โ
2
=
=8
1
81. Change both mixed numbers to improper
fractions, and then multiply by the reciprocal of
the divisor.
1 5 5 12
2 รท1 = รท
2 7 2 7
5 7
= โ
2 12
5โ
7
=
2 โ
12
35
11
, or 1
=
24
24
Copyright ยฉ 2020 Pearson Education, Inc.
10
Chapter R Prealgebra Review
82. Change both mixed numbers to improper
fractions, and then multiply by the reciprocal of
the divisor.
2 2 20 7
รท
2 รท1 =
9 5 9 5
20 5
=
โ
9 7
20 โ
5
=
9โ
7
100
37
=
, or 1
63
63
83. Change both mixed numbers to improper
fractions, and then multiply by the reciprocal of
the divisor.
5 15 21 47
2 รท1 = รท
8 32 8 32
21 32
= โ
8 47
21 โ
32
=
8 โ
47
21 โ
8 โ
4
=
8 โ
47
21 โ
4
=
47
84
37
=
, or 1
47
47
84. Change both mixed numbers to improper
fractions, and then multiply by the reciprocal of
the divisor.
3
4 23 9
รท
2 รท1 =
10 5 10 5
23 5
= โ
10 9
23 โ
5
=
2โ
5โ
9
23
5
= , or 1
18
8
85.
7 4 7 + 4 11
+ =
=
15 15
15
15
86.
2 5 2+5 7
+ =
=
9 9
9
9
87.
88.
7 1 7 +1
+ =
12 12 12
8
=
12
2โ
4
=
3โ
4
2
=
3
3 5 3+5 8 1
+ =
=
=
16 16
16
16 2
89. Since 9 = 3 โ
3, and 3 is prime, the LCD (least
common denominator) is 3 โ
3 = 9.
1 1 3 3
= โ
=
3 3 3 9
Now add the two fractions with the same
denominator.
5 1 5 3 8
+ = + =
9 3 9 9 9
4
1
and , first find the LCD. Since
15
5
15 = 3 โ
5 and 5 is prime, the LCD is 15.
4 1 4 1 3
+ = + โ
15 5 15 5 3
4 3
= +
15 15
4+3
=
15
7
=
15
90. To add
91. Since 8 = 2 โ
2 โ
2 and 6 = 2 โ
3, the LCD is
2 โ
2 โ
2 โ
3 = 24.
3 3 3 9
5 5 4 20
= โ
=
and = โ
=
8 8 3 24
6 6 4 24
Now add fractions with the same denominator.
3 5 9 20 29
5
+ =
+
= , or 1
24
8 6 24 24 24
92. Since 6 = 2 โ
3 and 9 = 3 โ
3, the LCD is
2 โ
3 โ
3 = 18.
5 5 3 15
2 2 2 4
and = โ
=
= โ
=
6 6 3 18
9 9 2 18
Now add fractions with the same denominator.
5 2 15 4 19
1
+ = + = , or 1
6 9 18 18 18
18
Copyright ยฉ 2020 Pearson Education, Inc.
R.1 Fractions 11
93. Since 9 = 3 โ
3 and 16 = 4 โ
4, the LCD is
3 โ
3 โ
4 โ
4 = 144.
5 5 16 80
3
3 9 27
= โ
=
= โ
=
and
9 9 16 144
16 16 9 144
Now add fractions with the same denominator.
5 3
80 27 107
+ =
+
=
9 16 144 144 144
94. Since 4 = 2 โ
2 and 25 = 5 โ
5, the LCD is
2 โ
2 โ
5 โ
5 = 100.
3 3 25 75
6
6 4 24
and
= โ
=
=
โ
=
4 4 25 100
25 25 4 100
Now add fractions with the same denominator.
3 6
75 24
99
+
=
+
=
4 25 100 100 100
95.
1
1 24 1 25
+ =
3 = 3+ =
8
8 8 8 8
1
1 8 1 9
2 = 2+ = + =
4
4 4 4 4
1
1 25 9
+
3 +2 =
8
4 8 4
Since 8 = 2 โ
2 โ
2 and 4 = 2 โ
2, the LCD is
2 โ
2 โ
2 or 8.
1
1 25 9 2
+ โ
3 +2 =
8
4 8 4 2
25 18
=
+
8 8
43
3
= , or 5
8
8
2
2 12 2 14
= 4+ = + =
3
3 3 3 3
1
1 12 1 13
2 = 2+ = + =
6
6 6 6 6
Since 6 = 2 โ
3, the LCD is 6.
2
1 14 2 13
4 +2 = โ
+
3
6 3 2 6
28 13
=
+
6 6
41
5
= , or 6
6
6
Since 4 = 2 โ
2, and 5 is prime, the LCD is
2 โ
2 โ
5 = 20.
1
4 13 5 9 4
3 +1 = โ
+ โ
4 5 4 5 5 4
65 36
=
+
20 20
101
1
=
, or 5
20
20
1
3
and 1 , first change to improper
3
4
fractions then find the LCD, which is 12.
3 1 23 4
5 +1 =
+
4 3 4 3
23 3 4 4
= โ
+ โ
4 3 3 4
69 16
=
+
12 12
85
1
= , or 7
12
12
98. To add 5
99.
7 2 7โ2 5
โ =
=
9 9
9
9
100.
8 3 8โ3 5
โ =
=
11 11 11 11
101.
96. 4
1
1 12 1 13
97. 3 = 3 + = + =
4
4 4 4 4
4
4 5 4 9
1 = 1+ = + =
5
5 5 5 5
102.
13 3 13 โ 3
โ =
15 15
15
10
=
15
2โ
5 2
=
=
3โ
5 3
11 3 11 โ 3
โ =
12 12
12
8
=
12
2โ
4 2
=
=
3โ
4 3
103. Since 12 = 4 โ
3 (12 is a multiple of 3), the LCD
is 12.
1 4 4
โ
=
3 4 12
Now subtract fractions with the same
denominator.
7 1 7 4
3 1โ
3 1
โ = โ =
=
=
12 3 12 12 12 4 โ
3 4
Copyright ยฉ 2020 Pearson Education, Inc.
12
Chapter R Prealgebra Review
104. Since 6 = 3 โ
2 (6 is a multiple of 2), the LCD
is 6.
1 3 3
โ
=
2 3 6
Now subtract fractions with the same
denominator.
5 1 5 3 2 1โ
2 1
โ = โ = =
=
6 2 6 6 6 3โ
2 3
105. Since 12 = 2 โ
2 โ
3 and 9 = 3 โ
3, the LCD is
2 โ
2 โ
3 โ
3 = 36.
7
7 3 21
1 4 4
= โ
=
and โ
=
12 12 3 36
9 4 36
Now subtract fractions with the same
denominator.
7 1 21 4 17
โ =
โ
=
12 9 36 36 36
106.
11 1 11 3 1 4 The LCD of 12
โ = โ
โ โ
16 12 16 3 12 4 and 16 is 48.
33 4
=
โ
48 48
29
=
48
3
3 16 3 19
= 4+ = + =
4
4 4 4 4
2
2 5 2 7
1 = 1+ = + =
5
5 5 5 5
Since 4 = 2 โ
2, and 5 is prime, the LCD is
2 โ
2 โ
5 = 20.
3 2 19 5 7 4
4 โ1 = โ
โ โ
4 5 4 5 5 4
95 28
=
โ
20 20
67
7
= , or 3
20
20
107. 4
108. Change both numbers to improper fractions
then add, using 45 as the common denominator.
4 4 19 13
3 โ1 = โ
5 9 5 9
19 9 13 5
= โ
โ โ
5 9 9 5
171 65
=
โ
45 45
106
16
, or 2
=
45
45
1
1 24 1 25
= 6+ =
+ =
4
4 4 4 4
1
1 15 1 16
5 = 5+ = + =
3
3 3 3 3
Since 4 = 2 โ
2, and 3 is prime, the LCD is
2 โ
2 โ
3 = 12.
1
1 25 16
6 โ5 =
โ
4
3 4 3
25 3 16 4
=
โ
โ โ
4 3 3 4
75 64
=
โ
12 12
11
=
12
109. 6
110.
1
1 15 1 16
5 = 5+ = + =
3
3 3 3 3
1
1 8 1 9
4 = 4+ = + =
2
2 2 2 2
2 and 3 are prime, so the LCD is 2 โ
3 = 6.
1
1 16 2 9 3
5 โ4 = โ
โ โ
3
2 3 2 2 3
32 27
=
โ
6
6
5
=
6
2
2 72 2 74
= 8+ =
+ =
9
9 9 9 9
2
2 12 2 14
4 = 4+ = + =
3
3 3 3 3
Since 9 = 3 โ
3, and 3 is prime, the LCD is
3 โ
3 = 9.
2
2 74 14 3
8 โ4 =
โ โ
9
3 9 3 3
74 42
=
โ
9
9
32
5
= , or 3
9
9
111. 8
Copyright ยฉ 2020 Pearson Education, Inc.
R.1 Fractions 13
5
5 84 5 89
= 7+ =
+ =
12
12 12 12 12
5
5 24 5 29
4 = 4+ =
+ =
6
6 6 6 6
Since 12 = 2 โ
2 โ
3 and 6 = 2 โ
3, the LCD is
2 โ
2 โ
3 = 12.
5
5 89 29 2
7 โ4 =
โ โ
12
6 12 6 2
89 58
=
โ
12 12
31
7
= , or 2
12
12
112. 7
113. Observe that there are 24 dots in the entire
figure, 6 dots in the triangle, 12 dots in the
rectangle, and 2 dots in the overlapping
region.
(a)
12 1
= of all the dots are in the rectangle.
24 2
(b)
6 1
= of all the dots are in the triangle.
24 4
(c)
(d)
2 1
= of the dots in the triangle are in the
6 3
overlapping region.
2 1
= of the dots in the rectangle are in
12 6
the overlapping region.
1
of 36, so Maureen got a hit in
3
1
exactly of her at-bats.
3
114. (a) 12 is
1
of 11, so Chase got a
2
1
hit in just less than of his at-bats.
2
(b) 5 is a little less than
1
of 11, so Chase got
10
1
a home run in just less than
of his at10
bats.
(c) 1 is a little less than
1
1
of 16, and 10 is of 20, so Joe and
2
2
1
Greg each got hits of the time they were
2
at bat.
(e) 8 is
115. Multiply the number of cups of water per
serving by the number of servings.
3
3 8
โ
8 = โ
4
4 1
3โ
8
=
4 โ
1
24
=
4
= 6 cups
For 8 microwave servings, 6 cups of water will
be needed.
1
2
tsp, or
4
8
1
tsp, of salt. Six stove-top servings require
2
4
tsp, or tsp, of salt. Five is halfway between 4
8
3
2
4
and 6, and is halfway between and .
8
8
8
Therefore, 5 stove-top servings would require
3
tsp of salt.
8
116. Four stove-top servings require
117. The difference in length is found by
subtracting.
1
1 13 17
3 โ2 = โ
4
8 4 8
13 2 17
= โ
โ
LCD = 8
4 2 8
26 17
=
โ
8 8
9
1
= , or 1
8
8
1
The difference is 1 inches.
8
1
of 40, so Christine
4
1
got a hit in just less than of her at-bats.
4
(d) 9 is a little less than
Copyright ยฉ 2020 Pearson Education, Inc.
14
Chapter R Prealgebra Review
118. The difference in length is found by
subtracting.
1 4 17
4โ2 = โ
8 1 8
4 8 17
= โ
โ
LCD = 8
1 8 8
32 17
=
โ
8 8
15
7
= , or 1
8
8
7
The difference is 1 inches.
8
119. The difference between the two measures is
found by subtracting, using 16 as the LCD.
3 3 3 4 3
โ = โ
โ
4 16 4 4 16
12 3
= โ
16 16
12 โ 3
=
16
9
=
16
9
The difference is
inch.
16
120. The difference between the two measures is
found by subtracting, using 16 as a common
denominator.
9 3 9 3 2
โ = โ โ
16 8 16 8 2
9 6
= โ
16 16
9โ6
=
16
3
=
16
3
inch.
The difference is
16
121. The perimeter is the sum of the measures of the
5 sides.
3
1
7
5
196 + 98 + 146 + 100 + 76
4
2
8
8
6
4
7
5
= 196 + 98 + 146 + 100 + 76
8
8
8
8
6+4+7+5
= 196 + 98 + 146 + 100 + 76 +
8
22 ๏ฆ 22
6
3๏ถ
= 616 +
=2 =2 ๏ท
๏ง
8 ๏จ 8
8
4๏ธ
3
= 618 feet
4
3
The perimeter is 618 feet.
4
122. To find the perimeter of a triangle, add the
lengths of the three sides.
1
1
1
2
4
1
5 + 7 + 10 = 5 + 7 + 10
4
2
8
8
8
8
7
= 22
8
7
The perimeter of the triangle is 22 feet.
8
123. Divide the total board length by 3.
5
125 3
15 รท 3 =
รท
8
8 1
125 1
=
โ
8 3
125 โ
1
=
8โ
3
125
5
=
, or 5
24
24
The length of each of the three pieces must be
5
inches.
5
24
124. Divide the total amount of tomato sauce by the
number of servings.
1
7 7 7 1 7 โ
1 1
2 รท7 = รท = โ
=
=
3
3 1 3 7 3โ
7 3
1
For 1 serving of barbecue sauce, cup of
3
tomato sauce is needed.
Copyright ยฉ 2020 Pearson Education, Inc.
R.1 Fractions 15
125. To find the number of cakes the caterer can
1
3
make, divide 15 by 1 .
2
4
1 3 31 7
15 รท 1 = รท
2 4 2 4
31 4
= โ
2 7
31 โ
2 โ
2
=
2โ
7
62
6
= , or 8
7
7
There is not quite enough sugar for 9 cakes.
The caterer can make 8 cakes with some sugar
left over.
126. Divide the total amount of fabric by the amount
of fabric needed to cover one chair.
2
1 71 9
23 รท 2 = รท
4 3 4
3
71 4
= โ
3 9
71 โ
4
=
3โ
9
284
14
, or 10
=
27
27
Kyla can cover 10 chairs. There will be some
fabric left over.
127. Multiply the amount of fabric it takes to make
one costume by the number of costumes.
3
19 7
2 โ
7 = โ
8
8 1
19 โ
7
=
8 โ
1
133
5
, or 16 yd
=
8
8
5
For 7 costumes, 16 yards of fabric would be
8
needed.
128. Multiply the amount of sugar for one batch
times the number of batches.
2
8 4
2 โ
4 = โ
3
3 1
8โ
4
=
3 โ
1
32
2
= , or 10
3
3
2
10 cups of sugar are required to make four
3
batches of cookies.
129. Subtract the heights to find the difference.
1
1 21 57
10 โ 7 = โ
2
8 2 8
21 4 57
= โ
โ
LCD = 8
2 4 8
84 57
=
โ
8
8
27
3
=
, or 3
8
8
3
The difference in heights is 3 inches.
8
3
11
from
using 16 as the LCD.
8
16
11 3 11 3 2
โ = โ โ
16 8 16 8 2
11 6
= โ
16 16
5
=
16
3
5
11
inch smaller than
inch.
Thus, inch is
8
16
16
130. Subtract
11
10
1
can be rounded to
= .
100
100 10
Multiply by the total number of foreign-born
people in the U.S., approximately 40 million.
1
1 40 4 โ
10 4
โ
40 = โ
=
= = 4,
10
10 1 1 โ
10 1
There were approximately 4 million (or
4,000,000) foreign-born people in the U.S. who
were born in Europe.
For the actual number:
2
11
11 40 11 โ
2 โ
20 22
โ
40 =
โ
=
=
, or 4
5
100
100 1
5 โ
20 โ
1
5
The actual number who were born in Europe
2
was 4 million (or 4,400,000) people.
5
131. A share of
132. Multiply the fraction representing the U.S.
foreign-born population from Latin America,
13
, by the total number of foreign-born people
25
in the U.S., approximately 40 million.
4
13
13 40 13 โ
5 โ
8 104
โ
40 =
โ
=
=
, or 20
5
25
25 1
5 โ
5 โ
1
5
4
There were approximately 20 million (or
5
20,800,000) foreign-born people in the U.S.
who were born in Latin America.
Copyright ยฉ 2020 Pearson Education, Inc.
16
Chapter R Prealgebra Review
133. The sum of the fractions representing the U.S.
foreign-born population from Latin America,
Asia, or Europe is
13 29 11 13 4 29 11
+
+
=
โ
+
+
25 100 100 25 4 100 100
52 + 29 + 11
=
100
92
=
100
23 โ
4
=
25 โ
4
23
= .
25
So the fraction representing the U.S. foreignborn population from other regions is
23 25 23
1โ
=
โ
25 25 25
2
= .
25
134. The sum of the fractions representing the U.S.
foreign-born population from Latin America or
Asia is
13 29 13 4 29
+
=
โ
+
25 100 25 4 100
52 + 29
=
100
81
=
.
100
14
1 98
135. Estimate each fraction.
is about ,
is
26
2 99
100
90
is about 2,
is about 3, and
about 1,
51
31
13
1
is about .
27
2
Therefore, the sum is approximately
1
1
+ 1 + 2 + 3 + = 7.
2
2
The correct choice is C.
202
99
is about 4,
is
50
100
21
75
1
is about , and
is about 2.
about 1,
40
36
2
Therefore, the product is approximately
1
4 โ
1 โ
โ
2 = 4
2
The correct choice is B.
R.2 Decimals and Percents
Classroom Examples, Now Try Exercises
1. (a) 0.15 =
15
100
9
1000
(b) 0.009 =
(c) 2.5 = 2
N1. (a) 0.8 =
5 25
=
10 10
8
10
(b) 0.431 =
431
1000
(c) 2.58 = 2
2. (a)
58 258
=
100 100
42.830
71.000
+ 3.074
116.904
(b)
32.50
โ 21.72
10.78
N2. (a)
68.900
42.720
+ 8.973
120.593
(b)
351.800
โ 2.706
349.094
3. (a)
30.2
1 decimal place
ร 0.052 3 decimal places
136. Estimate each fraction.
604
โ
1510
1+ 3 = 4
1.5704 4 decimal places
(b)
0.06 2 decimal places
ร 0.12 2 decimal places
12
โ
6
2+2 = 4
0.0072 4 decimal places
Copyright ยฉ 2020 Pearson Education, Inc.
R.2 Decimals and Percents 17
N3. (a)
9.32 2 decimal places
ร 1.4 1 decimal place
3728
โ
932
2 +1 = 3
13.048 3 decimal places
(b)
0.6 1 decimal place
ร 0.004 3 decimal places
24
1+ 3 = 4
0.0024 4 decimal places
4. (a) To change the divisor 0.37 into a whole
number, move each decimal point two
places to the right. Move the decimal point
straight up and divide as with whole
numbers.
14.8
37 547.6
37
177
148
296
296
0
Therefore, 5.476 รท 0.37 = 14.8.
(b) To change the divisor 3.1 into a whole
number, move each decimal point one place
to the right. Move the decimal point straight
up and divide as with whole numbers.
1.21
31 37.60
31
66
62
40
31
9
We carried out the division to 2 decimal
places so that we could round to 1 decimal
place. Therefore, 3.76 รท 3.1 โ 1.2.
N4. (a) To change the divisor 14.9 into a whole
number, move each decimal point one place
to the right. Move the decimal point straight
up and divide as with whole numbers.
30.3
149 4514.7
447
447
447
0
Therefore, 451.47 รท 14.9 = 30.3.
(b) To change the divisor 1.3 into a whole
number, move each decimal point one place
to the right. Move the decimal point straight
up and divide as with whole numbers.
5.641
13 73.340
65
83
78
54
52
20
13
7
We carried out the division to 3 decimal
places so that we could round to 2 decimal
places. Therefore, 7.334 รท 1.3 โ 5.64.
5. (a) Move the decimal point three places to the
right.
19.5 ร 1000 = 19,500
(b) Move the decimal point one place to the
left.
960.1 รท 10 = 96.01
N5. (a) Move the decimal point one place to the
right.
294.72 ร 10 = 2947.2
(b) Move the decimal point two places to the
left. Insert a 0 in front of the 4 to do this.
4.793 รท 100 = 0.04793
Copyright ยฉ 2020 Pearson Education, Inc.
18
Chapter R Prealgebra Review
6. (a) Divide 3 by 50. Add a decimal point and as
many 0s as necessary.
0.06
50 3.00
3 00
0
1
7. (a) 5 % = 5.25%
4
5.25
=
100
= 0.0525
(b) 200% =
3
= 0.06.
Therefore,
50
(b) Divide 11 by 1. Add a decimal point and as
many 0s as necessary.
0.090909…
11 1.000000…
99
100
N7. (a) 23% =
200
= 2.00, or 2
100
23
= 0.23
100
(b) 350% =
350
= 3.50, or 3.5
100
8. (a) 0.06 = 0.06 โ
100% = 6%
(b) 1.75 = 1.75 โ
100% = 175%
99
100
N8. (a) 0.31 = 0.31 โ
100% = 31%
99
(b) 1.32 = 1.32 โ
100% = 132%
1
Note that the pattern repeats. Therefore,
1
= 0.09, or about 0.091.
11
N6. (a) Divide 20 by 17. Add a decimal point and
as many 0s as necessary.
0.85
20 17.00
160
100
100
0
Therefore,
17
= 0.85.
20
(b) Divide 2 by 9. Add a decimal point and as
many 0s as necessary.
0.222…
9 2.000…
18
20
18
20
18
2
Note that the pattern repeats. Therefore,
2
= 0.2, or 0.222.
9
9. (a) 85% = 0.85
(b) 110% = 1.10, or 1.1
(c) 0.30 = 30%
(d) 0.165 = 16.5%
N9. (a) 52% = 0.52
(b) 2% = 02% = 0.02
(c) 0.45 = 45%
(d) 3.5 = 3.50 = 350%
65
100
In lowest terms,
65 13 โ
5 13
=
=
100 20 โ
5 20
10. (a) 65% =
(b) 1.5% =
1.5 1.5 10
15
3
=
โ
=
=
100 100 10 1000 200
20
100
In lowest terms,
20 1 โ
20 1
=
=
100 5 โ
20 5
N10. (a) 20% =
160
100
In lowest terms,
160 8 โ
20 8
3
=
= , or 1
100 5 โ
20 5
5
(b) 160% =
Copyright ยฉ 2020 Pearson Education, Inc.
R.2 Decimals and Percents 19
11. (a)
Exercises
3
3
= โ
100%
50 50
3 100
%
= โ
50 1
3 โ
50 โ
2
%
=
50
= 6%
1. 367.9412
(a) Tens: 6
(b) Tenths: 9
(c) Thousandths: 1
(d) Ones: 7
1 1
(b) = โ
100%
3 3
1 100
= โ
%
3 1
100
=
%
3
1
= 33 %, or 33.3%
3
N11. (a)
(b)
(e) Hundredths: 4
2. Answers will vary. One example is 5243.0164.
3. 46.249
(a) 46.25
(b) 46.2
(c) 46
(d) 50
6
6
=
โ
100%
25 25
6 100
=
โ
%
25 1
6 โ
25 โ
4
=
%
25
= 24%
4. (a) 0.889
(b) 0.444
(c) 0.976
(d) 0.865
7 7
= โ
100%
9 9
7 100
= โ
%
9 1
700
=
%
9
7
= 77 %, or 77.7%
9
12. The discount is 30% of $69. The word of here
means multiply.
30% of 69
โ
โ โ
0.30 โ
69 = 20.7
The discount is $20.70. The sale price is found
by subtracting.
$69.00 โ $20.70 = $48.30
N12. The discount is 60% of $120. The word of here
means multiply.
60% of 120
โ
โ
โ
0.60 โ
120 = 72
The discount is $72. The sale price is found by
subtracting.
$120.00 โ $72 = $48
5. 0.4 =
4
10
6. 0.6 =
6
10
7. 0.64 =
64
100
8. 0.82 =
82
100
9. 0.138 =
138
1000
10. 0.104 =
104
1000
11. 0.043 =
43
1000
12. 0.087 =
87
1000
13. 3.805 = 3
805 3805
=
1000 1000
14. 5.166 = 5
166 5166
=
1000 1000
Copyright ยฉ 2020 Pearson Education, Inc.
20
Chapter R Prealgebra Review
15.
16.
25.320
109.200
+ 8.574
26.
143.094
27.
20.418
90.527
32.430
+ 589.800
712.757
17.
28.73
โ 3.12
28.
19.
20.
43.50
โ 28.17
29.
2+2 = 4
22.41 2 decimal places
ร 33 0 decimal places
โ
739.53 2 decimal places
30.
3.87
15.00
+ 2.90
8.20
1.09
+ 12.00
32.560
47.356
+ 1.800
75.200
123.960
+ 3.897
203.057
25.
โ
345.10
โ 56.31
81.716
24.
34.04 2 decimal places
ร 0.56 2 decimal places
6723
6723
18.000
โ 2.789
15.211
2+0 = 2
55.76 2 decimal places
ร 72 0 decimal places
โ
11152
39032
2+0 = 2
4014.72 2 decimal places
31.
0.2
1 decimal place
ร 0.03 2 decimal places
6
1+ 2 = 3
0.006 3 decimal places
21.29
23.
โ
1+1 = 2
19.0624 4 decimal places
21.77
22.
128
1152
15.33
288.79
21.
1 decimal place
1 decimal place
20424
17020
46.88
โ 13.45
33.43
12.8
ร 9.1
116.48 2 decimal places
25.61
18.
29.000
โ 8.582
32.
0.07 2 decimal places
ร 0.004 3 decimal places
28
2+3 = 5
0.00028 5 decimal places
7.15
33. 11 78.65
77
16
11
55
55
0
Copyright ยฉ 2020 Pearson Education, Inc.
R.2 Decimals and Percents 21
5.24
34. 14 73.36
70
33
28
56
56
0
8.44
527 4447.88
4216
2318
2108
2108
2108
35. To change the divisor 11.6 into a whole
number, move each decimal point one place to
the right. Move the decimal point straight up
and divide as with whole numbers.
2.8
116 324.8
232
092 8
0928
000 0
Therefore, 32.48 รท 11.6 = 2.8 .
36. To change the divisor 17.4 into a whole
number, move each decimal point one place to
the right. Move the decimal point straight up
and divide as with whole numbers.
4.9
174 852.6
696
156 6
156 6
000 0
Therefore, 85.26 รท 17.4 = 4.9 .
37. To change the divisor 9.74 into a whole
number, move each decimal point two places to
the right. Move the decimal point straight up
and divide as with whole numbers.
2.05
974 1996.70
1948
4870
4870
0
Therefore, 19.967 รท 9.74 = 2.05.
38. To change the divisor 5.27 into a whole
number, move each decimal point two places to
the right. Move the decimal point straight up
and divide as with whole numbers.
0
Therefore, 44.4788 รท 5.27 = 8.44.
39. Move the decimal point one place to the right.
123.26 ร 10 = 1232.6
40. Move the decimal point one place to the right.
785.91ร 10 = 7859.1
41. Move the decimal point two places to the right.
57.116 ร 100 = 5711.6
42. Move the decimal point two places to the right.
82.053 ร 100 = 8205.3
43. Move the decimal point three places to the
right.
0.094 ร 1000 = 94
44. Move the decimal point three places to the
right.
0.025 ร 1000 = 25
45. Move the decimal point one place to the left.
1.62 รท 10 = 0.162
46. Move the decimal point one place to the left.
8.04 รท 10 = 0.804
47. Move the decimal point two places to the left.
124.03 รท 100 = 1.2403
48. Move the decimal point two places to the left.
490.35 รท 100 = 4.9035
49. Move the decimal point three places to the left.
23.29 รท 1000 = 023.29 รท 1000 = 0.02329
50. Move the decimal point three places to the left.
59.8 รท 1000 = 059.8 รท 1000 = 0.0598
51. Convert from a decimal to a percent.
0.01 = 0.01 โ
100% = 1%
Fraction in Lowest Terms
(or Whole Number)
Decimal
Percent
1
100
0.01
1%
Copyright ยฉ 2020 Pearson Education, Inc.
22
Chapter R Prealgebra Review
52. Convert from a percent to a decimal.
2
2% =
= 0.02
100
Fraction in Lowest Terms
(or Whole Number)
Decimal
Percent
1
50
0.02
2%
53. Convert from a percent to a fraction.
5
5% =
100
In lowest terms,
5
1โ
5
1
=
=
100 20 โ
5 20
Fraction in Lowest Terms
(or Whole Number)
Decimal
Percent
1
20
0.05
5%
56. Convert the percent to a decimal first.
20% = 0.20, or 0.2
Convert from a percent to a fraction.
20
20% =
100
In lowest terms,
20 1 โ
20 1
=
=
100 5 โ
20 5
Fraction in Lowest Terms
(or Whole Number)
Decimal
Percent
1
5
0.2
20%
57. Convert to a decimal first. Divide 1 by 4. Add a
decimal point and as many 0s as necessary.
0.25
4 1.00
8
20
20
54. Convert to a decimal first. Divide 1 by 10.
Move the decimal point one place to the left.
1 รท 10 = 0.1
Convert the decimal to a percent.
0.1 = 0.1 โ
100% = 10%
Fraction in Lowest Terms
(or Whole Number)
Decimal
Percent
1
10
0.1
10%
0
Convert the decimal to a percent.
0.25 = 0.25 โ
100% = 25%
Fraction in Lowest Terms
(or Whole Number)
Decimal
Percent
1
4
0.25
25%
58. Convert to a decimal first. Divide 1 by 3. Add a
decimal point and as many 0s as necessary.
0.33…
3 1.00…
9
55. Convert the decimal to a percent.
0.125 = 0.125 โ
100% = 12.5%
Fraction in Lowest Terms
(or Whole Number)
Decimal
Percent
1
8
0.125
12.5%
10
9
1
Note that the pattern repeats. Therefore,
1
= 0.3.
3
Convert the decimal to a percent.
1
0.333 = 0.333 โ
100% = 33.3%, or 33 %
3
Copyright ยฉ 2020 Pearson Education, Inc.
R.2 Decimals and Percents 23
Fraction in Lowest Terms
(or Whole Number)
1
3
Decimal
Percent
0.3
33.3%
or
1
33 %
3
59. Convert the percent to a decimal first.
50% = 0.50, or 0.5
Convert from a percentl to a fraction.
50
50% =
100
In lowest terms,
50 1 โ
50 1
=
=
100 2 โ
50 2
Decimal
Percent
1
2
0.5
50%
60. Divide 2 by 3. Add a decimal point and as
many 0s as necessary.
0.66…
3 2.00…
18
20
18
2
Note that the pattern repeats. Therefore,
2
= 0.6.
3
2
3
Fraction in Lowest Terms
(or Whole Number)
Decimal
Percent
3
4
0.75
75%
62. Convert the decimal to a percent.
1.0 = 1.0 โ
100% = 100%
Fraction in Lowest Terms
(or Whole Number)
Fraction in Lowest Terms
(or Whole Number)
61. Convert the decimal to a percent first.
0.75 = 0.75 โ
100% = 75%
Convert from a percent to a fraction.
75
75% =
100
In lowest terms,
75 3 โ
25 3
=
=
100 4 โ
25 4
Decimal
Percent
0.6
66.6%
or
2
66 %
3
Fraction in Lowest Terms
(or Whole Number)
Decimal
Percent
1
1.0
100%
63. Divide 21 by 5. Add a decimal point and as
many 0s as necessary.
4.2
5 21.0
20
010
010
000
64. Divide 9 by 5. Add a decimal point and as
many 0s as necessary.
1.8
5 9.0
5
40
40
0
Copyright ยฉ 2020 Pearson Education, Inc.
24
Chapter R Prealgebra Review
65. Divide 9 by 4. Add a decimal point and as
many 0s as necessary.
2.25
4 9.00
8
10
8
20
20
0
66. Divide 15 by 4. Add a decimal point and as
many 0s as necessary.
3.75
4 15.00
12
30
28
20
20
0
67. Divide 3 by 8. Add a decimal point and as
many 0s as necessary.
0.375
8 3.000
24
60
56
40
40
0
68. Divide 7 by 8. Add a decimal point and as
many 0s as necessary.
0.875
8 7.000
64
60
56
40
40
0
69. Divide 5 by 9. Add a decimal point and as
many 0s as necessary.
0.555…
9 5.000…
45
50
45
50
45
5
Note that the pattern repeats. Therefore,
5
= 0.5, or about 0.556.
9
70. Divide 8 by 9. Add a decimal point and as
many 0s as necessary.
0.888…
9 8.000…
72
80
72
80
72
8
Note that the pattern repeats. Therefore,
8
= 0.8, or about 0.889.
9
71. Divide 1 by 6. Add a decimal point and as
many 0s as necessary.
0.166…
6 1.000…
6
40
36
40
36
4
Note that the pattern repeats. Therefore,
1
= 0.16, or about 0.167.
6
Copyright ยฉ 2020 Pearson Education, Inc.
R.2 Decimals and Percents 25
72. Divide 5 by 6. Add a decimal point and as
many 0s as necessary.
0.833…
6 5.000…
48
20
18
20
18
2
Note that the pattern repeats. Therefore,
5
= 0.83, or about 0.833.
6
73. 54% = 0.54
74. 39% = 0.39
75. 7% = 07% = 0.07
78. 189% = 1.89
79. 2.4% = 02.4% = 0.024
80. 3.1% = 03.1% = 0.031
1
82. 5 % = 5.5% = 05.5% = 0.055
2
86. 0.83 = 83%
87. 0.02 = 2%
88. 0.08 = 8%
89. 0.004 = 0.4%
90. 0.005 = 0.5%
91. 1.28 = 128%
92. 2.35 = 235%
93. 0.40 = 40%
98. 47% =
47
100
15
100
In lowest terms,
15
3โ
5
3
=
=
100 20 โ
5 20
99. 15% =
35
100
In lowest terms,
35
7 โ
5
7
=
=
100 20 โ
5 20
100. 35% =
8
100
In lowest terms,
8
2โ
4
2
=
=
100 25 โ
4 25
102. 8% =
1
81. 6 % = 6.25% = 06.25% = 0.0625
4
85. 0.79 = 79%
51
100
101. 2% =
77. 117% = 1.17
84. 0.9% = 00.9% = 0.009
97. 51% =
2
100
In lowest terms,
2
1โ
2
1
=
=
100 50 โ
2 50
76. 4% = 04% = 0.04
83. 0.8% = 00.8% = 0.008
96. 10 = 10.00 = 1000%
140
100
In lowest terms,
140 7 โ
20 7
2
=
= , or 1
100 5 โ
20 5
5
103. 140% =
180
100
In lowest terms,
180 9 โ
20 9
4
=
= , or 1
100 5 โ
20 5
5
104. 180% =
7.5 7.5 10
75
=
โ
=
100 100 10 1000
In lowest terms,
75
3 โ
25
3
=
=
1000 40 โ
25 40
105. 7.5% =
94. 0.6 = 0.60 = 60%
95. 6 = 6.00 = 600%
Copyright ยฉ 2020 Pearson Education, Inc.
26
Chapter R Prealgebra Review
2.5 2.5 10
25
=
โ
=
100 100 10 1000
In lowest terms,
25
1 โ
25
1
=
=
1000 40 โ
25 40
106. 2.5% =
107.
108.
109.
110.
111.
115.
4 4
4 100
4 โ
5 โ
20
%=
% = 80%
= โ
100% = โ
5 5
5 1
5
13 13
= โ
100%
6
6
13 100
= โ
%
6 1
13 โ
2 โ
50
%
=
2โ
3
= 216.6%
31 31
= โ
100%
9
9
31 100
= โ
%
9 1
3100
%
=
9
= 344.4%
3
3
=
โ
100%
25 25
3 100
=
โ
%
25 1
3 โ
4 โ
25
=
%
25
= 12%
116.
7
7
= โ
100%
50 50
7 100
= โ
%
50 1
7 โ
2 โ
50
=
%
50
= 14%
117. The word of here means multiply.
50% of 320
โ
โ
โ
0.50 โ
320 = 160
118. The word of here means multiply.
25% of 120
โ
โ
โ
0.25 โ
120 = 30
9
9
=
โ
100%
20 20
9 100
=
โ
%
20 1
9 โ
5 โ
20
=
%
20
= 45%
119. The word of here means multiply.
6% of 80
โ
โ โ
0.06 โ
80 = 4.8
120. The word of here means multiply.
5% of 70
โ
โ โ
0.05 โ
70 = 3.5
2
2
= โ
100%
11 11
2 100
= โ
%
11 1
200
=
%
11
= 18.18%
121. The word of here means multiply.
14% of 780
โ
โ
โ
0.14 โ
780 = 109.2
112.
4 4
4 100
400
= โ
100% = โ
%=
% = 44.4%
9 9
9 1
9
113.
9 9
9 100
9 โ
4 โ
25
= โ
100% = โ
%=
% = 225%
4 4
4 1
4
114.
8 8
8 100
8 โ
5 โ
20
= โ
100% = โ
%=
% = 160%
5 5
5 1
5
122. The word of here means multiply.
26% of 480
โ
โ
โ
0.26 โ
480 = 124.8
Copyright ยฉ 2020 Pearson Education, Inc.
R.2 Decimals and Percents 27
123. The tip is 20% of $89. The word of here means
multiply.
20% of $89
โ
โ
โ
0.20 โ
$89 = $17.80
The tip is $17.80. The total bill is found by
adding.
$89 + $17.80 = $106.80
124. The raise is 7% of $15. The word of here means
multiply.
7% of $15
โ
โ
โ
0.07 โ
$15 = $1.05
The amount of the raise is $1.05 per hour. The
new hourly rate is found by adding.
$15 + $1.05 = $16.05
129. First, find the portion of the circle graph that
represents โOther.โ
100% โ ( 26% + 25% + 19% + 15% ) = 15%
The portion of the circle graph showing the
number of travelers from โOtherโ countries is
15% of the circle.
130. The portion of the circle graph showing the
number of travelers from โOtherโ countries is
15% of the circle. Find 15% of 76 million.
15% of 76 million
โ
โ
โ
0.15 โ
76 million = 11.4 million,
or approximately 11,400,000 travelers.
125. The discount is 15% of $795. The word of here
means multiply.
15% of $795
โ
โ
โ
0.15 โ
$795 = $119.25
The amount of the discount is $119.25. The
sale price is found by subtracting.
$795 โ $119.25 = $675.75
126. The discount is 20% of $597. The word of here
means multiply.
20% of $597
โ
โ
โ
0.20 โ
$597 = $119.40
The amount of the discount is $119.40. The
sale price is found by subtracting.
$597 โ $119.40 = $477.60
127. The portion of the circle graph showing the
number of travelers from Canada is 26% of the
circle. Find 26% of 76 million.
26% of 76 million
โ
โ
โ
0.26 โ
76 million = 19.76 million,
or approximately 19,760,000 travelers.
128. The portion of the circle graph showing the
number of travelers from Mexico is 25% of the
circle. Find 25% of 76 million.
25% of 76 million
โ
โ
โ
0.25 โ
76 million = 19 million,
or approximately 19,000,000 travelers.
Copyright ยฉ 2020 Pearson Education, Inc.
28
Chapter 1 The Real Number System
Chapter 1
The Real Number System
1.1 Exponents, Order of Operations, and
Inequality
N2. (a) 15 โ 2 โ
6
= 15 โ 12 Multiply.
Subtract.
=3
(b) 8 + 2 ( 5 โ 1)
= 8 + 2 ( 4 ) Subtract inside parentheses.
Classroom Examples, Now Try Exercises
1. (a) 92 = 9 โ
9 = 81
2
(c) (0.5) = 0.5 โ
0.5 = 0.25
N1. (a) 62 = 6 โ
6 = 36
3
4 4 4 64
๏ฆ4๏ถ
(b) ๏ง ๏ท = โ
โ
=
5
5๏ด
5 125
๏จ5๏ธ
๏ฑ
๏ด๏ฒ
๏ณ
4
is used as a factor 3 times.
5
= 36 โ 35
Multiply.
=1
Subtract.
= 9 [12]
= 108
(b)
(b) 18 + 2 ( 6 โ 3)
= 18 + 2 ( 3) Subtract inside parentheses.
(c) 7 โ
6 โ 3 ( 8 + 1)
= 7 โ
6 โ 3 ( 9 ) Add inside parentheses.
= 42 โ 27
Multiply.
= 15
Subtract.
(d) 2 + 32 โ 5 โ
2
= 2 + 9 โ 5 โ
2 Apply exponents.
= 2 + 9 โ 10 Multiply.
= 11 โ 10
Add.
=1
Subtract.
Divide/multiply.
Subtract.
Add.
= 9 [36 โ 2(12) ] Add inside parentheses.
= 9 [36 โ 24]
2. (a) 10 โ 6 รท 2
= 10 โ 3 Divide.
=7
Subtract.
Apply exponents.
Multiply.
3. (a) 9 [36 โ 2(4 + 8) ]
(c) (0.7) = 0.7 โ
0.7 = 0.49
Add.
Add.
(d) 8 โ
10 รท 4 โ 23 + 3 โ
42
= 8 โ
10 รท 4 โ 8 + 3 โ
16
= 80 รท 4 โ 8 + 3 โ
16
= 20 โ 8 + 48
= 12 + 48
= 60
2
= 24
= 16
= 6 ( 6 ) โ 7 โ
5 Add inside parentheses.
1
is used as a factor 4 times.
2
Multiply.
Multiply.
(c) 6 ( 2 + 4 ) โ 7 โ
5
4
1 1 1 1 1
๏ฆ1๏ถ
(b) ๏ง ๏ท = โ
โ
โ
=
2
2๏ด
2๏ฒ๏ด
2๏ณ
2 16
๏จ ๏ธ
๏ฑ
= 18 + 6
= 8+8
2(7 + 8) + 2
3โ
5 +1
2(15) + 2
=
3โ
5 +1
30 + 2
=
15 + 1
32
=
16
=2
Multiply inside brackets.
Subtract inside brackets.
Multiply.
Add inside parentheses.
Multiply.
Add.
Divide.
N3. (a) 7 [3(3 โ 1) + 4]
= 7 [3(2) + 4]
= 7 [ 6 + 4]
= 7 [10]
= 70
Copyright ยฉ 2020 Pearson Education, Inc.
Subtract inside parentheses.
Multiply inside brackets.
Add inside brackets.
Multiply.
1.1 Exponents, Order of Operations, and Inequality 29
(b)
9(14 โ 4) โ 2
4 + 3โ
6
9(10) โ 2
=
Subtract inside parentheses.
4 + 3โ
6
90 โ 2
=
Multiply.
4 + 18
88
=
Subtract and add.
22
=4
Divide.
4. (a) The statement 12 > 6 is true because 12 is
greater than 6. Note that the inequality
symbol points to the lesser number.
(b) The statement 28 โ 4 โ
7 is false because 28
is equal to 4 โ
7.
(c) The statement
1
โค 0.1 is true because
10
1
= 0.1
10
N5. (a) โTen is not equal to eight minus twoโ is
written as 10 โ 8 โ 2.
(b) โFifty is greater than fifteenโ is written as
50 > 15.
(c) โEleven is less than or equal to twentyโ is
written as 11 โค 20.
6. 9 โค 15 is equivalent to 15 โฅ 9.
N6. 8 8.
Exercises
1. False; 32 = 3 โ
3 = 9.
2. False; 1 raised to any power is 1.
Here, 13 = 1 โ
1 โ
1 = 1.
3. False; a number raised to the first power is that
number, so 31 = 3.
(d) Write the fractions with a common
1 1
denominator. The statement < is
3 4
4
3
4 โ
2 is false because 5 is
less than 8.
(c) The statement
(c) โTwo is greater than or equal to twoโ is
written as 2 โฅ 2.
1
โค 0.25 is true because
4
1
= 0.25
4
(d) Write the fractions with a common
5 7
is
denominator. The statement >
9 11
55 63
> .
equivalent to the statement
99 99
Because 55 is less than 63, the original
statement is false.
5. (a) โNine is equal to eleven minus twoโ is
written as 9 = 11 โ 2.
4. False; 62 means that 6 is used as a factor 2
times, so 62 = 6 โ
6 = 36.
5. False; the common error leading to 42 is adding
4 to 3 and then multiplying by 6. One must
follow the rules for order of operations.
4 + 3(8 โ 2)
= 4 + 3(6)
= 4 + 18
= 22
6. False; multiplications and divisions are
performed in order from left to right.
12 รท 2 โ
3
= 6โ
3
= 18
7. Additions and subtractions are performed in
order from left to right.
18 โ
๏ป 2+
๏ป3
1
2
8. Multiplications and divisions are performed in
order from left to right, and then additions and
subtractions are performed in order from left
to right.
28 โ
๏ป6รท
๏ป2
2
1
(b) โFourteen is greater than twelveโ is written
as 14 > 12.
Copyright ยฉ 2020 Pearson Education, Inc.
30
Chapter 1 The Real Number System
9. Multiplications and divisions are performed in
order from left to right, and then additions and
subtractions are performed in order from left
to right.
2๏ป
โ
8โ
๏ป6รท
๏ป3
1 1 1
๏ฆ1๏ถ
26. ๏ง ๏ท = โ
=
3
3 3 9
๏จ ๏ธ
10. Multiplications and divisions are performed in
order from left to right, and then additions and
subtractions are performed in order from left to
right. If grouping symbols are present, work
within them first, starting with the innermost.
40 +
๏ป 6(
๏ป 1)
๏ป 3โ
3 3 3 27
๏ฆ3๏ถ
28. ๏ง ๏ท = โ
โ
=
4
4 4 4 64
๏จ ๏ธ
11. Multiplications and divisions are performed in
order from left to right, and then additions and
subtractions are performed in order from left to
right. If grouping symbols are present, work
within them first, starting with the innermost.
3๏ป
โ
5โ
๏ป 2(
๏ป 2)
๏ป4+
3
1
3 2
4 3
2
2
1
1
12. Apply all exponents. Then, multiplications and
divisions are performed in order from left to
right, and additions and subtractions are
performed in order from left to right.
3
9โ
๏ป 2๏ป +
๏ป 3 ๏ปโ
4
3 1 4
2
13. 7 2 = 7 โ
7 = 49
14. 82 = 8 โ
8 = 64
15. 122 = 12 โ
12 = 144
16. 142 = 14 โ
14 = 196
17. 43 = 4 โ
4 โ
4 = 64
18. 53 = 5 โ
5 โ
5 = 125
3
19. 10 = 10 โ
10 โ
10 = 1000
20. 113 = 11 โ
11 โ
11 = 1331
21. 34 = 3 โ
3 โ
3 โ
3 = 81
22. 64 = 6 โ
6 โ
6 โ
6 = 1296
23. 45 = 4 โ
4 โ
4 โ
4 โ
4 = 1024
24. 35 = 3 โ
3 โ
3 โ
3 โ
3 = 243
2
1 1 1
๏ฆ1๏ถ
25. ๏ง ๏ท = โ
=
6 6 36
๏จ6๏ธ
2
4
2 2 2 2 16
๏ฆ2๏ถ
27. ๏ง ๏ท = โ
โ
โ
=
3 3 3 3 81
๏จ3๏ธ
3
29.
( 0.6 )2 = 0.6 โ
0.6 = 0.36
30.
( 0.9 )2 = 0.9 โ
0.9 = 0.81
31.
( 0.4 )3 = 0.4 โ
0.4 โ
0.4 = 0.064
32.
( 0.5)4 = 0.5 โ
0.5 โ
0.5 โ
0.5 = 0.0625
33. The multiplication should be performed before
the addition.
8 + 2 โ
3 = 8 + 6 Multiply.
= 14
Add.
The correct value of the expression is 14.
34. When cubing 2, the correct value is 2 โ
2 โ
2 = 8 ,
not 2 โ
3 = 6 .
16 โ 23 + 5 = 16 โ 8 + 5 Apply exponents.
Subtract.
= 8+5
Add.
= 13
The correct value of the expression is 13.
35. 64 รท 4 โ
2 = 16 โ
2 Divide.
= 32
Multiply.
36. 250 รท 5 โ
2 = 50 โ
2 Divide.
= 100
Multiply.
37. 13 + 9 โ
5 = 13 + 45 Multiply.
= 58
Add.
38. 11 + 7 โ
6 = 11 + 42 Multiply.
= 53
Add.
39. 25.2 โ 12.6 รท 4.2 = 25.2 โ 3 Divide.
= 22.2
Subtract.
40. 12.4 โ 9.3 รท 3.1 = 12.4 โ 3 Divide.
= 9.4
Subtract.
41. 9 โ
4 โ 8 โ
3 = 36 โ 24 Multiply.
= 12
Subtract.
Copyright ยฉ 2020 Pearson Education, Inc.
1.1 Exponents, Order of Operations, and Inequality 31
42. 11 โ
4 + 10 โ
3 = 44 + 30 Multiply.
= 74
Add.
43.
44.
1 2 2 11 1 22
โ
+ โ
= +
4 3 5 3 6 15
5 44
=
+
30 30
49
19
, or 1
=
30
30
9 2 4 5 3 4
โ
+ โ
= +
4 3 5 3 2 3
9 8
= +
6 6
17
5
= , or 2
6
6
Multiply.
LCD = 30
Add.
Multiply.
LCD = 6
Add.
45. 20 โ 4 โ
3 + 5 = 20 โ 12 + 5 Multiply.
= 8+5
Subtract.
Add.
= 13
46. 18 โ 7 โ
2 + 6 = 18 โ 14 + 6 Multiply.
Subtract.
= 4+6
Add.
= 10
47. 10 + 40 รท 5 โ
2 = 10 + 8 โ
2 Divide.
= 10 + 16 Multiply.
= 26
Add.
48. 12 + 64 รท 8 โ 4 = 12 + 8 โ 4 Divide.
Add.
= 20 โ 4
Subtract.
= 16
49. 18 โ 2(3 + 4)
= 18 โ 2(7) Add inside parentheses.
Multiply.
= 18 โ 14
Subtract.
=4
50. 30 โ 3 ( 4 + 2 )
= 30 โ 3 ( 6 ) Add inside parentheses.
= 30 โ 18
Multiply.
= 12
Subtract.
51. 3(4 + 2) + 8 โ
3 = 3 โ
6 + 8 โ
3 Add.
= 18 + 24
Multiply.
Add.
= 42
53. 18 โ 42 + 3 = 18 โ 16 + 3 Apply exponents.
Subtract.
= 2+3
Add.
=5
54. 22 โ 23 + 9 = 22 โ 8 + 9 Apply exponents.
Subtract.
= 14 + 9
Add.
= 23
55. 2 + 3[5 + 4(2)] = 2 + 3[5 + 8]
= 2 + 3[13]
= 2 + 39
= 41
Multiply.
Add.
Multiply.
Add.
56. 5 + 4 ๏ซ๏ฉ1 + 7 ( 3) ๏ป๏น = 5 + 4 [1 + 21] Multiply.
= 5 + 4 [ 22]
= 5 + 88
= 93
( )
57. 5 ๏ฉ3 + 4 22 ๏น = 5[3 + 4(4)]
๏ซ
๏ป
= 5(3 + 16)
= 5(19)
= 95
Add.
Multiply.
Add.
Apply exponents.
Multiply.
Add.
Multiply.
( )
58. 6 ๏ฉ 2 + 8 33 ๏น
๏ซ
๏ป
= 6 [ 2 + 8 โ
27 ]
= 6(2 + 216)
= 6 โ
218
= 1308
Apply exponents.
Multiply.
Add.
Multiply.
59. 32 [(11 + 3) โ 4]
= 32 [14 โ 4] Add inside parentheses.
= 32 [10]
Subtract.
= 9[10]
Apply exponents.
= 90
Multiply.
60. 42 [(13 + 4) โ 8]
= 42 [17 โ 8] Add inside parentheses.
= 42 [9]
= 16 [9]
= 144
52. 9 (1 + 7 ) + 2 โ
5 = 9 โ
8 + 2 โ
5 Add.
= 72 + 10
Multiply.
= 82
Add.
Copyright ยฉ 2020 Pearson Education, Inc.
Subtract.
Apply exponents.
Multiply.
32
Chapter 1 The Real Number System
61. Simplify the numerator and denominator
separately, and then divide.
(
)
6 32 โ 1 + 8
8โ2
2
6(9 โ 1) + 8
8โ4
6(8) + 8
=
4
48 + 8
=
4
56
=
= 14
4
67. 10 โ 7 โ 3 = 6
10 โ (7 โ 3) = 10 โ 4 = 6
=
68. 8 + 22 = 100
(8 + 2 )2 = 102 = 10 โ
10 = 100
62. Simplify the numerator and denominator
separately, and then divide.
(
)
2 82 โ 4 + 8
29 โ 33
=
=
66. 2 โ
8 โ 1 โ
3 = 42
2 โ
( 8 โ 1) โ
3 = 2 โ
7 โ
3 = 14 โ
3 = 42
69. 9 โ
3 โ 11 โค 16
27 โ 11 โค 16
16 โค 16
The statement is true since 16 = 16.
70. 6 โ
5 โ 12 โค 18
30 โ 12 โค 18
18 โค 18
The statement is true since 18 = 18.
2 ( 64 โ 4 ) + 8
29 โ 27
2 ( 60 ) + 8
2
120 + 8
=
2
128
=
= 64
2
63. Simplify the numerator and denominator
separately, and then divide.
4(6 + 2) + 8(8 โ 3) 4(8) + 8(5)
=
6(4 โ 2) โ 22
6(2) โ 22
4(8) + 8(5)
=
6(2) โ 4
32 + 40
=
12 โ 4
72
=
=9
8
64. Simplify the numerator and denominator
separately, and then divide.
6(5 + 1) โ 9(1 + 1) 6(6) โ 9(2)
=
5(8 โ 6) โ 23
5(2) โ 23
36 โ 18
=
10 โ 8
18
=
=9
2
65. 3 โ
6 + 4 โ
2 = 60
Listed below are some possibilities. Use trial
and error until you get the desired result.
(3 โ
6) + 4 โ
2 = 18 + 8 = 26 โ 60
(3 โ
6 + 4) โ
2 = 22 โ
2 = 44 โ 60
3 โ
(6 + 4 โ
2) = 3 โ
14 = 42 โ 60
3 โ
(6 + 4) โ
2 = 3 โ
10 โ
2 = 30 โ
2 = 60
71. 5 โ
11 + 2 โ
3 โค 60
55 + 6 โค 60
61 โค 60
The statement is false since 61 is greater
than 60.
72. 9 โ
3 + 4 โ
5 โฅ 48
27 + 20 โฅ 48
47 โฅ 48
The statement is false since 47 is less than 48.
73. 0 โฅ 12 โ
3 โ 6 โ
6
0 โฅ 36 โ 36
0โฅ0
The statement is true since 0 = 0.
74. 10 โค 13 โ
2 โ 15 โ
1
10 โค 26 โ 15
10 โค 11
The statement is true since 10 72
[12 + 10] โ
3 > 72
[22] โ
3 > 72
66 > 72
The statement is false since 66 is less than 72.
78. 2 โ
[7 โ
5 โ 3(2)] โค 58
2 โ
[35 โ 6] โค 58
2[29] โค 58
58 โค 58
The statement is true since 58 = 58.
79.
80.
3 + 5(4 โ 1)
โฅ3
2 โ
4 +1
3 + 5(3)
โฅ3
8 +1
3 + 15
โฅ3
9
18
โฅ3
9
2โฅ3
The statement is false since 2 is less than 3.
7 ( 3 + 1) โ 2
3+ 5โ
2
7 ( 4) โ 2
2(5 + 1) โ 3(1 + 1)
5(8 โ 6) โ 4 โ
2
2(6) โ 3(2)
3โฅ
5(2) โ 8
12 โ 6
3โฅ
10 โ 8
6
3โฅ
2
3โฅ3
The statement is true since 3 = 3.
81. 3 โฅ
82. 7 โค
3(8 โ 3) + 2(4 โ 1)
9(6 โ 2) โ 11( 5 โ 2 )
3(5) + 2(3)
9(4) โ 11(3)
15 + 6
7โค
36 โ 33
21
7โค
3
7โค7
The statement is true since 7 = 7.
7โค
83. โ5 < 17โ means โfive is less than seventeen.โ
The statement is true.
84. โ8 2.50โ means โtwo and five-tenths is
greater than two and fifty-hundredths.โ The
statement is false.
94. โ1.80 > 1.8โ means โone and eighty-hundredths
is greater than one and eight-tenths.โ The
statement is false.
95. โFifteen is equal to five plus tenโ is written
as 15 = 5 + 10.
(c) 85% of 9.5 is 0.85(9.5) = 8.075.
Walking at 5 mph is associated with
8.0 METs, which is the table value closest
to 8.075.
(d) Substitute โ55โ for โageโ in the expression
for men.
14.7 โ 55 โ
0.11
14.7 โ 55 โ
0.11 = 14.7 โ 6.05 Multiply.
96. โTwelve is equal to twenty minus eightโ is
written as 12 = 20 โ 8.
= 8.65
Subtract.
85% of 8.65 is 0.85(8.65) = 7.3525.
Swimming is associated with 7.0 METs,
which is the table value closest to 7.3525.
97. โNine is greater than five minus fourโ is written
as 9 > 5 โ 4.
98. โTen is greater than six plus oneโ is written
as 10 > 6 + 1.
110. Answers will vary.
99. โSixteen is not equal to nineteenโ is written
as 16 โ 19.
111. The states that had a number greater than 12.6
are Alaska (16.4), Texas (15.2), California
(22.5), and Idaho (19.7).
100. โThree is not equal to fourโ is written as 3 โ 4.
101. โOne-half is less than or equal to two-fourthsโ
1 2
is written as โค .
2 4
102. โOne-third is less than or equal to three-ninthsโ
1 3
is written as โค .
3 9
103. 5 5 when the inequality
symbol is reversed.
104. 30 > 9 becomes 9 becomes < when the inequality
5 4
4 5
symbol is reversed.
5 3
3 5
106.
when the inequality
4 2
2 4
symbol is reversed.
107. 2.5 โฅ 1.3 becomes 1.3 โค 2.5 when the
inequality symbol is reversed.
108. 4.1 โค 5.3 becomes 5.3 โฅ 4.1 when the
inequality symbol is reversed.
109. (a) Substitute โ40โ for โageโ in the expression
for women.
14.7 โ 40 โ
0.13
(b) 14.7 โ 40 โ
0.13 = 14.7 โ 5.2 Multiply.
= 9.5
Subtract.
112. The states that had a number that was at most
15.2 are Texas (15.2), Virginia (12.6), Maine
(12.4), and Missouri (12.1).
113. The states that had a number not less than 12.6,
which is the same as greater than or equal to
12.6, are Alaska (16.4), Texas (15.2),
California (22.5), Virginia (12.6), and Idaho
(19.7).
114. The states that had a number less than 13.0 are
Virginia (12.6), Maine (12.4), and Missouri
(12.1).
1.2 Variables, Expressions, and Equations
Classroom Examples, Now Try Exercises
1. (a) 16 p โ 8 = 16 โ
3 โ 8
= 48 โ 8
=40
(b)
2 p3 = 2 โ
33
Replace p with 3.
Multiply.
Subtract.
Replace p with 3.
= 2 โ
27 Cube 3.
= 54
Multiply.
N1. (a) 9 x โ 5 = 9 โ
6 โ 5
= 54 โ 5
= 49
(b)
Replace x with 6.
Multiply.
Subtract.
4 x 2 = 4 โ
62 Replace x with 6.
= 4 โ
36 Square 6.
= 144
Copyright ยฉ 2020 Pearson Education, Inc.
Multiply.
1.2 Variables, Expressions, and Equations 35
(b) โA number divided by 7โ translates as
x
x รท 7, or .
7
2. (a) 4 x + 5 y = 4 โ
6 + 5 โ
9
Multiply.
= 24 + 45
Add.
= 69
4x โ 2 y 4 โ
6 โ 2 โ
9
=
(b)
6 +1
x +1
24 โ 18
=
Multiply.
6 +1
6
=
Subtract and add .
7
2
2
2
(c) โThe difference between 9 and a numberโ
translates as 9 โ x. Thus, โthe product of 3
and the difference between 9 and a numberโ
translates as 3(9 โ x).
4. (a) 8 p โ 10 = 5
?
8 โ
2 โ 10 = 5
2
(c) 2 x + y = 2 โ
6 + 9
= 2 โ
36 + 81 Use exponents.
= 72 + 81
Multiply.
= 153
Add.
?
16 โ 10 = 5
2
2
2
2
(c) 4 x โ y = 4 โ
4 โ 7
= 4 โ
16 โ 49 Use exponents.
= 64 โ 49
Multiply.
= 15
Subtract.
3. (a) โThe difference ofโ indicates subtraction.
Using x as the variable to represent the
number, โthe difference of 48 and a
numberโ translates as 48 โ x .
(b) โDivided byโ indicates division. Using x as
the variable to represent the number, โ6
divided by a numberโ translates as 6 รท x or
6
.
x
(c) โThe sum of a number and 5โ suggests a
number plus 5. Using x as the variable to
represent the number, โ9 multiplied by the
sum of a number and 5โ translates as
9 ( x + 5) .
Multiply.
6 = 5 False
The number 2 is not a solution of the
equation.
(b) 0.1( x + 3) = 0.8
N2. (a) 3x + 4 y = 3 โ
4 + 4 โ
7
= 12 + 28
Multiply.
= 40
Add.
6x โ 2 y 6 โ
4 โ 2 โ
7
=
(b)
2y โ 9
2โ
7 โ 9
24 โ 14
=
Multiply.
14 โ 9
10
=2
Subtract; reduce.
=
5
Replace p with 2.
?
0.1(5 + 3) = 0.8
Replace x with 5.
?
0.1(8) = 0.8
Add.
0.8 = 0.8 True
The number 5 is a solution of the equation.
N4.
8k + 5 = 61
?
8 โ
7 + 5 = 61 Replace k with 7.
?
56 + 5 = 61 Multiply.
61 = 61 True
The number 7 is a solution of the equation.
5. Using x as the variable to represent the number,
โthree times a number is subtracted from 21,
giving 15โ translates as 21 โ 3x = 15. Now try
each number from the set {0, 2, 4, 6, 8, 10}.
?
x = 0 : 21 โ 3 ( 0 ) =15
21 = 15 False
?
x = 2 : 21 โ 3 ( 2 ) =15
15 = 15 True
?
x = 4 : 21 โ 3 ( 4 ) =15
9 = 15 False
Similarly, x = 6, 8, or 10 result in false
statements. Thus, 2 is the only solution.
N3. (a) Using x as the variable to represent the
number, โthe sum of a number and 10โ
translates as x + 10, or 10 + x.
Copyright ยฉ 2020 Pearson Education, Inc.
36
Chapter 1 The Real Number System
N5. Using x as the variable to represent the number,
โthe sum of a number and nine is equal to the
difference between 25 and the numberโ
translates as x + 9 = 25 โ x. Now try each
number from the set {0, 2, 4, 6, 8, 10}.
?
x = 4 : 4 + 9 = 25 โ 4
13 = 21
False
?
x = 6 : 6 + 9 = 25 โ 6
15 = 19
False
?
x = 8 : 8 + 9 = 25 โ 8
17 = 17
True
Similarly, x = 0, 2, or 10 result in false
statements. Thus, 8 is the only solution.
6. (a)
(b)
3x โ 1
has no equality symbol, so this is an
5
expression.
3x
= 1 has an equality symbol, so this is an
5
equation.
N6. (a) 2 x + 5 = 6 has an equality symbol, so this is
an equation.
(b) 2 x + 5 โ 6 has no equality symbol, so this is
an expression.
Exercises
1. The expression 8x 2 means 8 โ
x โ
x. The correct
choice is B.
6. There is no equality symbol in 6 x + 7 or
6 x โ 7, so those are expressions. 6 x = 7 and
6 x โ 7 = 0 have equality symbols, so those are
equations.
7. The exponent refers only to the 4.
5 x 2 = 5 โ
42
= 5 โ
16
= 80
The correct value is 80.
8. Addition in the numerator comes before
division.
x + 3 10 + 3
=
5
5
13
=
5
13
The correct value is
.
5
9. (a) x + 7 = 4 + 7
= 11
(b) x + 7 = 6 + 7
= 13
10. (a) x โ 3 = 4 โ 3
=1
(b) x โ 3 = 6 โ 3
=3
11. (a) 4 x = 4 โ
4 = 16
(b) 4 x = 4 โ
6 = 24
2. If x = 2 and y = 1, then the value of xy is
2 โ
1 = 2. The correct choice is C.
12. (a) 6 x = 6 โ
4 = 24
3. The sum of 15 and a number x is represented by
the expression 15 + x. The correct choice is A.
13. (a) 5 x โ 4 = 5 โ
4 โ 4
= 20 โ 4
= 16
4. 7 less than a number x is represented by the
expression x โ 7 . The correct choice is D.
5. Try each number in the equation 3x โ 1 = 5 .
?
x = 0 : 3 โ
0 โ 1= 5
?
0 โ 1= 5
โ1 = 5 False
?
x = 2 : 3 โ
2 โ 1= 5
?
6 โ 1= 5
5 = 5 False
(b) 6 x = 6 โ
6 = 36
(b) 5 x โ 4 = 5 โ
6 โ 4
= 30 โ 4
= 26
14. (a) 7 x โ 9 = 7 โ
4 โ 9
= 28 โ 9
= 19
(b) 7 x โ 9 = 7 โ
6 โ 9
= 42 โ 9
= 33
Copyright ยฉ 2020 Pearson Education, Inc.
1.2 Variables, Expressions, and Equations 37
15. (a) 4 x 2 = 4 โ
42
= 4 โ
16
= 64
(b) 4 x 2 = 4 โ
62
= 4 โ
36
= 144
2
2
16. (a) 5 x = 5 โ
4
= 5 โ
16
= 80
(b) 5 x 2 = 5 โ
62
= 5 โ
36
= 180
17. (a)
(b)
18. (a)
x +1 4 +1
=
3
3
5
=
3
x +1 6 +1
=
3
3
7
=
3
x+2 4+2
=
5
5
6
=
5
x+2 6+2
=
(b)
5
5
8
=
5
19. (a)
(b)
3x โ 5 3 โ
4 โ 5
=
2x
2โ
4
12 โ 5
=
8
7
=
8
3x โ 5 3 โ
6 โ 5
=
2x
2โ
6
18 โ 5
=
12
13
=
12
20. (a)
(b)
4x โ1 4 โ
4 โ1
=
3x
3โ
4
16 โ 1
=
12
15 5
=
=
12 4
4x โ1 4 โ
6 โ1
=
3x
3โ
6
24 โ 1
=
18
23
=
18
21. (a) 3x 2 + x = 3 โ
42 + 4
= 3 โ
16 + 4
= 48 + 4 = 52
(b) 3x 2 + x = 3 โ
62 + 6
= 3 โ
36 + 6
= 108 + 6 = 114
22. (a) 2 x + x 2 = 2 โ
4 + 42
= 8 + 16
= 24
(b) 2 x + x 2 = 2 โ
6 + 62
= 12 + 36
= 48
23. (a) 6.459 x = 6.459 โ
4
= 25.836
(b) 6.459 x = 6.459 โ
6
= 38.754
24. (a) 3.275 x = 3.275 โ
4
= 13.1
(b) 3.275 x = 3.275 โ
6
= 19.65
25. (a) 8 x + 3 y + 5 = 8 โ
2 + 3 โ
1 + 5
= 16 + 3 + 5
= 19 + 5
= 24
(b) 8 x + 3 y + 5 = 8 โ
1 + 3 โ
5 + 5
= 8 + 15 + 5
= 23 + 5
= 28
Copyright ยฉ 2020 Pearson Education, Inc.
38
Chapter 1 The Real Number System
26. (a) 4 x + 2 y + 7 = 4(2) + 2(1) + 7
= 8+ 2+7
= 17
31. (a)
(b) 4 x + 2 y + 7 = 4(1) + 2(5) + 7
= 4 + 10 + 7
= 21
27. (a) 3( x + 2 y ) = 3(2 + 2 โ
1)
= 3(2 + 2)
= 3(4)
= 12
(b) 3( x + 2 y ) = 3(1 + 2 โ
5)
= 3(1 + 10)
= 3(11)
(b)
32. (a)
= 33
28. (a) 2(2 x + y ) = 2 ๏ฉ๏ซ 2 ( 2 ) + 1๏น๏ป
= 2 ( 4 + 1)
= 2 (5)
= 10
(b)
(b) 2(2 x + y ) = 2 ๏ฉ๏ซ 2 (1) + 5๏น๏ป
= 2 ( 2 + 5)
= 2 (7)
= 14
29. (a) x +
4
4
= 2+
1
y
= 2+4
33. (a)
=6
(b) x +
30. (a) y +
4
4
= 1+
5
y
5 4
= +
5 5
9
=
5
8
8
= 1+
2
x
= 1+ 4
(b)
x y 2 1
+ = +
2 3 2 3
6 2
= +
6 6
8 4
= =
6 3
x y 1 5
+ = +
2 3 2 3
3 10
= +
6 6
13
=
6
x y 2 1
+ = +
5 4 5 4
8
5
=
+
20 20
13
=
20
x y 1 5
+ = +
5 4 5 4
4 25
=
+
20 20
29
=
20
2 x + 4 y 2 โ
2 + 4 โ
1
=
5 x + 2 y 5 โ
2 + 2 โ
1
4+4
=
10 + 2
8
=
12
2
=
3
2 x + 4 y 2 โ
1 + 4 โ
5
=
5 x + 2 y 5 โ
1 + 2 โ
5
2 + 20
=
5 + 10
22
=
15
=5
(b) y +
8
8
= 5+
x
1
= 5+8
= 13
Copyright ยฉ 2020 Pearson Education, Inc.
1.2 Variables, Expressions, and Equations 39
34. (a)
(b)
7 x + 5 y 7(2) + 5(1)
=
8x + y
8(2) + 1
14 + 5
=
16 + 1
19
=
17
7 x + 5 y 7(1) + 5(5)
=
8x + y
8(1) + 5
7 + 25
=
8+5
32
=
13
35. (a) 3x 2 + y 2 = 3 โ
22 + 12
= 3โ
4 +1
= 12 + 1
= 13
(b) 3x 2 + y 2 = 3 โ
12 + 52
= 3 โ
1 + 25
= 3 + 25
= 28
36. (a) 4 x 2 + 2 y 2 = 4 โ
22 + 2 โ
12
= 4 โ
4 + 2 โ
1
= 16 + 2
= 18
(b) 4 x 2 + 2 y 2 = 4 โ
12 + 2 โ
52
= 4 โ
1 + 2 โ
25
= 4 + 50
= 54
37. (a)
3x + y 2
3 โ
2 + 12
=
2 x + 3 y 2 โ
2 + 3 โ
1
3โ
2 +1
=
4+3
6 +1
=
7
7
=
7
=1
(b)
38. (a)
3x + y 2
3 โ
1 + 52
=
2 x + 3 y 2 โ
1 + 3 โ
5
3 โ
1 + 25
=
2 + 15
3 + 25
=
17
28
=
17
x2 + 1
22 + 1
=
4 x + 5 y 4 ( 2 ) + 5 (1)
4 +1
8+5
5
=
13
=
(b)
12 + 1
x2 + 1
=
4 x + 5 y 4 (1) + 5 ( 5 )
1+1
4 + 25
2
=
29
=
39. (a) 0.841x 2 + 0.32 y 2
= 0.841 โ
22 + 0.32 โ
12
= 0.841 โ
4 + 0.32 โ
1
= 3.364 + 0.32
= 3.684
(b) 0.841x 2 + 0.32 y 2
= 0.841 โ
12 + 0.32 โ
52
= 0.841 โ
1 + 0.32 โ
25
= 0.841 + 8
= 8.841
40. (a) 0.941x 2 + 0.25 y 2
2
= 0.941( 2 ) + 0.25 (1)
2
= 0.941( 4 ) + 0.25 (1)
= 3.764 + 0.25
= 4.014
(b) 0.941x 2 + 0.25 y 2
2
= 0.941(1) + 0.25 ( 5 )
2
= 0.941(1) + 0.25 ( 25 )
= 0.941 + 6.25
= 7.191
Copyright ยฉ 2020 Pearson Education, Inc.
40
Chapter 1 The Real Number System
41. โTwelve times a numberโ translates as 12 โ
x,
or 12x.
57.
?
2 โ
3 + 3(3 โ 2) = 14
42. โFifteen times a numberโ translates as 15 โ
x,
or 15x.
2 โ
3 + 3 โ
1 = 14
?
6 + 3 = 14
9 = 14
False
Because substituting 3 for y results in a false
statement, 3 is not a solution of the equation.
44. โSix added to a numberโ translates as x + 6.
58.
6 x + 2 ( x + 3) = 14; 2
46. โSeven subtracted from a numberโ translates
as x โ 7.
6 ( 2 ) + 2(2 + 3) = 14
47. โA number subtracted from sevenโ translates
as 7 โ x.
6 ( 2 ) + 2 ( 5 ) = 14
?
?
12 + 10 = 14
22 = 14
False
The false result shows that 2 is not a solution of
the equation.
49. โThe difference between a number and 8โ
translates as x โ 8.
59.
6 p + 4 p + 9 = 11;
1
5
?
1
1
1
6 โ
+ 4 โ
+ 9 = 11 Let p = .
5
5
5
?
6 4
+ + 9 = 11
5 5
?
10
+ 9 = 11
5
18
.
51. โ18 divided by a numberโ translates as
x
52. โA number divided by 18โ translates as
Let x = 2.
?
48. โA number subtracted from fourโ translates
as 4 โ x.
50. โThe difference between 8 and a numberโ
translates as 8 โ x.
Let y = 3.
?
43. โAdded toโ indicates addition. โNine added to a
numberโ translates as x + 9.
45. โTwo subtracted from a numberโ translates
as x โ 2.
2 y + 3( y โ 2) = 14; 3
x
.
18
53. โThe product of 6 and four less than a numberโ
translates as 6 ( x โ 4 ) .
?
2 + 9 = 11
11 = 11 True
54. โThe product of 9 and five more than a
numberโ translates as 9 ( x + 5 ) .
The true result shows that
55. 4m + 2 = 6;1
equation.
?
4 โ
1 + 2 = 6
Let m = 1.
60.
?
4+2 = 6
6=6
True
Because substituting 1 for m results in a true
statement, 1 is a solution of the equation.
56.
2r + 6 = 8; 1
?
2(1) + 6 = 8
Let r = 1.
?
2+6 =8
8=8
True
The true result shows that 1 is a solution of the
equation.
2 x + 3x + 8 = 20;
12
5
?
๏ฆ 12 ๏ถ ๏ฆ 12 ๏ถ
2 ๏ง ๏ท + 3 ๏ง ๏ท + 8 = 20
๏จ 5๏ธ ๏จ 5๏ธ
24 36 40 ?
+ +
= 20
5
5
5
100 ?
= 20
5
20 = 20
The true result shows that
the equation.
Copyright ยฉ 2020 Pearson Education, Inc.
1
is a solution of the
5
Let x =
12
.
5
True
12
is a solution of
5

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