Chapter 4€¦ · · 2008-10-24Chapter 4 Section 4.1 Practice Problems 1. In the fraction 11, 2...

$Page 1: Chapter 4€¦ · · 2008-10-24Chapter 4 Section 4.1 Practice Problems 1. In the fraction 11, 2 the numerator is 11 and the denominator is 2. 2. In the fraction 10, 17 y the numerator$
135

Chapter 4

Section 4.1

Practice Problems

1. In the fraction 11,2

the numerator is 11 and the

denominator is 2.

2. In the fraction 10 ,17y

the numerator is 10y and

the denominator is 17.

3. 3 out of 8 equal parts are shaded: 38



10


16

7. answers may vary; for example,


9. number of planets farther 5number of planets in our solar system 8

→→

58

of the planets in our solar system are farther

from the Sun than Earth.

10. Each part is 13

of a whole and there are 8 parts

shaded, or 2 wholes and 2 more parts. 8 2; 23 3

11. Each part is 14


shaded, or 1 whole and 1 more part. 5 1; 14 4

12. a.

1∞0

b. 1s0

c. 10 46

13. a. 1 2 3h0

b. 1 2@0

c. 0 1 or 77

14. 9 19=

15. 6 16−

=−

16. 0 01=

−

17. 4 41=

18. 130−

is undefined.

19. 13 131−

= −

20. a. 2 7 5 2 35 2 3757 7 7 7

⋅ + += = =

b. 2 3 6 2 18 2 2063 3 3 3

⋅ + += = =

c. 9 10 10 9 100 9 10910

10 10 10 10⋅ + +

= = =

d. 1 5 4 1 20 1 2145 5 5 5

⋅ + += = =

Copyright © 2008 Pearson Education, Inc., publishing as Pearson Prentice Hall

Chapter 4: Fractions and Mixed Numbers ISM: Prealgebra

136

21. a. 1

5 954

9 415 5=

b. 2

9 2318

5

23 529 9=

c. 12

4 484880

48 124=

d. 4

13 625210

62 10413 13

=

e. 7

7 5149

2

51 277 7=

f. 1

20 2120

1

21 1120 20

=

Vocabulary and Readiness Check

1. The number 1731

is called a fraction. The number

31 is called its denominator and 17 is called its numerator.

2. If we simplify each fraction, 9 1,9−

=−

0 0,4=

−

and we say 4

0−

is undefined.

3. The fraction 83

is called an improper fraction,

the fraction 38

is called a proper fraction, and

3108

is called a mixed number.

4. The value of an improper fraction is always ≥1 and the value of a proper fraction is always <1.

Exercise Set 4.1

1. In the fraction 1 ,2


denominator is 2. Since 1 < 2, the fraction is proper.



denominator is 4. Since 1 < 4, the fraction is proper.



denominator is 3. Since 10 > 3, the fraction is improper.



denominator is 21. Since 53 > 21, the fraction is improper.



denominator is 15. Since 15 ≥ 15, the fraction is improper.


ISM: Prealgebra Chapter 4: Fractions and Mixed Numbers

137



denominator is 26. Since 26 ≥ 26, the fraction is improper.

7. 1 out of 3 equal parts is shaded: 13


9. Each part is 14


shaded, or 2 wholes and 3 more parts.

a. 114

b. 324

10. Each part is 14



a. 104

b. 224

11. Each part is 16



a. 236

b. 536

12. Each part is 13



a. 113

b. 233


12




12



19. Each part is 13


shaded, or 1 whole and 1 more part.

a. 43

b. 113

20. Each part is 15


shaded, or 1 whole and 1 more part.

a. 65

b. 115

21. Each part is 12


shaded, or 5 wholes and 1 more part.

a. 112

b. 152



138

22. Each part is 15



a. 235

b. 345



25. 5 of 8 equal parts are shaded: 58








33. freshmen 42students 131

→→

42131

of the students are freshmen.

34. men 22employees 63

→→

2263

of the employees are men.

35. a. number of students not freshmen 131 4289

= −=

b. not freshmen 89students 131

→→

89131

of the students are not freshmen.

36. a. number of women employees 63 2241

= −=

b. women 41employees 63

→→

4163

of the employees are women.

37. born in Virginia 8U.S. presidents 43

→→

843

of U.S. presidents were born in Virginia.

38. planets with longer days 4number of planets in solar system 8

→→

48

of the planets in our solar system have longer

days than Earth.

39. number turned into hurricanes 15number of tropical storms 28

→→

1528

of the tropical storms turned into hurricanes.

40. 5 of 12 inches is 5

12 of a foot.

41. 11 of 31 days of March is 1131

of the month.



139

42. 37 of 60 minutes is 3760

of an hour.

43. number of sophomores 10number of students in class 31

→→

1031

of the class is sophomores.

44. number of boys 9number on team 20

→→

920

of the team is boys.

45. There are 50 states total. 33 states contain federal Indian reservations.

a. 3350

of the states contain federal Indian reservations.

b. 50 − 33 = 17 17 states do not contain federal Indian reservations.

c. 1750

of the states do not contain federal Indian reservations.

46. There are 50 states total. Consumer fireworks are legal in 40 states.

a. Consumer fireworks are legal in 4050

of the states.

b. 50 − 40 = 10 Consumer fireworks are illegal in 10 states.

c. Consumer fireworks are illegal in 1050

of the states.

47. a. blue 21total 50

→→

2150

of the marbles are blue.

b. 50 − 21 = 29 29 of the marbles are red.

c. red 29total 50

→→

2950

of the marbles are red.



140

48. a. number of watercolors 15total number of pieces 37

→→

1537

of the pieces are watercolor paintings.

b. number of oil paintings 17total number of pieces 37

→→

1737

of the pieces are oil paintings.

c. 37 − 15 − 17 = 5 There are 5 sculptures.

d. numbers of sculptures 5total number of pieces 37

→→

537

of the pieces are sculptures.

49.

~

1 2 30

50. a

1 2 3 40

51. ¢

1 20

52. X

1 20

53.

U

1 20

54. ∫

10

55. g

1 2 3 40

56.

œ

1 20

57. 12 112

=

58. 3 13−

=−



141

59. 5 5

1−

= −

60. 20 201−

= −

61. 0 02=

−

62. 0 08=

−

63. 8 18−

=−

64. 14 114−

=−

65. 9 is undefined

0−

66. 7 is undefined

0−

67. 3 31=

68. 5 15=

69. 1 3 2 1 6 1 723 3 3 3

⋅ + += = =

70. 13 17 1 13 17 13 30117 17 17 17

⋅ + += = =

71. 3 5 3 3 15 3 1835 5 5 5

⋅ + += = =

72. 5 9 2 5 18 5 2329 9 9 9

⋅ + += = =

73. 5 8 6 5 48 5 5368 8 8 8

⋅ + += = =

74. 3 8 7 3 56 3 5978 8 8 8

⋅ + += = =

75. 6 7 11 6 77 6 83117 7 7 7

⋅ + += = =

76. 2 5 12 2 60 2 62125 5 5 5

⋅ + += = =

77. 7 20 9 7 180 7 187920 20 20 20

⋅ + += = =

78. 14 27 10 14 270 14 2841027 27 27 27

⋅ + += = =

79. 2 3 166 2 498 2 5001663 3 3 3

⋅ + += = =

80. 2 7 114 2 798 2 8001147 7 7 7

⋅ + += = =

81. 3

5 17152

17 235 5=

82. 1

7 1376

13 617 7=

83. 4

8 3732

5

37 548 8=

84. 7

9 6463

1

64 179 9=



142

85. 3

15 47452

47 2315 15

=

86. 5

12 6560

5

65 5512 12

=

87. 15

15 22515

75750

225 1515

=

88. 14

14 196145656

0

196 1414

=

89. 1

175 182175

7

182 71175 175

=

90. 1

143 149143

6

149 61143 143

=

91. 6

112 737672

65

737 656112 112

=

92. 7

123 90186140

901 407123 123

=

93. 23 3 3 9= ⋅ =

94. 34 4 4 4 64= ⋅ ⋅ =

95. 35 5 5 5 125= ⋅ ⋅ =

96. 43 3 3 3 3 81= ⋅ ⋅ ⋅ =

97. 11 11 112 2 2

−− = =

−

98. 21 21 214 4 4

−− = =

−

99. 13 13 13

15 15 15−

= = −−

100. 45 45 4557 57 57

−= = −

−

101. answers may vary

102.

103.

104. 1 is not close to 8, so 18

is closer to 0 than to 1.

178

rounded to the nearest whole number is 7.

105. 11 is close to 12, so 1112

is close to 1. 11512

rounded to the nearest whole number is 6.



143

106. 1591 + 498 + 959 + 5 = 3053 959 of the 3053 stores are named Old Navy: 9593053

107. 87 + 57 + 20 + 7 = 171 57 of the 171 licensees are universities or

colleges: 57171

108. 1651 + 634 = 2285 1651 of the 2285 affiliates are located in the

United States: 16512285

109. 5 + 3 + 2 + 1 + 1 + 1 = 13 Five of the 13 USMC training centers are in

California: 5 .

13

Section 4.2

Practice Problems

1. a. 30 2 15

2 3 5

= ⋅↓ ↓⋅ ⋅

b.

3

56 2 28

2 2 14

2 2 2 7 2 7

= ⋅↓ ↓⋅ ⋅

↓ ↓ ↓

⋅ ⋅ ⋅ = ⋅

c.

3 2

72 2 36

2 2 18

2 2 2 9

2 2 2 3 3 2 3

= ⋅↓ ↓⋅ ⋅

↓ ↓ ↓⋅ ⋅ ⋅

↓ ↓ ↓ ↓

⋅ ⋅ ⋅ ⋅ = ⋅

2.

2

60 2 30

2 2 15

2 2 3 5 2 3 5

= ⋅↓ ↓⋅ ⋅

↓ ↓ ↓

⋅ ⋅ ⋅ = ⋅ ⋅

3. 11

3 333 993 297

The prime factorization of 297 is 33 11.⋅

4. 30 15 2 245 15 3 3

⋅= =

⋅

5. 39 3 13 13 1351 3 17 17 17x x x x⋅ ⋅ ⋅= = =

⋅

6. 9 3 3

50 2 5 5⋅

− = −⋅ ⋅

Since 9 and 50 have no common factors, 9

50− is

already in simplest form.

7. 49 7 7 7

112 7 16 16⋅

= =⋅

8. 64 4 16 1620 4 5 5

⋅− = − = −

⋅

9. 3

27 7

7 8 856a a a a a

a aa⋅ ⋅ ⋅

= =⋅ ⋅ ⋅

10. 79

is in simplest form.

21 3 7 727 3 9 9

⋅= =

⋅

Since 79

and 2127

both simplify to 7 ,9

they are

equivalent.

11. Not equivalent, since the cross products are not equal: 13 ⋅ 5 = 65 and 4 ⋅ 18 = 72

12. 6 parks in Washington 2 3 3

58 national parks 2 29 29⋅

= =⋅

329

of the national parks are in Washington

state.

Calculator Explorations

1. 128 4224 7

=



144

2. 231 7396 12

=

3. 340 20459 27

=

4. 999 37

1350 50=

5. 432 8810 15

=

6. 225 5315 7

=

7. 54 2243 9

=

8. 455 35689 53

=


1. The number 40 equals 2 ⋅ 2 ⋅ 2 ⋅ 5. Since each factor is prime, we call 2 ⋅ 2 ⋅ 2 ⋅ 5 the prime factorization of 40.

2. A natural number, other than 1, that is not prime is called a composite number.

3. A natural number that has exactly two different factors, 1 and itself, is called a prime number.

4. In 11 ,48

since 11 and 48 have no common factors

other than 1, 1148

is in simplest form.

5. Fractions that represent the same portion of a whole are called equivalent fractions.

6. In the statement 5 15 ,

12 36= 5 ⋅ 36 and 12 ⋅ 15 are

called cross products.

Exercise Set 4.2

1.

2

20 2 10

2 2 5 2 5

= ⋅↓

⋅ ⋅ = ⋅

2.

2

12 2 6

2 2 3 2 3

= ⋅↓ ↓⋅ ⋅ = ⋅

3.

4

48 2 24

2 2 12

2 2 2 6

2 2 2 2 3 2 3

= ⋅↓ ↓⋅ ⋅

↓ ↓ ↓⋅ ⋅ ⋅

↓ ↓ ↓ ↓⋅ ⋅ ⋅ ⋅ = ⋅

4.

2

75 3 25

3 5 5 3 5

= ⋅↓ ↓⋅ ⋅ = ⋅

5.

4

81 9 9

3 3 3 3 3

= ⋅

⋅ ⋅ ⋅ =

6.

6

64 8 8

2 4 2 4

2 2 2 2 2 2 2

= ⋅

⋅ ⋅ ⋅↓

⋅ ⋅ ⋅ ⋅ ⋅ =

7.

4

162 2 81

2 9 9

2 3 3 3 3 2 3

= ⋅↓

⋅ ⋅↓⋅ ⋅ ⋅ ⋅ = ⋅

8.

7

128 2 64

2 2 32

2 2 2 16

2 2 2 2 8

2 2 2 2 2 4

2 2 2 2 2 2 2 2

= ⋅↓ ↓⋅ ⋅

↓ ↓ ↓⋅ ⋅ ⋅

↓ ↓ ↓ ↓⋅ ⋅ ⋅ ⋅

↓ ↓ ↓ ↓ ↓⋅ ⋅ ⋅ ⋅ ⋅

↓ ↓ ↓ ↓ ↓ ↓

⋅ ⋅ ⋅ ⋅ ⋅ ⋅ =

9. 110 2 55

2 5 11 2 5 11

= ⋅↓ ↓⋅ ⋅ = ⋅ ⋅



145

10. 130 2 65

2 5 13 2 5 13

= ⋅↓ ↓⋅ ⋅ = ⋅ ⋅

11. 85 = 5 ⋅ 17

12. 93 = 3 ⋅ 31

13.

4

240 2 120

2 2 60

2 2 2 30

2 2 2 2 15

2 2 2 2 3 5 2 3 5

= ⋅↓ ↓⋅ ⋅

↓ ↓ ↓⋅ ⋅ ⋅

↓ ↓ ↓ ↓⋅ ⋅ ⋅ ⋅

↓ ↓ ↓ ↓ ↓

⋅ ⋅ ⋅ ⋅ ⋅ = ⋅ ⋅

14.

2

836 2 418

2 2 209

2 2 11 19 2 11 19

= ⋅↓ ↓⋅ ⋅

↓ ↓ ↓⋅ ⋅ ⋅ = ⋅ ⋅

15.

2 2

828 2 414

2 2 207

2 2 3 69

2 2 3 3 23 2 3 23

= ⋅↓ ↓⋅ ⋅

↓ ↓ ↓⋅ ⋅ ⋅

↓ ↓ ↓ ↓⋅ ⋅ ⋅ ⋅ = ⋅ ⋅

16.

3 2

504 2 252

2 2 126

2 2 2 63

2 2 2 3 21

2 2 2 3 3 7 2 3 7

= ⋅↓ ↓⋅ ⋅

↓ ↓ ↓⋅ ⋅ ⋅

↓ ↓ ↓ ↓⋅ ⋅ ⋅ ⋅

↓ ↓ ↓ ↓ ↓

⋅ ⋅ ⋅ ⋅ ⋅ = ⋅ ⋅

17. 3 3 1 1

12 3 4 4⋅

= =⋅

18. 5 5 1 1

30 5 6 6⋅

= =⋅

19. 4 2 2 2 242 2 3 7 3 7 21x x x x⋅ ⋅ ⋅= = =

⋅ ⋅ ⋅

20. 9 3 3 3 348 3 16 16 16y y y y⋅ ⋅ ⋅= = =

⋅

21. 14 2 7 716 2 8 8

⋅= =

⋅

22. 22 2 11 1134 2 17 17

⋅= =

⋅

23. 20 2 10 230 3 10 3

⋅= =

⋅

24. 70 7 10 780 8 10 8

⋅= =

⋅

25. 35 5 7 750 5 10 10a aa a

⋅ ⋅= =

⋅ ⋅

26. 25 5 5 555 5 11 11z zz z

⋅ ⋅= =

⋅ ⋅

27. 63 9 7 781 9 9 9

⋅− = − = −

⋅

28. 21 7 3 349 7 7 7

⋅− = − = −

⋅

29. 230 5 6 5 5

36 6 6 6 6x x x x xx x

⋅ ⋅ ⋅ ⋅= = =

⋅ ⋅

30. 245 5 9 9 9

5 16 16 1680b b

b b b bb⋅ ⋅

= = =⋅ ⋅ ⋅ ⋅

31. 27 3 3 364 4 4 4

⋅ ⋅=

⋅ ⋅


is


32. 32 2 2 2 2 263 3 3 7

⋅ ⋅ ⋅ ⋅=

⋅ ⋅


is




146

33. 25 5 5 5 540 5 8 8 8xy x y x xy y

⋅ ⋅ ⋅ ⋅= = =

⋅ ⋅

34. 36 6 6 6 642 6 7 7 7y yyz y z z z

⋅ ⋅= = =

⋅ ⋅ ⋅ ⋅

35. 40 8 5 564 8 8 8

⋅− = − = −

⋅

36. 28 4 7 760 4 15 15

⋅− = − = −

⋅

37. 3 2 236 3 12 3 3

24 2 12 2 2x y x x x y y x x y x yxy x y

⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅= = =

⋅ ⋅ ⋅

38. 2

3 260 5 12 5 5

3 12 336 3a b a a b a a

a b b b b bab b⋅ ⋅ ⋅ ⋅ ⋅

= = =⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅

39. 90 30 3 3

120 30 4 4⋅

= =⋅

40. 60 30 2 2

150 30 5 5⋅

= =⋅

41. 40 5 8 5 564 8 8 8 8xy x yxyz x y z z z

⋅ ⋅ ⋅= = =

⋅ ⋅ ⋅ ⋅ ⋅

42. 28 7 4 7 760 15 4 15 15abc a b c b bac a c

⋅ ⋅ ⋅ ⋅ ⋅= = =

⋅ ⋅ ⋅

43. 66 22 3 3308 22 14 14

⋅= =

⋅

44. 65 5 13 5234 18 13 18

⋅= =

⋅

45. 55 5 11 11 11

85 5 17 17 17y y y y⋅

− = − = − = −⋅ ⋅ ⋅

46. 78 6 13 13 13

90 6 15 15 15x x x x⋅

− = − = − = −⋅ ⋅ ⋅

47. 189 7 27 7216 8 27 8z zz z

⋅ ⋅= =

⋅ ⋅

48. 144 18 8 8162 18 9 9

y yy y

⋅ ⋅= =

⋅ ⋅

49. 3 4 2

4 2

2

224 14 161 1616

141

14

a b c a a a b b b b c ca b b b b c cab c

a a

a

⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅=

⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅⋅ ⋅

=

=

50. 4 3 3

3 3 3270 18 15

1 151518

118

x y z x x x x y y y z z zx x x y y y z z zx y z

x

x

⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅=

⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅⋅

=

=

51. Equivalent, since the cross products are equal: 2 ⋅ 12 = 24 and 6 ⋅ 4 = 24












63. 2 hours 1 2 18 hours 4 2 4

⋅= =

⋅

2 hours represents 14

of a work shift.



147

64. 200 caps 200 1 12000 caps 200 10 10

⋅= =

⋅

200 caps represents 1

10 of the total caps.

65. 2640 feet 2640 1 15280 feet 2640 2 2

⋅= =

⋅

2640 feet represents 12

of a mile.

66. 20 centimeters 1 20 1

100 centimeters 5 20 5⋅

= =⋅

20 centimeters is 15

of a meter.

67. a. 15 3 5 350 10 5 10

⋅= =

⋅

310

of the states can claim at least one Ritz-

Carlton hotel.

b. 50 − 15 = 35 35 states do not have a Ritz-Carlton hotel.

c. 35 7 5 750 10 5 10

⋅= =

⋅

710

of the states do not have a Ritz-Carlton

hotel.

68. a. 10 2 5 578 2 39 39

⋅= =

⋅

539

of the national monuments can be found

in New Mexico.

b. 78 − 10 = 68 68 monuments are found outside New Mexico.

c. 68 2 34 3478 2 39 39

⋅= =

⋅

3439

of the national monuments are found

outside New Mexico.

69. 10 inches 2 5 524 inches 2 12 12

⋅= =

⋅

512

of the wall is concrete.

70. 10 5 2 235 5 7 7

⋅= =

⋅

27

of the students made an A on the first test.

71. a. 50 − 22 = 28 28 states do not have this type of Web site.

b. 28 2 14 1450 2 25 25

⋅= =

⋅

1425

of the states do not have this type of

Web site.

72. a. 22,000 − 10,000 = 12,000 $12,000 was not covered by his trade-in.

b. $12,000 2000 6 6$22,000 2000 11 11

⋅= =

⋅

611

of the purchase price was not covered

by the trade-in.

73. 291 individuals 291449 individuals 449

=

291449

of the individuals who had flown in space

were American.

74. 4500 employees 4500 1 1

18,000 employees 4500 4 4⋅

= =⋅

14

of the employees work at the Hallmark

headquarters in Kansas City, Missouri.

75. 3 3( 3) 27 3

9 9 9x − −

= = = −

76. 3 3( 5) 125 25

5 5 5y − −

= = = −

77. 2y = 2(−7) = −14

78. −5a = −5(−4) = 20



148



81. 3975 3 1325 36625 5 1325 5

⋅= =

⋅

82. 9506 7 1358 7

12,222 9 1358 9⋅

= =⋅

83. 36 blood donors have blood type A Rh-positive. 36 donors 4 9 9

100 donors 4 25 25⋅

= =⋅

925

of blood donors have type A Rh-positive

blood type.

84. 37 + 7 = 44 blood donors have an O blood type. 44 donors 4 11 11

100 donors 4 25 25⋅

= =⋅

1125

of blood donors have an O blood type.

85. 3 + 1 = 4 blood donors have an AB blood type. 4 donors 4 1 1

100 donors 4 25 25⋅

= =⋅

125

of blood donors have an AB blood type.

86. 9 + 1 = 10 blood donors have a B blood type. 10 donors 1 10 1100 donors 10 10 10

⋅= =

⋅

110

of blood donors have a B blood type.

87.

34,020 2 17,010

2 2 8505

2 2 3 2835

2 2 3 3 945

2 2 3 3 3 315

2 2 3 3 3 3 105

2 2 3 3 3 3 3 35

2 2 3 3 3 3 3 5 7

= ⋅↓ ↓⋅ ⋅

↓ ↓ ↓⋅ ⋅ ⋅

↓ ↓ ↓ ↓⋅ ⋅ ⋅ ⋅

↓ ↓ ↓ ↓ ↓⋅ ⋅ ⋅ ⋅ ⋅

↓ ↓ ↓ ↓ ↓ ↓⋅ ⋅ ⋅ ⋅ ⋅ ⋅

↓ ↓ ↓ ↓ ↓ ↓ ↓⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅

↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅

2 534,020 2 3 5 7= ⋅ ⋅ ⋅

88.

131,625 3 43,875

3 3 14,625

3 3 3 4875

3 3 3 3 1625

3 3 3 3 5 325

3 3 3 3 5 5 65

3 3 3 3 5 5 5 3

= ⋅↓ ↓⋅ ⋅

↓ ↓ ↓⋅ ⋅ ⋅

↓ ↓ ↓ ↓⋅ ⋅ ⋅ ⋅

↓ ↓ ↓ ↓ ↓⋅ ⋅ ⋅ ⋅ ⋅

↓ ↓ ↓ ↓ ↓ ↓⋅ ⋅ ⋅ ⋅ ⋅ ⋅

↓ ↓ ↓ ↓ ↓ ↓ ↓⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅

4 3131,625 3 5 13= ⋅ ⋅



91. no; answers may vary

92. The piece representing education is labeled 1 ,

10

so 1

10 of entering college freshmen plan to

major in education.

93. The piece representing biological sciences is

labeled 2 ,25

so 225

of entering college freshmen

plan to major in the biological sciences.





149

96. 8691, 786, 2235, 105, 222, 900, and 1470 are divisible by 3 because the sum of each number’s digits is divisible by 3. 786, 22, 222, 900, and 1470 are divisible by 2 because they are even numbers. 786, 222, 900, and 1470 are divisible by both 2 and 3.

97. 1235, 2235, 85, 105, 900, and 1470 are divisible by 5 because each number ends with a 0 or 5. 8691, 786, 2235, 105, 222, 900, and 1470 are divisible by 3 because the sum of each number’s digits is divisible by 3. 2235, 105, 900, and 1470 are divisible by both 3 and 5.

98. 6; answers may vary


Section 4.3

Practice Problems

1. 3 5 3 5 157 11 7 11 77

⋅⋅ = =

⋅

2. 1 1 1 1 13 9 3 9 27

⋅⋅ = =

⋅

3. 6 7 6 7 2 3 7 3 3

77 8 77 8 7 11 2 4 11 4 44⋅ ⋅ ⋅

⋅ = = = =⋅ ⋅ ⋅ ⋅ ⋅

4. 4 3 4 3 1 4 3 1 1

27 8 27 8 3 9 4 2 9 2 18⋅ ⋅ ⋅

⋅ = = = =⋅ ⋅ ⋅ ⋅ ⋅

5. 1 11 1 11 112 28 2 28 56

⋅⎛ ⎞⋅ − = − = −⎜ ⎟ ⋅⎝ ⎠

6. 4 33 4 33 4 3 11 3

11 16 11 16 11 4 4 4⋅ ⋅ ⋅⎛ ⎞⎛ ⎞− − = = =⎜ ⎟⎜ ⎟ ⋅ ⋅ ⋅⎝ ⎠⎝ ⎠

7. 1 3 25 1 3 256 10 16 6 10 16

1 3 5 52 3 2 5 16

1 52 2 165

64

⋅ ⋅⋅ ⋅ =

⋅ ⋅⋅ ⋅ ⋅

=⋅ ⋅ ⋅ ⋅⋅

=⋅ ⋅

=

8. 2 3 2 33 2 3 2 1 1

y y y y⋅ ⋅⋅ = = =

⋅ ⋅

9. 3 3

2 2 2 2a b a b a a a b a

b b a a bb a b a⋅ ⋅ ⋅ ⋅

⋅ = = =⋅ ⋅ ⋅⋅

10. a. 33 3 3 3 3 3 3 27

4 4 4 4 4 4 4 64⋅ ⋅⎛ ⎞ = ⋅ ⋅ = =⎜ ⎟ ⋅ ⋅⎝ ⎠

b. 24 4 4 4 4 16

5 5 5 5 5 25⋅⎛ ⎞ ⎛ ⎞ ⎛ ⎞− = − ⋅ − = =⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⋅⎝ ⎠ ⎝ ⎠ ⎝ ⎠

11. 8 2 8 9 8 9 2 4 9 4 9 367 9 7 2 7 2 7 2 7 7

⋅ ⋅ ⋅ ⋅÷ = ⋅ = = = =

⋅ ⋅

12. 4 1 4 2 4 2 89 2 9 1 9 1 9

⋅÷ = ⋅ = =

⋅

13. 10 2 10 94 9 4 2

10 94 22 5 94 2

5 94

454

− ÷ = − ⋅

⋅= −

⋅⋅ ⋅

= −⋅⋅

= −

= −

14.

33

3

3

2

3 3 554 4 1

3 14 53 1

4 53 1

4 53 1

4 53

20

y y yy

yy

yyyy y y

y y

y

÷ = ÷

= ⋅

⋅=

⋅⋅ ⋅

=⋅ ⋅ ⋅ ⋅

⋅=

⋅ ⋅ ⋅

=



150

15.

2 9 7 2 9 73 14 15 3 14 15

2 3 3 73 7 2 153 77 153 157 7

3 157 74549

⋅⎛ ⎞ ⎛ ⎞− ⋅ ÷ = − ÷⎜ ⎟ ⎜ ⎟⋅⎝ ⎠ ⎝ ⎠⋅ ⋅⎛ ⎞= − ÷⎜ ⎟⋅ ⋅⎝ ⎠

⎛ ⎞= − ÷⎜ ⎟⎝ ⎠⎛ ⎞= − ⋅⎜ ⎟⎝ ⎠

⋅= −

⋅

= −

16. a. 3 9 3 9 274 2 4 2 8

xy ⋅= − ⋅ = − = −

⋅

b.

3 94 23 24 93 24 91 3 2

2 2 3 31

2 316

x y÷ = − ÷

= − ⋅

⋅= −

⋅⋅ ⋅

= −⋅ ⋅ ⋅

= −⋅

= −

17.

924

9 928 4

2 9 91 8 4

2 9 91 8 4

2 9 91 2 4 4

9 9 True4 4

x = −

⎛ ⎞− −⎜ ⎟⎝ ⎠⎛ ⎞⋅ − −⎜ ⎟⎝ ⎠

⋅− −

⋅⋅

− −⋅ ⋅

− = −

Yes, 98

− is a solution of the equation.

18. 1 1 60 1 60 1 6 1060 106 6 1 6 1 6 1

⋅ ⋅ ⋅⋅ = ⋅ = = =

⋅ ⋅

Thus, there are 10 roller coasters in Hershey Park.


1. To multiply two fractions, we write

.a c a cb d b d

⋅⋅ =

⋅

2. Two numbers are reciprocals of each other if their product is 1.

3. The expression 32 2 2 2

7 7⋅ ⋅

= while

32 2 2 2.7 7 7 7

⎛ ⎞ = ⋅ ⋅⎜ ⎟⎝ ⎠

4. Every number has a reciprocal expect 0.

5. To divide two fractions, we write .a c a db d b c

⋅÷ =

⋅

6. The word “of” indicates multiplication.

Exercise Set 4.3

1. 6 3 6 3 18

11 7 11 7 77⋅

⋅ = =⋅

2. 5 7 5 7 359 4 9 4 36

⋅⋅ = =

⋅

3. 2 5 2 5 2 5 5 57 8 7 8 7 2 4 7 4 28

⋅ ⋅− ⋅ = − = − = − = −

⋅ ⋅ ⋅ ⋅

4. 4 1 4 1

15 20 15 204 1

15 4 51

15 5175

⋅⋅ − = −

⋅⋅

= −⋅ ⋅

= −⋅

= −

5. 1 2 1 2 12 15 2 15 15

⋅− ⋅ − = =

⋅

6. 3 11 3 11 3 11 1 1

11 12 11 12 11 3 4 4⋅ ⋅ ⋅

− ⋅ − = = =⋅ ⋅ ⋅



151

7. 18 36 18 3620 99 20 99

2 9 2 182 2 5 9 11185 111855

x x

x

x

x

⋅⋅ =

⋅⋅ ⋅ ⋅ ⋅

=⋅ ⋅ ⋅ ⋅⋅

=⋅

=

8. 5 64 5 64 5 32 2

32 100 32 100 32 2 5 10 10y y y y⋅ ⋅ ⋅ ⋅

⋅ = = =⋅ ⋅ ⋅ ⋅

9. 2 2 2

2 1 3 1 3 1 334 1 4 1 4 4

a a aa ⋅⋅ = ⋅ = =

⋅

10. 3

3

3

3

3

3

2 2 663 3 1

2 63 1

2 3 23 1

2 21

4

yy

y

y

y

y

− ⋅ = − ⋅

⋅= −

⋅⋅ ⋅ ⋅

= −⋅

⋅ ⋅= −

= −

11. 3 2 3 2 2

3 3x y x y x x x y y x x x

x y y y x y yy y x⋅ ⋅ ⋅ ⋅ ⋅ ⋅

⋅ = = = =⋅ ⋅ ⋅⋅

12. 3 3 3 3

2 2

1

1

1

a b a bb a b a

a bb b b a a a

a a b b

a b

⋅⋅ =

⋅⋅ ⋅

=⋅ ⋅ ⋅ ⋅ ⋅

=⋅ ⋅ ⋅

=

13. 80 09⋅ =

14. 11 0 012

⋅ =

15. 17 4 17 420 5 20 5

17 45 4 5175 51725

y yy y

yy

⋅− ⋅ = −

⋅⋅ ⋅

= −⋅ ⋅ ⋅

= −⋅

= −

16. 13 5 13 520 6 20 6

13 54 5 6134 61324

x xx x

xx

⋅− ⋅ = −

⋅⋅ ⋅

= −⋅ ⋅ ⋅

= −⋅

= −

17. 11 1 5 11 1 520 7 22 20 7 22

11 1 55 2 2 7 11 2

12 2 7 2156

⋅ ⋅⋅ ⋅ =

⋅ ⋅⋅ ⋅

=⋅ ⋅ ⋅ ⋅ ⋅

=⋅ ⋅ ⋅

=

18. 27 10 16 27 10 1632 13 30 32 13 30

3 3 3 10 1616 2 13 10 33 32 13926

⋅ ⋅⋅ ⋅ =

⋅ ⋅⋅ ⋅ ⋅ ⋅

=⋅ ⋅ ⋅ ⋅⋅

=⋅

=

19. 31 1 1 1 1 1 1 1

5 5 5 5 5 5 5 125⋅ ⋅⎛ ⎞ ⎛ ⎞⎛ ⎞⎛ ⎞= = =⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⋅ ⋅⎝ ⎠ ⎝ ⎠⎝ ⎠⎝ ⎠

20. 28 8 8 8 8 64

9 9 9 9 9 81⋅⎛ ⎞ ⎛ ⎞⎛ ⎞= = =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⋅⎝ ⎠ ⎝ ⎠⎝ ⎠

21. 22 2 2 2 2 4

3 3 3 3 3 9⋅⎛ ⎞− = − ⋅− = =⎜ ⎟ ⋅⎝ ⎠



152

22. 41 1 1 1 1

2 2 2 2 21 1 1 1

2 2 2 21

16

⎛ ⎞ ⎛ ⎞⎛ ⎞⎛ ⎞⎛ ⎞− = − − − −⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠⎝ ⎠⎝ ⎠

⋅ ⋅ ⋅=

⋅ ⋅ ⋅

=

23. 32 1 2 2 2 1

3 2 3 3 3 22 2 2 13 3 3 22 2 13 3 34

27

⎛ ⎞ ⎛ ⎞⎛ ⎞⎛ ⎞− ⋅ = − − − ⋅⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠⎝ ⎠

⋅ ⋅ ⋅= −

⋅ ⋅ ⋅⋅ ⋅

= −⋅ ⋅

= −

24. 33 1 3 3 3 1

4 3 4 4 4 33 3 3 14 4 4 33 3 14 4 4964

⎛ ⎞ ⎛ ⎞⎛ ⎞⎛ ⎞− ⋅ = − − − ⋅⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠⎝ ⎠

⋅ ⋅ ⋅= −

⋅ ⋅ ⋅⋅ ⋅

= −⋅ ⋅

= −

25. 2 5 2 6 2 6 2 2 3 2 2 43 6 3 5 3 5 3 5 5 5

⋅ ⋅ ⋅ ⋅÷ = ⋅ = = = =

⋅ ⋅

26. 5 3 5 4 5 4 5 4 5 58 4 8 3 8 3 4 2 3 2 3 6

⋅ ⋅÷ = ⋅ = = = =

⋅ ⋅ ⋅ ⋅

27. 6 12 6 5

15 5 15 126 5

15 126 5 1

5 3 6 21

3 216

− ÷ = − ⋅

⋅= −

⋅⋅ ⋅

= −⋅ ⋅ ⋅

= −⋅

= −

28. 4 8 4 3

15 3 15 84 3

15 84 3 1

5 3 4 21

5 21

10

− ÷ − = − ⋅−

⋅=

⋅⋅ ⋅

=⋅ ⋅ ⋅

=⋅

=

29. 8 8 2 8 2 169 2 9 9 9

xx x x

⋅− ÷ = − ⋅ = − = −

⋅

30. 10 4 10 511 5 11 4

10 511 42 5 511 2 25 511 22522

xx

x

x

x

x

÷ − = ⋅−

⋅= −

⋅⋅ ⋅ ⋅

= −⋅ ⋅

⋅ ⋅= −

⋅

= −

31. 11 3 11 11 11 11 12120 11 20 3 20 3 60y y y y⋅÷ = ⋅ = =

⋅

32. 9 2 9 9 9 9 8120 9 20 2 20 2 40z z z z⋅÷ = ⋅ = =

⋅

33.

2 2 443 3 1

2 13 42 13 4

2 13 2 21

3 216

− ÷ = − ÷

= − ⋅

⋅= −

⋅⋅

= −⋅ ⋅

= −⋅

= −



153

34.

5 5 10106 6 1

5 16 105 1

6 105 1

6 5 21

6 21

12

− ÷ = − ÷

= − ⋅

⋅= −

⋅⋅

= −⋅ ⋅

= −⋅

= −

35. 2

2

2

1 5 15 5 5

15 515 515 5

25

xx xx

xxx xxx

x

÷ = ⋅

⋅=

⋅⋅ ⋅

=⋅ ⋅⋅

=⋅

=

36. 3 3

2 3 2 23 9 3 3 3

9 3 3 39y y y y y y

y yy y y y⋅ ⋅ ⋅ ⋅

÷ = ⋅ = = =⋅ ⋅ ⋅⋅

37. 2 5 2 5 103 9 3 9 27

⋅⋅ = =

⋅

38. 8 5 8 5 1 8 5 1 1

15 32 15 32 5 3 8 4 3 4 12⋅ ⋅ ⋅

⋅ = = = =⋅ ⋅ ⋅ ⋅ ⋅

39. 23 5 3 6 3 6 18

7 6 7 5 7 5 35x x x x x x

x⋅

÷ = ⋅ = =⋅

40. 22 5 2 11 2 11 22

5 11 5 5 5 5 25y

y y y y y y⋅

÷ = ⋅ = =⋅

41. 16 8 16 1527 15 27 8

16 1527 88 2 3 5

3 9 82 59

109

yy y y

yy

yy

÷ = ⋅

⋅=

⋅⋅ ⋅ ⋅ ⋅

=⋅ ⋅ ⋅⋅

=

=

42. 12 4 12 721 7 21 4

12 721 44 3 7 17 3 4 11

y y yy

yyyy

÷ = ⋅

⋅=

⋅⋅ ⋅ ⋅ ⋅

=⋅ ⋅ ⋅ ⋅

=

43. 5 35 5 35 5 7 5 1 128 25 28 25 7 4 5 5 4

⋅ ⋅ ⋅ ⋅− ⋅ = − = − = −

⋅ ⋅ ⋅ ⋅

44. 24 5 24 5 8 3 5 1 145 8 45 8 5 3 3 8 3

⋅ ⋅ ⋅ ⋅⋅ − = − = − = −

⋅ ⋅ ⋅ ⋅

45. 23 3 3 3 3 9

4 4 4 4 4 16⋅⎛ ⎞− = − ⋅− = =⎜ ⎟ ⋅⎝ ⎠

46. 51 1 1 1 1 1

2 2 2 2 2 21 1 1 1 1

2 2 2 2 21

32

⎛ ⎞ ⎛ ⎞⎛ ⎞⎛ ⎞⎛ ⎞⎛ ⎞− = − − − − −⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠⎝ ⎠⎝ ⎠⎝ ⎠

⋅ ⋅ ⋅ ⋅= −

⋅ ⋅ ⋅ ⋅

= −

47. 2 3 2 3

21 1

x y x y x x y y y x y y xyy x y x y x

⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅⋅ = = = =

⋅ ⋅ ⋅

48. 3 3

2 3 2 3 2b a b a b a a a a a

a a b b b b ba b a b b⋅ ⋅ ⋅ ⋅

⋅ = = = =⋅ ⋅ ⋅ ⋅ ⋅⋅

49. 2 7 2 7 11 7 11 777

11 1 11 1 2 1 2 2⋅

÷ = ÷ = ⋅ = =⋅



154

50. 1 100 11002 1 2

100 21 1

100 21 1

200

− ÷ = − ÷

= − ⋅

⋅= −

⋅= −

51.

2 2

2

2

3312 1 12

3 121

3 1213 1213 12136

x x xx

xx

xxxx x

x

x

− ÷ = − ÷

= − ⋅

⋅= −

⋅⋅ ⋅

= −⋅ ⋅⋅

= −⋅

= −

52.

22

2

2

14 7 1473 1 3

7 31 14

7 31 147 31 2 7

31 232

x x xx

xx

xx

x xx

x

x

− ÷ = − ÷

= − ⋅

⋅= −

⋅⋅ ⋅ ⋅

= −⋅ ⋅ ⋅⋅

= −⋅

= −

53. 2 7 3 2 2 37 2 4 7 7 4

2 2 37 7 4

2 2 37 7 2 2

37 73

49

⎛ ⎞ ⎛ ⎞÷ ⋅ = ⋅ ⋅⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⋅ ⋅=

⋅ ⋅⋅ ⋅

=⋅ ⋅ ⋅

=⋅

=

54. 1 5 1 1 5 122 6 12 2 6 1

1 5 122 6 1

1 5 6 22 6 1

1 51

5

⎛ ⎞ ⎛ ⎞⋅ ÷ = ⋅ ⋅⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⋅ ⋅=

⋅ ⋅⋅ ⋅ ⋅

=⋅ ⋅

⋅=

=

55. 2

2

2

19 19 9963 63 1

19 963 1

19 99 7 1

197 1

197

yyy y

yyy yy

y

y

− ⋅ = − ⋅

⋅= −

⋅⋅ ⋅ ⋅

= −⋅ ⋅ ⋅⋅

= −⋅

= −

56. 2

2

2

31 16 311624 1 24

16 311 24

2 8 311 8 3

2 311 3

623

aaa a

aaa a

aa

a

⋅ − = ⋅−

⋅= −

⋅⋅ ⋅ ⋅ ⋅

= −⋅ ⋅ ⋅

⋅ ⋅= −

⋅

= −

57. 2 6 2 6 2 2 3 2 2 43 11 3 11 3 11 11 11

⋅ ⋅ ⋅ ⋅− ⋅− = = = =

⋅ ⋅

58. 1 6 1 6 65 7 5 7 35

⋅− ⋅ − = =

⋅

59. 4 3 4 16 4 16 4 8 2 4 2 88 16 8 3 8 3 8 3 3 3

⋅ ⋅ ⋅ ⋅÷ = ⋅ = = = =

⋅ ⋅

60. 9 16 9 15 9 15 1352 15 2 16 2 16 32

⋅÷ = ⋅ = =

⋅



155

61. 2 2

2

21 14 21 2510 25 10 14

21 2510 14

3 7 5 52 5 2 7

3 52 2

154

x x x yy y y x

x yy xx x yy x

x

x

÷ = ⋅

⋅=

⋅⋅ ⋅ ⋅ ⋅ ⋅ ⋅

=⋅ ⋅ ⋅ ⋅ ⋅⋅ ⋅

=⋅

=

62. 2 2

2

17 13 17 1824 18 24 13

17 1824 13

17 6 36 4 13

17 34 13

5152

y y y xx x x y

y xx yy y xx y

y

y

÷ = ⋅

⋅=

⋅⋅ ⋅ ⋅ ⋅ ⋅

=⋅ ⋅ ⋅ ⋅⋅ ⋅

=⋅

=

63. 3 2 1 3 214 3 1 4 3

1 4 21 3 3

1 4 21 3 389

⎛ ⎞ ⎛ ⎞÷ ⋅ = ÷ ⋅⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞= ⋅ ⋅⎜ ⎟⎝ ⎠⋅ ⋅

=⋅ ⋅

=

64.

2 5 33 2 53311 9 1 11 9

33 11 51 2 9

33 11 52 9

3 11 11 52 3 3

11 11 52 3

6056

⎛ ⎞ ⎛ ⎞÷ ⋅ = ÷ ⋅⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞= ⋅ ⋅⎜ ⎟⎝ ⎠

⋅ ⋅=

⋅⋅ ⋅ ⋅

=⋅ ⋅⋅ ⋅

=⋅

=

65.

3 3 33

3

3

3

3

30302 2 1

12 30

12 30

12 30

12 30160

a a aa

aa

aa

a a aa a a

÷ = ÷

= ⋅

⋅=

⋅⋅ ⋅ ⋅

=⋅ ⋅ ⋅ ⋅

=⋅

=

66. 2 3 2

3

3

2

3

2

3 15 3155 1 5

15 51 3

15 51 3

c c cc

cc

cc

÷ = ÷

= ⋅

⋅=

⋅

5 3 51 3

5 51

25

c c cc c

c

c

⋅ ⋅ ⋅ ⋅ ⋅=

⋅ ⋅ ⋅⋅ ⋅

=

=

67. 2 2ab c ab c a b b c bc ab c ab c a b

⋅ ⋅ ⋅ ⋅⋅ = = =

⋅ ⋅ ⋅

68. 3 3 2

2 2ac b ac b a c b b b b b bb b a a c a aa c b a c

⋅ ⋅ ⋅ ⋅ ⋅ ⋅⋅ = = = =

⋅ ⋅ ⋅⋅

69.

1 2 5 1 2 52 3 6 2 3 6

1 53 61 63 51 63 51 2 3

3 51 2

525

⋅⎛ ⎞ ⎛ ⎞⋅ ÷ = ÷⎜ ⎟ ⎜ ⎟⋅⎝ ⎠ ⎝ ⎠

= ÷

= ⋅

⋅=

⋅⋅ ⋅

=⋅

⋅=

=



156

70.

3 8 2 3 8 24 9 5 4 9 5

3 4 2 24 3 3 5

2 23 52 53 22 53 253

⋅⎛ ⎞ ⎛ ⎞⋅ ÷ = ÷⎜ ⎟ ⎜ ⎟⋅⎝ ⎠ ⎝ ⎠⋅ ⋅⎛ ⎞= ÷⎜ ⎟⋅ ⋅⎝ ⎠

= ÷

= ⋅

⋅=

⋅

=

71.

4 4 3 4 4 37 5 7 7 5 7

4 127 354 357 124 357 124 7 57 4 353

⋅⎛ ⎞ ⎛ ⎞− ÷ ⋅ = − ÷⎜ ⎟ ⎜ ⎟⋅⎝ ⎠ ⎝ ⎠

= − ÷

= − ⋅

⋅= −

⋅⋅ ⋅

= −⋅ ⋅

= −

72.

5 4 5 5 4 58 7 16 8 7 16

5 4 58 7 4 45 58 285 288 5

5 288 5

5 7 44 2 572

⋅⎛ ⎞ ⎛ ⎞÷ ⋅− = ÷ −⎜ ⎟ ⎜ ⎟⋅⎝ ⎠ ⎝ ⎠⋅⎛ ⎞= ÷ −⎜ ⎟⋅ ⋅⎝ ⎠

⎛ ⎞= ÷ −⎜ ⎟⎝ ⎠⎛ ⎞= ⋅ −⎜ ⎟⎝ ⎠⋅

= −⋅⋅ ⋅

= −⋅ ⋅

= −

73. a. 2 5 2 5 2 5 15 6 5 6 5 2 3 3

xy ⋅ ⋅= ⋅ = = =

⋅ ⋅ ⋅

b. 2 5 2 6 2 6 125 6 5 5 5 5 25

x y ⋅÷ = ÷ = ⋅ = =

⋅

74. a. 8 1 8 1 2 4 1 2 1 29 4 9 4 9 4 9 9

xy ⋅ ⋅ ⋅ ⋅= ⋅ = = = =

⋅ ⋅

b. 8 1 8 4 8 4 329 4 9 1 9 1 9

x y ⋅÷ = ÷ = ⋅ = =

⋅

75. a. 4 9 4 9 365 11 5 11 55

xy ⋅= − ⋅ = − = −

⋅

b. 4 9 4 11 4 11 445 11 5 9 5 9 45

x y ⋅÷ = − ÷ = − ⋅ = − = −

⋅

76. a. 7 1 7 1 76 2 6 2 12

xy ⋅= ⋅− = − = −

⋅

b. 7 16 27 26 1

7 26 17 2

2 3 173

x y÷ = ÷−

= ⋅−

⋅= −

⋅⋅

= −⋅ ⋅

= −

77.

536

5 5318 6

3 5 51 18 6

3 5 51 18 63 5 53 6 6

5 5 True6 6

x = −

⎛ ⎞− −⎜ ⎟⎝ ⎠

⋅ − −

⋅− −

⋅⋅

− −⋅

− = −

Yes, 5

18− is a solution to the equation.



157

78.

2 63 11

2 9 63 11 112 9 63 11 11

2 611

2 3 611 11

6 6 True11 11

y =

⋅

⋅⋅

⋅ 3⋅33⋅11

⋅

=

Yes, 911

is a solution of the equation.

79. 1 12 10

1 2 12 5 101 2 12 5 10

1 1 False5 10

z− =

− ⋅

⋅−

⋅

− =

No, 25

is not a solution of the equation.

80. 153

3 155 3

5 3 11 5 35 3 11 5 3

3 1 False1 3

x =

⋅

⋅

⋅⋅

=

No, 35

is not a solution of the equation.

81. 1 1 of 200 2004 4

1 2004 11 200

4 11 4 50

4 1501

50

= ⋅

= ⋅

⋅=

⋅⋅ ⋅

=⋅

=

=

82. 1 1 of 2005 5

1

1

401

40

= ⋅200

1 200= ⋅5 1⋅200

=5⋅1⋅5⋅ 40

=5⋅1

=

=

83.

5 5 of 246 6

5

5

5

201

20

= ⋅24

5 24= ⋅6 1⋅ 24

=6⋅1⋅6 ⋅ 4

=6 ⋅1⋅ 4

=1

=

=

84.

5 5 of 248 8

5

5

5

151

15

= ⋅24

5 24= ⋅8 1⋅24

=8⋅1⋅8⋅3

=8⋅1⋅3

=1

=

=



158

85. 4 4 of 800 80025 25

4 80025 14 80025 1

4 25 3225 1

4 321

128

= ⋅

= ⋅

⋅=

⋅⋅ ⋅

=⋅

⋅=

=

128 of the students would be expected to major in business.

86. 7 7 of 7660

10 10

7

7

7

5362

= ⋅7660

7 7660= ⋅10 1⋅7660

=10 ⋅1⋅766 ⋅10

=10 ⋅1⋅766

=1

=

The money received was $5362 million.

87. 2 2 of 21705 5

2

2

2

868

= ⋅2170

2 2170= ⋅5 1⋅2170

=5⋅1⋅5⋅ 434

=5⋅1⋅ 434

=1

=

He hiked 868 miles.

88.

3 3 of 816 16

3

3

3232

= ⋅8

3 8= ⋅16 1⋅8

=16⋅1

⋅8=8⋅2 ⋅1

=⋅1

=

The screw sinks 32

inches deep after 8 turns.

89. 1 3 1 3 1 3 3 of 2 8 2 8 2 8 16

⋅= ⋅ = =

⋅

The radius of the circle is 3 inch.

16

90. 1 1 of 30005 5

1

1

600

= ⋅3000

1 3000= ⋅5 1⋅3000

=5⋅1⋅5⋅600

=5⋅1

=

The diet can contain 600 calories from fat per day.

91. 2 2 of 27573 3

2

2

2

1838

= ⋅2757

2 2757= ⋅3 1⋅2757

=3⋅1⋅3⋅919

=3⋅1⋅919

=1

=

The sale price is $1838.



159

92. 3 3 of 102,000

50 50

3

3

3

6120

= ⋅102,000

3 102,000= ⋅50 1⋅102,000

=50 ⋅1

⋅50 ⋅ 2040=

50⋅1⋅ 2040

=1

=

The Gonzales family pays $6120 to the real estate companies.

93. 1 1 of 180 1809 9

1 1809 11 180

9 11 9 20

9 1201

20

= ⋅

= ⋅

⋅=

⋅⋅ ⋅

=⋅

=

=

20 contestants participated more than once.

94. 23 23 of 854,00050 50

23

23

392,840

= ⋅854,000

23 854,000= ⋅50 1

⋅50 ⋅17,080=

50 ⋅1⋅17,080

=1

=

They might believe there were 392,840 people between the ages of 16 and 24.

95. 5 5 1area length width

14 141 ⋅1

= ⋅ = ⋅ = =5 14 ⋅5

The area is 1

14 square foot.

96. 1 1 3area length width2 16

3 ⋅3= ⋅ = ⋅ = =

8 2⋅8

The area is 3

16 square mile.

97. 8 8 12,00012,00025 25 1

8 25 48025 1

8 4801

3840

⋅ = ⋅

⋅ ⋅=

⋅⋅

=

=

The family drove 3840 miles for work.

98. 3 3 of 12,00025 25

3

3

3

1440

= ⋅12,000

3 12,000= ⋅25 1⋅12,00

=25⋅1⋅ 25⋅480

=25⋅1⋅ 480

=1

=

The family drove 1440 miles for shopping.

99. 1 1 of 12,0005 5

1

2400

= ⋅12,000

1 12,000= ⋅5 1⋅5⋅ 2400

=5⋅1

=

The family drove 2400 miles for family business.

100. 1

1001

120

1 12,000⋅12,000 = ⋅

100 1⋅120 ⋅100

=100⋅1

=

The family drove 120 miles for medical.

101. 277698

201+

102. 8114269

922+

103. 968772196

−



160

104. 882773109

−



107. 42 125 7 42 125 625 36 6 25 36 7

42 125 625 36 76 7 5 25 625 6 6 7

5

⋅ ÷ = ⋅ ⋅

⋅ ⋅=

⋅ ⋅⋅ ⋅ ⋅ ⋅

=⋅ ⋅ ⋅

=

108.

2 2

2

2

8 39 8 1 8 39 8 113 16 9 2 13 16 9 2

8 13 3 2 4 113 8 2 3 3 24 13 24 4 13 3 24 4 13 3 2

16 19 2

16 29 1

16 29 1

329

⋅ ⋅⎛ ⎞ ⎛ ⎞⋅ ⋅ ÷ = ÷⎜ ⎟ ⎜ ⎟⋅ ⋅⎝ ⎠ ⎝ ⎠⋅ ⋅ ⋅ ⋅⎛ ⎞= ÷⎜ ⎟⋅ ⋅ ⋅ ⋅⎝ ⎠

⎛ ⎞= ÷⎜ ⎟⎝ ⎠⎛ ⎞⎛ ⎞= ÷⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

⋅⎛ ⎞= ÷⎜ ⎟⋅⎝ ⎠

= ÷

= ⋅

⋅=

⋅

=

109. 3 3 of 290,810,000 290,810,00025 25

3 290,810,00025 13 290,810,000

25 13 25 11,632, 400

25 13 11,632, 400

134,897, 200

= ⋅

= ⋅

⋅=

⋅⋅ ⋅

=⋅

⋅=

=

The population of California is approximately 34,897,200.

110. 32 32 of 10,600265 265

32

32

32

1280

= ⋅10,600

32 10,600= ⋅265 1⋅10,600

=265⋅1⋅ 265⋅ 40

=265⋅1⋅ 40

=1

=

There were 1280 news/talk stations.

111. 34

of 36 on the first bus and 13

of 30 on the

second bus are girls. 3 14 3

3 1

3 1

3 101

27 1037

1 3 36 30⋅36 + ⋅30 = ⋅ + ⋅

3 4 1 1⋅36 ⋅30

= +4 ⋅1 3⋅1⋅ 4 ⋅9 ⋅3⋅10

= +4 ⋅1 3⋅1⋅9

= +1

= +=

There are 37 girls on the two buses.

112.

42 of fleet 252113

42fleet 252113

2521

252

42

6

678

=

= ÷

113= ⋅

42⋅113

=1⋅42⋅6 ⋅113

=1⋅42⋅113

=1

=

There were 678 aircraft in the FedEx Express air fleet.

The Bigger Picture

1. 2 2 163 27

8 ⋅8⋅ = =9 3⋅9



161

2. 2 8 23 9 3

2

2

34

9− ÷ = − ⋅

8⋅9

= −3⋅8⋅3⋅3

= −3⋅ 2 ⋅ 4

= −

3. 12 12 3 41 3

1 ⋅1 ⋅ 4 ⋅1⋅ = = =9 1⋅9 1⋅3⋅3

4. 9 15 92 8 2

9 82 153 3 2 4

2 3 53 45

125

x x xx

xxxx

8÷ = ⋅

15⋅

=⋅⋅ ⋅ ⋅ ⋅

=⋅ ⋅ ⋅

⋅=

=

5. 80 ÷ 20 ⋅ 2 = 4 ⋅ 2 = 8

6. 2 33 2 3 3 2 2 2 72⋅ = ⋅ ⋅ ⋅ ⋅ =

7. 11 5 4 11 5 4 11 5 4 1 1 120 8 33 20 8 33 4 5 8 3 11 8 3 24

⋅ ⋅ ⋅ ⋅ ⋅⋅ ⋅ = = = =

⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅

8. 1 20 1 20 2 20 2 4020 402 1 2 1 1 1 1 1

⋅÷ = ÷ = ⋅ = = =

⋅

9. 3 33 4(18 16) 3 4(2)3 4(8)3 3235

+ − = += += +=

10. 100 − 76 = 24

11. 12 7 712 7 7 7 7

12 012 012 12

0

xx

xx

x

− + =− + − = −

− =−

=− −

=

12. 9 14 6 329 6 14 6 6 32

3 14 323 14 14 32 14

3 183 183 3

6

n nn n n n

nn

nn

n

− = −− − = − −

− = −− + = − +

= −−

=

= −

Section 4.4

Practice Problems

1. 6 2 6 2 8

13 13 13 13+

+ = =

2. 5 1 5 1 6 2 3 38 8 8 8 2 4 4

+ ⋅+ = = = =

⋅

3. 20 6 7 20 6 7 33 or 311 11 11 11 11

+ ++ + = =

4. 11 6 11 6 512 12 12 12

−− = =

5. 7 2 7 2 5 1 5 1

15 15 15 15 3 5 3− ⋅

− = = = =⋅

6. 8 4 8 4 4 4 or

17 17 17 17 17− + −

− + = = −

7. 2 7 2 75 5 5

y y−− =

8. 4 6 3 4 6 3 5 5 or

11 11 11 11 11 11− − −

− − = = −

9. 10 5 10 5 5 5 or 12 12 12 12 12

x y − + −+ = − + = = −



162

10. 3 3 3 3perimeter

20 20 20 203 3 3 3

2012203 45 435

= + + +

+ + +=

=

⋅=

⋅

=

The perimeter is 35

mile.

11. 13 11 13 11 2 1 2 14 4 4 4 2 2 2

− ⋅− = = = =

⋅

He ran 12

mile farther on Monday.

12. 16 is a multiple of 8, so the LCD of 78

and 1116

is 16.

13. 30 ⋅ 1 = 30 Not a multiple of 25. 30 ⋅ 2 = 60 Not a multiple of 25. 30 ⋅ 3 = 90 Not a multiple of 25. 30 ⋅ 4 = 120 Not a multiple of 25. 30 ⋅ 5 = 150 A multiple of 25.

The LCD of 2325

and 130

is 150.

14. 40 = 2 ⋅ 2 ⋅ 2 ⋅ 5 108 = 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 3 LCD = 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 5 = 1080

The LCD of 340

− and 11

108 is 1080.

15. 20 = 2 ⋅ 2 ⋅ 5 24 = 2 ⋅ 2 ⋅ 2 ⋅ 3 45 = 3 ⋅ 3 ⋅ 5 LCD = 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 5 = 360

The LCD of 7 1, ,20 24

and 1345

is 360.

16. y = y 11 = 11

The LCD of 7y

and 6

11 is 11y.

17. 7 7 7 7 7 498 8 7 8 7 56

⋅= ⋅ = =

⋅

18. 1 1 5 1 5 54 4 5 4 5 20

⋅= ⋅ = =

⋅

19. 3 3 6 3 6 187 7 6 7 6 42x x x x⋅= ⋅ = =

⋅

20. 4 6 4 6 2441 6 1 6 6

⋅= ⋅ = =

⋅

21. 9 9 9 9 9 814 4 9 4 9 36x x x x

⋅= ⋅ = =

⋅


1. The fractions 911

and 1311

are called like

fractions while 34

and 13

are called unlike

fractions.

2. a c a cb b b

++ = and .a c a c

b b b−

− =

3. As long as b is not 0, .a a ab b b−

= = −−

4. The distance around a figure is called its perimeter.

5. The smallest positive number divisible by all the denominators of a list of fractions is called the least common denominator (LCD).


Exercise Set 4.4

1. 5 2 5 2 7

11 11 11 11+

+ = =

2. 9 2 9 2 11

17 17 17 17+

+ = =

3. 2 4 2 4 6 3 2 29 9 9 9 3 3 3

+ ⋅+ = = = =

⋅



163

4. 3 2 3 2 5 1 5 1

10 10 10 10 2 5 2+ ⋅

+ = = = =⋅

5. 6 1 6 1 5 1 5 120 20 20 20 4 5 4

− + − ⋅− + = = = − = −

⋅

6. 3 1 3 1 2 1 2 18 8 8 8 4 2 4

− + − ⋅− + = = = − = −

⋅

7. 3 4 3 ( 4) 7 1 7 1

14 14 14 14 2 7 2− + − − ⋅⎛ ⎞− + − = = = − = −⎜ ⎟ ⋅⎝ ⎠

8. 5 7 5 ( 7) 12 1 12 124 24 24 24 2 12 2

− + − − ⋅⎛ ⎞− + − = = = − = −⎜ ⎟ ⋅⎝ ⎠

9. 2 4 2 4 6 2 3 2

9 9 9 9 3 3 3x x x x x x+ ⋅

+ = = = =⋅ ⋅

10. 3 2 3 2 5 1 5 1

10 10 10 10 2 5 2y y y y y y+ ⋅

+ = = = =⋅ ⋅

11. 7 3 2 7 3 218 18 18 18

21822 9

9

x x x x x x

x

x

x

− + +− + + =

−=

⋅= −

⋅

= −

12. 7 3 1 7 3 115 15 15 15

31533 5

5

z z z z z z

z

z

z

− + +− + + =

−=

⋅= −

⋅

= −

13. 10 4 10 4 611 11 11 11

−− = =

14. 9 5 9 5 4

13 13 13 13−

− = =

15. 7 1 7 1 6 3 2 38 8 8 8 4 2 4

− ⋅− = = = =

⋅

16. 5 1 5 1 4 2 2 26 6 6 6 3 2 3

− ⋅− = = = =

⋅

17. 1 4 1 4 3 3y y y y y

− −− = = = −

18. 4 7 4 7 3 3z z z z z

− −− = = = −

19. 27 8 27 833 33 33 33

27 833

19331933

⎛ ⎞− − − = − +⎜ ⎟⎝ ⎠

− +=

−=

= −

20. 37 18 37 1845 45 45 45

37 1845

19451945

⎛ ⎞− − − = − +⎜ ⎟⎝ ⎠

− +=

−=

= −

21. 20 10 17 20 10 1721 21 21 21

721

1 73 713

− −− − =

−=

⋅= −

⋅

= −

22. 27 5 28 27 5 2828 28 28 28

628

3 214 23

14

− −− − =

−=

⋅= −

⋅

= −

23. 7 3 7 34 4 4a a −− =

24. 18 3 18 3

5 5 5b b −− =



164

25. 9 99 9 99 90 9 10 9

100 100 100 100 10 10 10− + ⋅

− + = = = =⋅

26. 15 85 15 85 70 7 10 7200 200 200 200 20 10 20

− + ⋅− + = = = =

⋅

27. 13 13 13 1328 28 28

26282 132 14

1314

x x x x

x

x

x

− −− − =

−=

⋅ ⋅= −

⋅

= −

28. 15 15 15 15

26 26 2630

262 15

2 1315

13

y y y

y

y

y

− −− − =

−=

⋅= −

⋅ ⋅

= −

29. 9 1 9 115 15 15x x ++ =

30. 2 7 2 715 15 15x x ++ =

31. 7 15 7 15 8 816 16 16 16 8 2 2x x x x x x x− − ⋅− = = = − = −

⋅

32. 3 15 3 15 12 3 4 3

16 16 16 16 4 4 4z z z z z z− − ⋅

− = = = − = −⋅ ⋅

33. 9 7 10 9 7 10 8 2 4 2

12 12 12 12 12 3 4 3− − − ⋅

− − = = = − = −⋅

34. 1 15 2 1 15 2 12 4 3 38 8 8 8 8 4 2 2

− + − ⋅− + = = = − = −

⋅

35. 3 2 3 2 3

4 4 4 4 4 4x x x x x x x x x+ − ++ − + = =

36. 9 2 5 4 9 2 5 48 8 8 8 8

128

4 34 2

32

y y y y y y y y

y

y

y

+ + −+ + − =

=

⋅ ⋅=

⋅

=

37. 3 2 3 2 54 4 4 4

x y ++ = + = =

38. 7 9 7 9 2 1 2 18 8 8 8 4 2 4

x y − − ⋅− = − = = = − = −

⋅

39. 1 3 1 3 1 3 25 5 5 5 5 5

x y − +⎛ ⎞− = − − − = − + = =⎜ ⎟⎝ ⎠

40. 1 5 1 5 4 2 2 26 6 6 6 2 3 3

x y − + ⋅+ = − + = = = =

⋅

41. 4 7 9 4 7 9 20 120 20 20 20 20

+ ++ + = = =

The perimeter is 1 inch.

42. 3 6 4 3 2 3 6 4 3 2 18

13 13 13 13 13 13 13+ + + +

+ + + + = =

The perimeter is 1813

feet.

43. 5 7 5 7 5 7 5 7 24 2

12 12 12 12 12 12+ + +

+ + + = = =

The perimeter is 2 meters.

44. 1 1 1 1 1 1 1 1 4 2 2 26 6 6 6 6 6 2 3 3

+ + + ⋅+ + + = = = =

⋅

The perimeter is 23

centimeter.

45. North America makes up 16100

of the world’s

land area, while South America makes up 12100

of the land area. 16 12 16 12 28 4 7 7100 100 100 100 4 25 25

+ ⋅+ = = = =

⋅

725

of the world’s land area is within North

America and South America.



165

46. Asia makes up 30

100 of the world’s land area,

while Africa makes up 20

100 of the world’s land

area. 30 20 30 20 50 1 50 1

100 100 100 100 2 50 2+ ⋅

+ = = = =⋅

12

of the world’s land area is within Asia and

Africa.

47. Antarctica makes up 9

100 of the world’s land

area, while Europe makes up 7

100 of the world’s

land area. 9 7 9 7 2 1 2 1

100 100 100 100 50 2 50− ⋅

− = = = =⋅

Antarctica’s land area is 150

greater than that of

Europe.

48. Asia makes up 30

100 of the world’s land area,

while Australia makes up 6

100 of the world’s

land area. 30 6 30 6 24 6 4 6

100 100 100 100 25 4 25− ⋅

− = = = =⋅

Asia’s land area is 625

greater than that of

Australia’s.

49. To find the remaining amount of track to be

inspected, subtract the 520

mile that has already

been inspected from the 1920

mile total that must

be inspected. 19 5 19 5 14 2 7 720 20 20 20 2 10 10

− ⋅− = = = =

⋅

710

of a mile of track remains to be inspected.

50. To find the remaining distance, subtract the 118

miles already run from the 168

total miles.

16 11 16 11 58 8 8 8

−− = =

He must run 58

mile more.

51. To find the fraction that had speed limits less

than 70 mph, subtract the 1750

that have 70 mph

speed limits from the 3750

that have speed limits

up to and including 70. 37 17 37 17 20 2 10 250 50 50 50 5 10 5

− ⋅− = = = =

⋅

25

of the states have speed limits that were less

than 70 mph.

52. To find the fraction treating pain other than

headaches, subtract the 750

for treating

headaches from the 3150

for some type of pain.

31 7 31 7 24 2 12 1250 50 50 50 2 25 25

− ⋅− = = = =

⋅

1225

of aspirin use is for treating pain other than

headaches.

53. Multiples of 9: 9, 18, 27, 36, 45, 54 Multiples of 15: 15, 30, 45, 60 LCD: 45

54. Multiples of 12: 12, 24, 36, 48, 60, 72 Multiples of 20: 20, 40, 60, 80 LCD: 60

55. Multiples of 36: 36, 72 Multiples of 24: 24, 48, 72 LCD: 72

56. Multiples of 15: 15, 30, 45, 60, 75, 90 Multiples of 90: 90, 180 LCD: 90



166

57. 6 = 2 ⋅ 3 15 = 3 ⋅ 5 25 = 5 ⋅ 5 LCD = 2 ⋅ 3 ⋅ 5 ⋅ 5 = 150

58. 4 = 2 ⋅ 2 14 = 2 ⋅ 7 20 = 2 ⋅ 2 ⋅ 5 LCD = 2 ⋅ 2 ⋅ 5 ⋅ 7 = 140

59. 24 = 2 ⋅ 2 ⋅ 2 ⋅ 3 x = x LCD = 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ x = 24x

60. y = y 70 = 2 ⋅ 5 ⋅ 7 LCD = 2 ⋅ 5 ⋅ 7 ⋅ y = 70y

61. 18 = 2 ⋅ 3 ⋅ 3 21 = 3 ⋅ 7 LCD = 2 ⋅ 3 ⋅ 3 ⋅ 7 = 126

62. 24 = 2 ⋅ 2 ⋅ 2 ⋅ 3 45 = 3 ⋅ 3 ⋅ 5 LCD = 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 5 = 360

63. 3 = 3 21 = 3 ⋅ 7 56 = 2 ⋅ 2 ⋅ 2 ⋅ 7 LCD = 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 7 = 168

64. 11 = 11 33 = 3 ⋅ 11 121 = 11 ⋅ 11 LCD = 3 ⋅ 11 ⋅ 11 = 363

65. 2 2 7 2 7 143 3 7 3 7 21

⋅= ⋅ = =

⋅

66. 5 5 4 5 4 206 6 4 6 4 24

⋅= ⋅ = =

⋅

67. 4 4 5 4 5 207 7 5 7 5 35

⋅= ⋅ = =

⋅

68. 3 3 20 3 20 605 5 20 5 20 100

⋅= ⋅ = =

⋅

69. 1 1 25 1 25 252 2 25 2 25 50

⋅= ⋅ = =

⋅

70. 1 1 10 1 10 105 5 10 5 10 50

⋅= ⋅ = =

⋅

71. 14 14 4 14 4 5617 17 4 17 4 68x x x x⋅= ⋅ = =

⋅

72. 19 19 6 19 6 11421 21 6 21 6 126z z z z⋅= ⋅ = =

⋅

73. 2 2 4 2 4 83 3 4 3 4 12y y y y⋅= ⋅ = =

⋅

74. 3 3 6 3 6 182 2 6 2 6 12x x x x⋅= ⋅ = =

⋅

75. 5 5 4 5 4 209 9 4 9 4 36

a a aa a a

⋅= ⋅ = =

⋅

76. 7 7 6 7 6 426 6 6 6 6 36

a a aa a a

⋅= ⋅ = =

⋅

77. Denmark: 89 89

100 100=

Finland: 9 9 10 90

10 10 10 100⋅

= =⋅

Israel: 24 24 4 9625 25 4 100

⋅= =

⋅

Spain: 23 23 4 9225 25 4 100

⋅= =

⋅

Japan: 17 17 4 6825 25 4 100

⋅= =

⋅

Norway: 91 91

100 100=

Singapore: 4 4 20 805 5 20 100

⋅= =

⋅

South Korea: 69 69

100 100=

India: 2 2 4 825 25 4 100

⋅= =

⋅

United States: 7 7 10 70

10 10 10 100⋅

= =⋅

78. 96

100 is the largest fraction, so Israel has the

largest fraction of cell phone users.



167

79. 8

100 is the smallest fraction, so India has the

smallest fraction of cell phone users.

80. 3 3 25 754 4 25 100= ⋅ =

Find those countries whose fraction is over 75 .

100

Denmark, Finland, Israel, Spain, Norway, and

Singapore have over 34

of the population using

cell phones.

81. 23 3 3 9= ⋅ =

82. 34 4 4 4 64= ⋅ ⋅ =

83. 35 5 5 5 125= ⋅ ⋅ =

84. 43 3 3 3 3 81= ⋅ ⋅ ⋅ =

85. 27 7 7 49= ⋅ =

86. 45 5 5 5 5 625= ⋅ ⋅ ⋅ =

87. 32 3 2 2 2 3 24⋅ = ⋅ ⋅ ⋅ =

88. 24 5 4 4 5 80⋅ = ⋅ ⋅ =

89. 2 9 2 9 117 7 7 7

++ = =

90. 3 1 3 1 2 1 2 14 4 4 4 2 2 2

− ⋅− = = = =

⋅



93. 16 12 7 20 30 6 9100 100 100 100 100 100 100

16 12 7 20 30 6 9100

1001001

+ + + + + +

+ + + + + +=

=

=

answers may vary

94. 3 3 4 3 3 4 2 1 2 18 8 8 8 8 4 2 4

+ − ⋅+ − = = = =

⋅

He is 14

mile from home.

95. 37 37 22 37 22 814165 165 22 165 22 3630x x x x⋅= ⋅ = =

⋅

96. 108 108 19 108 19 2052215 215 19 215 19 4085y y y y

⋅= ⋅ = =

⋅



99. 2 2 5 103 3 5 152 2 20 403 3 20 602 2 100 2003 3 100 300

⋅= =

⋅⋅

= =⋅⋅

= =⋅

a, b, and d are equivalent to 2 .3

100. False; 7 7 4 7 4 28

12 12 4 12 4 48⋅

= ⋅ = =⋅

Section 4.5

Practice Problems

1. The LCD of 7 and 21 is 21. 2 8 2 3 8 6 8 14 2 7 27 21 7 3 21 21 21 21 3 7 3

⋅ ⋅+ = + = + = = =

⋅ ⋅

2. The LCD of 6 and 9 is 18. 5 2 5 3 2 2 15 4 196 9 6 3 9 2 18 18 18y y y y y y y⋅ ⋅+ = + = + =

⋅ ⋅

3. The LCD of 5 and 20 is 20. 1 9 1 4 95 20 5 4 20

4 920 20

5201 54 514

⋅− + = − +

⋅

= − +

=

⋅=

⋅

=



168

4. The LCD of 7 and 10 is 70. 5 9 5 10 9 7 50 63 137 10 7 10 10 7 70 70 70

⋅ ⋅− = − = − = −

⋅ ⋅

5. The LCD of 8, 3, and 12 is 24. 5 1 1 5 3 1 8 1 28 3 12 8 3 3 8 12 2

15 8 224 24 24524

⋅ ⋅ ⋅− − = − −

⋅ ⋅ ⋅

= − −

=

6. Recall that 55 .1

= The LCD of 1 and 4 is 4.

5 5 4 20 201 4 1 4 4 4 4 4

y y y y⋅ −− = − = − =

⋅

7. The LCD of 8 and 20 is 40. Write each fraction as an equivalent fraction with a denominator of 40. 5 5 5 258 8 5 40

⋅= =

⋅

11 11 2 2220 20 2 40

⋅= =

⋅

Since 25 > 22, 25 22 ,40 40

> so 5 11 .8 20>

8. The LCD of 20 and 5 is 20. Write each fraction as an equivalent fraction with a denominator of 20.

1720

−

4 4 4 165 5 4 20

⋅− = − = −

⋅

Since −17 < −16, 17 16 ,20 20

− < − so 17 4 .20 5

− < −

9. The LCD of 11 and 9 is 99. 5 4 5 9 4 11 45 44 1

11 9 11 9 9 11 99 99 99x y ⋅ ⋅− = − = − = − =

⋅ ⋅

10. The LCD of 5, 10, and 15 is 30. 3 3 1 3 6 3 3 1 25 10 15 5 6 10 3 15 2

18 9 230 30 302930

⋅ ⋅ ⋅+ + = + +

⋅ ⋅ ⋅

= + +

=

The homeowner needs 2930

cubic yard of cement.

11. The LCD of 4 and 3 is 12. 3 2 3 3 2 4 9 8 14 3 4 3 3 4 12 12 12

⋅ ⋅− = − = − =

⋅ ⋅

The difference in length is 1 foot.

12


1. 1 2 37

16 5 80+ =

2. 3 2 2320 25 100

+ =

3. 4 7 959 8 72+ =

4. 9 5 16311 12 132

+ =

5. 10 12 39417 19 323

+ =

6. 14 15 25331 21 217

+ =


1. To add or subtract unlike fractions, we first write the fractions as equivalent fractions with a common denominator. The common denominator we use is called the least common denominator.

2. The LCD for 16

and 58

is 24.

3. 1 5 1 4 5 3 4 15 19.6 8 6 4 8 3 24 24 24+ = ⋅ + ⋅ = + =



169

4. 1 5 1 4 5 3 4 15 11.6 8 6 4 8 3 24 24 24− = ⋅ − ⋅ = − = −

5. x − y is an expression while 135

x = is an

equation.

6. Since −10 < −1, we know that 10 1 .13 13

− < −

Exercise Set 4.5

1. The LCD of 3 and 6 is 6. 2 1 2 2 1 4 1 53 6 3 2 6 6 6 6

⋅+ = + = + =

⋅

2. The LCD of 6 and 12 is 12. 5 1 5 2 1 10 1 116 12 6 2 12 12 12 12

⋅+ = + = + =

⋅

3. The LCD of 2 and 3 is 6. 1 1 1 3 1 2 3 2 12 3 2 3 3 2 6 6 6

⋅ ⋅− = − = − =

⋅ ⋅

4. The LCD of 3 and 4 is 12. 2 1 2 4 1 3 8 3 53 4 3 4 4 3 12 12 12

⋅ ⋅− = − = − =

⋅ ⋅

5. The LCD of 11 and 33 is 33. 2 2 2 3 2 6 2 4

11 33 11 3 33 33 33 33⋅

− + = − + = − + = −⋅

6. The LCD of 9 and 3 is 9. 5 1 5 1 3 5 3 29 3 9 3 3 9 9 9

⋅− + = − + = − + = −

⋅

7. The LCD of 14 and 7 is 14. 3 3 3 3 2 3 6 3

14 7 14 7 2 14 14 14⋅

− = − = − = −⋅

8. The LCD of 15 and 5 is 15. 2 2 2 2 3 2 6 4

15 5 15 5 3 15 15 15⋅

− = − = − = −⋅

9. The LCD of 35 and 7 is 35. 11 2 11 2 535 7 35 7 5

11 1035 352135

3 75 7

35

x x x x

x x

x

x

x

⋅+ = +

⋅

= +

=

⋅ ⋅=

⋅

=

10. The LCD of 5 and 25 is 25. 2 3 2 5 3 10 3 135 25 5 5 25 25 25 25y y y y y y y⋅+ = + = + =

⋅

11. The LCD of 1 and 12 is 12. 2 2 12 24 242

12 1 12 1 12 12 12 12 12y y y y y⋅ −

− = − = − = − =⋅

12. The LCD of 1 and 20 is 20. 55

20 1 205 201 20 2010020 20

10020

y y

y

y

y

− = −

⋅= −

⋅

= −

−=

13. The LCD of 12 and 9 is 36. 5 1 5 3 1 4 15 4 11

12 9 12 3 9 4 36 36 36⋅ ⋅

− = − = − =⋅ ⋅

14. The LCD of 12 and 18 is 36. 7 5 7 3 5 2 21 10 11

12 18 12 3 18 2 36 36 36⋅ ⋅

− = − = − =⋅ ⋅

15. The LCD of 1 and 7 is 7. 5 7 5 7 7 5 49 5 4477 1 7 1 7 7 7 7 7

⋅− + = − + = − + = − + = −

⋅

16. The LCD of 1 and 10 is 10. 7 10 710

10 1 1010 10 71 10 10

100 710 109310

− + = − +

⋅= − +

⋅

= − +

= −



170

17. The LCD of 11 and 9 is 99. 5 4 5 9 4 1111 9 11 9 9 11

45 4499 99

8999

a a a a

a a

a

⋅ ⋅+ = +

⋅ ⋅

= +

=

18. The LCD of 18 and 9 is 18. 7 2 7 2 2 7 4 1118 9 18 9 2 18 18 18x x x x x x x⋅+ = + = + =

⋅

19. The LCD of 3 and 6 is 6. 2 1 2 2 1 4 1 4 13 6 3 2 6 6 6 6y y y y⋅ −− = − = − =

⋅

20. The LCD of 6 and 12 is 12. 5 1 5 2 1 10 1 10 16 12 6 2 12 12 12 12z z z z⋅ −− = − = − =

⋅

21. The LCD of 2 and x is 2x. 1 3 1 3 2 6 62 2 2 2 2 2

x x xx x x x x x

⋅ ⋅ ++ = + = + =

⋅ ⋅

22. The LCD of 5 and x is 5x. 2 3 2 3 5 2 15 2 155 5 5 5 5 5

x x xx x x x x x

⋅ ⋅ ++ = + = + =

⋅ ⋅

23. The LCD of 11 and 33 is 33. 2 2 2 3 2 6 2 8

11 33 11 3 33 33 33 33⋅

− − = − − = − − = −⋅

24. The LCD of 9 and 3 is 9. 5 1 5 1 3 5 3 89 3 9 3 3 9 9 9

⋅− − = − − = − − = −

⋅

25. The LCD of 14 and 7 is 14. 9 3 9 3 2 9 6 3

14 7 14 7 2 14 14 14⋅

− = − = − =⋅

26. The LCD of 5 and 15 is 15. 4 2 4 3 2 12 2 10 2 5 25 15 5 3 15 15 15 15 3 5 3

⋅ ⋅− = − = − = = =

⋅ ⋅

27. The LCD of 35 and 7 is 35. 11 2 11 2 5 11 10 11 1035 7 35 7 5 35 35 35y y y y⋅ −− = − = − =

⋅

28. The LCD of 5 and 25 is 25. 2 3 2 5 3 10 3 10 35 25 5 5 25 25 25 25b b b b⋅ −− = − = − =

⋅

29. The LCD of 9 and 12 is 36. 1 5 1 4 5 3 4 15 119 12 9 4 12 3 36 36 36

⋅ ⋅− = − = − = −

⋅ ⋅

30. The LCD of 18 and 12 is 36. 5 7 5 2 7 3 10 21 11

18 12 18 2 12 3 36 36 36⋅ ⋅

− = − = − = −⋅ ⋅

31. The LCD of 15 and 12 is 60. 7 5 7 4 5 5

15 12 15 4 12 528 2560 603

601 320 3120

⋅ ⋅− = −

⋅ ⋅

= −

=

⋅=

⋅

=

32. The LCD of 8 and 20 is 40. 5 3 5 5 3 2 25 6 198 20 8 5 20 2 40 40 40

⋅ ⋅− = − = − =

⋅ ⋅

33. The LCD of 7 and 8 is 56. 5 1 5 8 1 7 40 7 337 8 7 8 8 7 56 56 56

⋅ ⋅− = − = − =

⋅ ⋅

34. The LCD of 13 and 10 is 130. 10 7 10 10 7 13 100 91 913 10 13 10 10 13 130 130 130

⋅ ⋅− = − = − =

⋅ ⋅

35. The LCD of 8 and 16 is 16. 7 3 7 2 3 14 3 178 16 8 2 16 16 16 16

⋅+ = + = + =

⋅

36. The LCD of 18 and 9 is 18. 7 2 7 2 2 7 4 11

18 9 18 9 2 18 18 18⋅

+ = + = + =⋅

37. 3 5 3 5 29 9 9 9

−− = = −

38. 1 4 1 4 3

13 13 13 13−

− = = −



171

39. The LCD of 5, 3, and 10 is 30. 2 1 3 2 6 1 10 3 35 3 10 5 6 3 10 10 3

12 10 930 30 301130

⋅ ⋅ ⋅− + − = − + −

⋅ ⋅ ⋅

= − + −

= −

40. The LCD of 3, 4, and 5 is 60. 1 1 2 1 20 1 15 2 123 4 5 3 20 4 15 5 12

20 15 2460 60 601160

⋅ ⋅ ⋅− − + = − − +

⋅ ⋅ ⋅

= − − +

= −

41. The LCD of 11 and 3 is 33. 5 5 3 11

11 3 11 3 3 1115 1133 3315 11

33

y y

y

y

⋅ ⋅+ = +

⋅ ⋅

= +

+=

42. The LCD of 13 and 26 is 26. 5 3 5 2 3 10 3 10 313 26 13 2 26 26 26 26z z z z⋅ ++ = + = + =

⋅

43. The LCD of 6 and 7 is 42. 5 3 5 7 3 6 35 18 536 7 6 7 7 6 42 42 42

⋅ ⋅− − = − − = − − = −

⋅ ⋅

44. The LCD of 2 and 29 is 58. 1 3 1 29 3 2 29 6 352 29 2 29 29 2 58 58 58

⋅ ⋅− − = − − = − − = −

⋅ ⋅

45. The LCD of 2, 4, and 16 is 16. 2 8 4 2

2 4 16 2 8 4 4 168 4 216 16 1614162 72 8

78

x x x x x x

x x x

x

x

x

⋅ ⋅+ + = + +

⋅ ⋅

= + +

=

⋅=

⋅

=

46. The LCD of 4, 8, and 16 is 16. 2 4 2 2

4 8 16 4 4 8 2 164 2 216 16 1681688 2

2

z z z z z z

z z z

z

z

z

⋅ ⋅+ + = + +

⋅ ⋅

= + +

=

⋅=

⋅

=

47. The LCD of 9 and 6 is 18. 7 1 7 2 1 3 14 3 119 6 9 2 6 3 18 18 18

⋅ ⋅− = − = − =

⋅ ⋅

48. The LCD of 16 and 8 is 16. 9 3 9 3 2 9 6 3

16 8 16 8 2 16 16 16⋅

− = − = − =⋅

49. The LCD of 3 and 13 is 39. 2 6 2 13 6 33 13 3 13 13 3

26 1839 39

4439

a a a a

a a

a

⋅ ⋅+ = +

⋅ ⋅

= +

=

50. The LCD of 4 and 7 is 28. 3 3 7 4 21 4 254 7 4 7 7 4 28 28 28y y y y y y y⋅ ⋅+ = + = + =

⋅ ⋅

51. The LCD of 30 and 12 is 60. 7 5 7 2 5 5 14 25 1130 12 30 2 12 5 60 60 60

⋅ ⋅− = − = − = −

⋅ ⋅

52. The LCD of 30 and 20 is 60. 7 3 7 2 3 330 20 30 2 20 3

14 960 605

601 5

12 51

12

⋅ ⋅− = −

⋅ ⋅

= −

=

⋅=

⋅

=



172

53. The LCD of 9 and y is 9y. 5 1 5 1 9 5 9 5 99 9 9 9 9 9

y y yy y y y y y

⋅ ⋅ ++ = + = + =

⋅ ⋅

54. The LCD of 12 and x is 12x. 1 5 1 5 12

12 12 1260

12 1260

12

xx x x

xx x

xx

⋅ ⋅− = −

⋅ ⋅

= −

−=

55. The LCD of 5, 4, and 2 is 20. 6 3 1 6 4 3 5 1 105 4 2 5 4 4 5 2 10

24 15 1020 20 201920

⋅ ⋅ ⋅− + = − +

⋅ ⋅ ⋅

= − +

=

56. The LCD of 5, 4, and 2 is 20. 6 3 1 6 4 3 5 1 105 4 2 5 4 4 5 2 10

24 15 1020 20 202920

⋅ ⋅ ⋅+ − = + −

⋅ ⋅ ⋅

= + −

=

57. The LCD of 5 and 9 is 45. 4 4 4 9 4 5 36 20 565 9 5 9 9 5 45 45 45

⋅ ⋅+ = + = + =

⋅ ⋅

58. The LCD of 12 and 24 is 24. 11 7 11 2 7 22 7 15 3 5 512 24 12 2 24 24 24 24 3 8 8

⋅ ⋅− = − = − = = =

⋅ ⋅

59. The LCD of 9x and 8 is 72x. 5 1 5 8 1 9

9 8 9 8 8 940 9

72 7240 9

72

xx x x

xx xxx

⋅ ⋅+ = +

⋅ ⋅

= +

+=

60. The LCD of 8 and 12x is 24x. 3 5 3 3 5 28 12 8 3 12 2

9 1024 249 10

24

xx x x

xx xxx

⋅ ⋅+ = +

⋅ ⋅

= +

+=

61. The LCD of 12, 24, and 6 is 24. 9 17 1 9 2 17 1 4

12 24 6 12 2 24 6 418 17 424 24 24524

⋅ ⋅− + − = − + −

⋅ ⋅

= − + −

= −

62. The LCD of 14, 7, and 2 is 14. 5 3 1 5 3 2 1 7

14 7 2 14 7 2 2 75 6 7

14 14 146

143 27 237

⋅ ⋅− + − = − + −

⋅ ⋅

= − + −

= −

⋅= −

⋅

= −

63. The LCD of 8, 7, and 14 is 56. 3 2 5 3 7 2 8 5 48 7 14 8 7 7 8 14 4

21 16 2056 56 5637 20

56

x x x x

x x

x

⋅ ⋅ ⋅+ − = + −

⋅ ⋅ ⋅

= + −

−=

64. The LCD of 10, 2, and 5 is 10. 9 1 9 1 5 210 2 5 10 2 5 5 2

9 5 210 10 1011 5

10

x x x x

x x

x

⋅ ⋅− + = − +

⋅ ⋅

= − +

−=



173


⋅= =

⋅

3 3 7 2110 10 7 70

⋅= =

⋅

Since 20 < 21, 20 21 ,70 70

< so 2 3 .7 10<


⋅= =

⋅

6 6 9 5411 11 9 99

⋅= =

⋅

Since 55 > 54, 55 54 ,99 99

> so 5 6 .9 11>

67. A positive fraction is greater than a negative fraction. 5 136 15> −


7 7 6 428 8 6 48

⋅− = − = −

⋅

5 5 8 406 6 8 48

⋅− = − = −

⋅

Since −42 < −40, 42 40 ,48 48

− < − so 7 5 .8 6

− < −


3 3 7 214 4 7 28

⋅− = − = −

⋅

11 11 2 2214 14 2 28

⋅− = − = −

⋅

Since −21 > −22, 21 22 ,28 28

− > − so 3 11.4 14

− > −


2 2 13 269 9 13 117

⋅− = − = −

⋅

3 3 9 2713 13 9 117

⋅− = − = −

⋅

Since −26 > −27, 26 27 ,

117 117− > − so

2 3 .9 13

− > −

71. The LCD of 3 and 4 is 12. 1 3 1 4 3 3 4 9 133 4 3 4 4 3 12 12 12

x y ⋅ ⋅+ = + = + = + =

⋅ ⋅

72. The LCD of 3 and 4 is 12. 1 3 1 4 3 3 4 9 53 4 3 4 4 3 12 12 12

x y ⋅ ⋅− = − = − = − = −

⋅ ⋅

73. 1 3 1 3 13 4 3 4 4

xy ⋅= ⋅ = =

⋅

74. 1 3 1 4 1 4 43 4 3 3 3 3 9

x y ⋅÷ = ÷ = ⋅ = =

⋅

75.

3 12 24 3

6 14 32 3 12 2 33 12 33 3 1 22 3 3 29 26 6116

y x ⎛ ⎞+ = +⎜ ⎟⎝ ⎠

= +

⋅= +

⋅

= +

⋅ ⋅= +

⋅ ⋅

= +

=

76. 1 32 23 4

2 33 42 4 3 33 4 4 38 9

12 121712

x y ⎛ ⎞+ = +⎜ ⎟⎝ ⎠

= +

⋅ ⋅= +

⋅ ⋅

= +

=



174

77. The LCD of 3 and 5 is 15. 4 1 4 1 4 3 1 5 4 3 1 55 3 5 3 5 3 3 5 5 3 3 5

12 5 12 515 15 15 153415

+ + + = ⋅ + ⋅ + ⋅ + ⋅

= + + +

=


or 42

15 centimeters.

78. The LCD of 8 and 2 is 8. 3 5 1 3 5 1 48 8 2 8 8 2 4

3 5 48 8 81284 34 23 1 or 12 2

⋅+ + = + +

⋅

= + +

=

⋅=

⋅

=

The perimeter is 32

miles or 112

miles.

79. The LCD of 4, 5, and 2 is 20. 1 1 1 3 1 5 1 4 1 10 3 54 5 2 4 4 5 5 4 2 10 4 5

5 4 10 1520 20 20 20342017 210 217 7 or 110 10

⋅ ⋅ ⋅ ⋅+ + + = + + +

⋅ ⋅ ⋅ ⋅

= + + +

=

⋅=

⋅

=


meters or 71

10 meters.

80. The LCD of 21 and 7 is 21. 10 1 10 1 10 1 3 10 1 321 7 21 7 21 7 3 21 7 3

10 3 10 321 21 21 2126 5 or 121 21

⋅ ⋅+ + + = + + +

⋅ ⋅

= + + +

=


yards or 5121

yards.

81. The sum of a number and 12

translates as 1 .2

x +

82. A number increased by 25

− translates as

2 .5

x ⎛ ⎞+ −⎜ ⎟⎝ ⎠

83. A number subtracted from 38

− translates as

3 .8x− −

84. The difference of a number and 720

translates as

7 .20

x −

85. 3 3 16 3 3 10 51

16 16 16 16 16 16 8− − = − − = =

The inner diameter is 5 inch.8

86. 3 1 1 3 2 1 14 8 8 4 2 8 8

6 1 18 8 84812

⋅− − = − −

⋅

= − −

=

=

The inner diameter is 12

inch.

87. The LCD of 10 and 100 is 100. 17 1 17 1 10 17 10 7100 10 100 10 10 100 100 100

⋅− = − = − =

⋅

The sloth can travel 7

100 mph faster in trees.

88. The LCD of 4 and 264 is 264. 1 5 1 66 5 66 5 614 264 4 66 264 264 264 264

⋅− = − = − =

⋅

A killer bee will chase a person 61

264 mile

farther than a domestic honeybee.



175

89. The LCD of 2, 16, and 32 is 32. 1 11 9 1 16 11 2 92 16 32 2 16 16 2 32

16 22 932 32 324732

+ + = ⋅ + ⋅ +

= + +

=

The total length is 47 inches.32

90. The LCD of 16 and 8 is 16. 11 5 11 11 5 2 1116 8 16 16 8 2 16

11 10 1116 16 1632162

+ + = + ⋅ +

= + +

=

=

The total width is 2 inches.

91. The LCD of 20 and 25 is 100. 13 4 13 5 4 4 65 16 4920 25 20 5 25 4 100 100 100

⋅ ⋅− = − = − =

⋅ ⋅

49100

of the American students ages 10 to 17

name math or science as their favorite subject in school.

92. The LCD of 100 and 50 is 100. 49 3 49 3 2 49 6 43

100 50 100 50 2 100 100 100⋅

− = − = − =⋅

43100

of the world’s mail is handled by the postal

service of the United States.

93. The LCD of 50 and 2 is 50. 13 1 13 1 2550 2 50 2 25

13 2550 5038502 192 251925

+ = + ⋅

= +

=

⋅=

⋅

=

The Pacific and Atlantic Oceans account for 1925

of the world’s water surface.

94. The LCD of 25 and 5 is 25. 1 1 1 1 5 1 5 625 5 25 5 5 25 25 25

+ = + ⋅ = + =

The Arctic and Indian Oceans account for 625

of

the world’s water surface.

95. The LCD of 25 and 10 is 50. 3 1 3 2 1 5 6 5 1125 10 25 2 10 5 50 50 50

⋅ ⋅+ = + = + =

⋅ ⋅

1150

of the beginning freshmen plan to major in

arts and humanities or education.

96. The LCD of 100 and 20 is 100. 3 3 3 3 5

100 20 100 20 53 15

100 10018

1002 92 50950

⋅+ = +

⋅

= +

=

⋅=

⋅

=

950

of the beginning freshmen plan to major in a

technical or professional field.

97. 4 25 4 21125 25 25 25

− = − =

2125

of the beginning freshmen do not plan to

major in business.

98. 1 10 1 91

10 10 10 10− = − =

910

of the beginning freshmen do not plan to

major in engineering.

99. −50 ÷ 5 ⋅ 2 = −10 ⋅ 2 = −20

100. 8 − 6 ⋅ 4 − 7 = 8 − 24 − 7 = −16 − 7 = −23

101. (8 − 6) ⋅ (4 − 7) = 2 ⋅ (−3) = −6

102. 50 ÷ (5 ⋅ 2) = 50 ÷ 10 = 5



176

103. a.

b. There seems to be an error.

c. 3 4 3 4 75 5 5 5

++ = =

14 7or not 10 10

⎛ ⎞⎜ ⎟⎝ ⎠

104. a.

b. There seems to be an error.

c. The LCD of 8 and 4 is 8. 5 3 5 3 2 5 6 18 4 8 4 2 8 8 8

⋅− = − = − = −

⋅

2not 4

⎛ ⎞⎜ ⎟⎝ ⎠

105. The LCD of 3, 4, and 540 is 540. 2 1 2 2 180 1 135 23 4 540 3 180 4 135 540

360 135 2540 540 540225 2540 540223540

− − = ⋅ − ⋅ −

= − −

= −

=

106. The LCD of 10, 200, and 3 is 600. 9 7 1 9 60 7 3 1 200

10 200 3 10 60 200 3 3 200540 21 200600 600 600519 200600 600319600

− − = ⋅ − ⋅ − ⋅

= − −

= −

=

107. The LCD of 55 and 1760 is 1760. 30 1000 30 32 100055 1760 55 32 1760

960 10001760 17601960176049 4044 404944

⋅+ = +

⋅

= +

=

⋅=

⋅

=

108. The LCD of 26 and 1352 is 1352. 19 968 19 52 96826 1352 26 52 1352

988 9681352 1352

201352

5 4338 4

5338

⋅− = −

⋅

= −

=

⋅=

⋅

=




⋅= =

⋅

51106

Since 51 > 48, 51 48 ,

106 106> so

51 24 .106 53

>

Standard mail accounted for the greater portion of the mail handled by volume.

Integrated Review


2. Each part is 14


shaded, or 1 whole and 1 more part: 5 1or 14 4



177

3. number that get fewer than 8 73total number of people 85

→→

7385

of people get fewer than 8 hours of sleep

each night.

4.

5. 11 111

= −−

6. 17 171=

7. 0 03=

−

8. 7 is undefined0

9. 65 = 5 ⋅ 13

10. 70 2 35

2 5 7

= ⋅↓ ↓⋅ ⋅

70 = 2 ⋅ 5 ⋅ 7

11. 315 3 105

3 3 35

3 3 5 7

= ⋅↓ ↓⋅ ⋅

↓ ↓ ↓⋅ ⋅ ⋅

2315 3 5 7= ⋅ ⋅

12. 441 3 147

3 3 49

3 3 7 7

= ⋅↓ ↓⋅ ⋅

↓ ↓ ↓⋅ ⋅ ⋅

2 2441 3 7= ⋅

13. 2 2 1 1

14 2 7 7⋅

= =⋅

14. 24 6 4 620 5 4 5

⋅= =

⋅

15. 56 14 4 1460 15 4 15

⋅− = − = −

⋅

16. 72 8 9 980 8 10 10

⋅− = − = −

⋅

17. 54 27 2 2135 27 5 5x x x⋅ ⋅= =

⋅

18. 90 30 3 3

240 30 8 8y y y⋅

= =⋅ ⋅

19. 3 2165 15 11 11 11

210 15 14 14 14z z z z z z zz z

⋅ ⋅ ⋅ ⋅ ⋅ ⋅= = =

⋅ ⋅

20. 2 3

2

245 35 735 11385

711

711

ab a ba a b b ba b

a b b

ab

⋅ ⋅ ⋅=

⋅ ⋅ ⋅ ⋅ ⋅ ⋅

=⋅ ⋅ ⋅

=

21. Not equivalent, since the cross products are not equal: 7 ⋅ 10 = 70, 8 ⋅ 9 = 72

22. Equivalent, since the cross products are equal: 10 ⋅ 18 = 180, 12 ⋅ 15 = 180

23. a. number not adjacent 2total number 50

→→

2 1 2 150 25 2 25

⋅= =

⋅ of the states are not

adjacent to any other states.

b. 50 − 2 = 48; 48 states are adjacent to other states.

c. 48 2 24 2450 2 25 25

⋅= =

⋅ of the states are adjacent

to other states.

24. a. number rated R 92total number 460

→→

92 92 1 1460 92 5 5

⋅= =

⋅ of the new films were

rated R.

b. 460 − 92 = 368; 368 of the new films were rated other than R.

c. 368 4 92 4460 5 92 5

⋅= =

⋅ of the new films were

rated other than R.



178

25. 5 = 5 6 = 2 ⋅ 3 LCM = 2 ⋅ 3 ⋅ 5 = 30

26. 2 = 2 14 = 2 ⋅ 7 LCM = 2 ⋅ 7 = 14

27. 6 = 2 ⋅ 3 18 = 2 ⋅ 3 ⋅ 3 30 = 2 ⋅ 3 ⋅ 3 ⋅ 5 LCM = 2 ⋅ 3 ⋅ 3 ⋅ 5 = 90

28. 7 7 4 7 4 289 9 4 9 4 36

⋅= ⋅ = =

⋅

29. 11 11 5 11 5 5515 15 5 15 5 75

⋅= ⋅ = =

⋅

30. 5 5 8 5 8 406 6 8 6 8 48

⋅= ⋅ = =

⋅

31. 1 3 1 3 45 5 5 5

++ = =

32. 1 3 1 3 2 25 5 5 5 5

− −− = = = −

33. 1 3 1 3 35 5 5 5 25

⋅⋅ = =

⋅

34. 1 3 1 5 1 5 15 5 5 3 5 3 3

⋅÷ = ⋅ = =

⋅

35. 2 5 2 6 2 6 2 2 3 2 2 43 6 3 5 3 5 3 5 5 5

⋅ ⋅ ⋅ ⋅÷ = ⋅ = = = =

⋅ ⋅

36. 2 5 2 5 2 5 5 53 6 3 6 3 2 3 3 3 9a a a

a a a⋅ ⋅ ⋅

⋅ = = = =⋅ ⋅ ⋅ ⋅ ⋅

37. The LCD of 3y and 6y is 6y. 2 5 2 2 5

3 6 3 2 62 2 5

3 2 64 5

6 61

6

y y y y

y y

y y

y

− = ⋅ −

⋅= −

⋅

= −

= −

38. The LCD of 3 and 6 is 6. 2 5 2 2 53 6 3 2 6

2 2 53 2 6

4 56 6

96

3 33 2

32

x x x x

x x

x x

x

x

x

+ = ⋅ +

⋅= +

⋅

= +

=

⋅ ⋅=

⋅

=

39. 1 7 1 7 17 18 7 18 18

⋅− ⋅ − = =

⋅

40. 4 3 4 3 4 3 4 49 7 9 7 3 3 7 3 7 21

⋅ ⋅− ⋅ − = = = =

⋅ ⋅ ⋅ ⋅

41. 2

2

2

2

7 7 668 8 1

7 18 67 1

8 67 1

8 67

48

z z zz

zz

zzzz z

z

− ÷ = − ÷

= − ⋅

⋅= −

⋅⋅ ⋅

= −⋅ ⋅ ⋅

= −

42. 9 9 5 9 1 9 1 95

10 10 1 10 5 10 5 50⋅

− ÷ = − ÷ = − ⋅ = − = −⋅

43. The LCD of 8 and 20 is 40. 7 1 7 5 1 2 35 2 378 20 8 5 20 2 40 40 40

⋅ ⋅+ = + = + =

⋅ ⋅

44. The LCD of 12 and 9 is 36. 5 1 5 3 1 4 15 4 11

12 9 12 3 9 4 36 36 36⋅ ⋅

− = − = − =⋅ ⋅

45. The LCD of 9, 18, and 3 is 18. 2 1 1 2 2 1 1 69 18 3 9 2 18 3 6

4 1 618 18 181118

⋅ ⋅+ + = + +

⋅ ⋅

= + +

=



179

46. The LCD of 10, 5, and 25 is 50. 3 6 3 5 10 6 210 5 25 10 5 5 10 25 2

15 10 1250 50 5025 1250 50

25 1250

y y y y

y y

y

y

⋅ ⋅ ⋅+ + = + +

⋅ ⋅ ⋅

= + +

= +

+=

47. 23

of a number translates as 23x⋅ or

2 .3x

48. The quotient of a number and 15

− translates as

1 .5

x ⎛ ⎞÷ −⎜ ⎟⎝ ⎠

49. A number subtracted from 89

− translates as

8 .9x− −

50. 6

11 increased by a number translates as

6 .11

x+

51. 2 2 153015303 3 1

2 15303 1

2 3 5103 1

2 5101

1020

⋅ = ⋅

⋅=

⋅⋅ ⋅

=⋅

⋅=

=

23

of 1530 is 1020.

52. 3 18 3184 1 4

18 41 3

18 41 33 6 4

1 36 41

24

÷ = ÷

= ⋅

⋅=

⋅⋅ ⋅

=⋅⋅

=

=

He can sell 24 lots.

53. The LCD of 8 and 16 is 16. 7 1 1 7 2 1 18 16 16 8 2 16 16

14 1 116 16 1612164 34 434

⋅− − = − −

⋅

= − −

=

⋅=

⋅

=

The inner diameter is 34

foot.

Section 4.6

Practice Problems

1. 71015

7 1 7 5 7 5 7 5 710 5 10 1 10 1 2 5 1 2

yy y y y y⋅ ⋅ ⋅

= ÷ = ⋅ = = =⋅ ⋅ ⋅

2.

131 1 12 6 2 3 63 3 32 2 44 3 4 3 3 4

3 16 69 8

12 12461

124 16 124 126 14 2 6

6 18 or 81

⋅⋅⋅ ⋅⋅ ⋅

+ +=

− −

+=

−

=

= ÷

= ⋅

⋅ ⋅=

⋅

=

3. The LCD of 2, 6, 4, and 3 is 12.

( )( )

1 11 12 62 6

3 2 3 24 3 4 3

1 12 63 24 3

12

12

12 12

12 126 29 88 or 81

++=

− −

⋅ + ⋅=

⋅ − ⋅+

=−

=



180

4. The LCD of 4, 5, and 1 is 20.

( )( )

33 344 4

5 55

20 20 154 201 20 20 120 1x xx x

⋅= = =

−− ⋅ − ⋅−

5. 32 8 8 54 462 2

3 27 27 27 27⎛ ⎞ − = − = − = −⎜ ⎟⎝ ⎠

6. 1 1 7 1 1 5 1 2 7 12 5 8 8 2 5 5 2 8 8

5 2 7 110 10 8 83 8

10 83 (1)

103

10

⋅ ⋅⎛ ⎞⎛ ⎞ ⎛ ⎞⎛ ⎞− + + = − + +⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⋅ ⋅⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞= − + +⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞= −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⎛ ⎞= −⎜ ⎟⎝ ⎠

= −

7. 3 3 3 25 5 10 3

3 3 25 5 2 33 15 545

xy− − = − − ⋅

⋅= − −

⋅ ⋅

= − −

= −


1. A fraction whose numerator or denominator or both numerator and denominator contain fractions is called a complex fraction.

2. To simplify 1 2 7 ,2 3 8

− + ⋅ which operation do we

perform first? multiplication

3. To simplify 1 2 7 ,2 3 8

− ÷ ⋅ which operation do we

perform first? division

4. To simplify 7 1 2 ,8 2 3⎛ ⎞⋅ −⎜ ⎟⎝ ⎠

which operation do we

perform first? subtraction

5. To simplify 21 1 9 3 ,

3 4 11 8⎛ ⎞÷ ⋅ +⎜ ⎟⎝ ⎠

which operation

do we perform first? addition

6. To simplify 239 ,

4⎛ ⎞− −⎜ ⎟⎝ ⎠

which operation do we

perform first? evaluate the exponential expression

Exercise Set 4.6

1. 1834

1 3 1 4 1 4 1 4 18 4 8 3 8 3 2 4 3 6

⋅ ⋅= ÷ = ⋅ = = =

⋅ ⋅ ⋅

2. 5

121512

5 15 5 12 5 12 1 5 12 112 12 12 15 12 15 12 5 3 3

⋅ ⋅ ⋅= ÷ = ⋅ = = =

⋅ ⋅ ⋅

3. 2327

2 2 2 7 2 7 73 7 3 2 3 2 3

⋅= ÷ = ⋅ = =

⋅

4. 925625

9 625 259 2525 69 2525 63 3 2525 2 332

= ÷

= ⋅

⋅=

⋅⋅ ⋅

=⋅ ⋅

=

5. 22749

2 4 2 9 2 9 2 927 9 27 4 27 4 3 9 2 2 6

x x x x x x⋅ ⋅ ⋅= ÷ = ⋅ = = =

⋅ ⋅ ⋅ ⋅

6. 31112

3 1 3 2 3 2 611 2 11 1 11 1 11

yy y y y⋅

= ÷ = ⋅ = =⋅


( )( )

3 23 24 54 5

31 312 5 2 5

3 24 5

312 5

20

20

20 20

20 2015 810 122322

⋅ ++=

+ ⋅ +

⋅ + ⋅=

⋅ + ⋅+

=+

=



181

8. The LCD of 6, 3, 2, and 9 is 18.

( )( )

7 27 26 36 3

3 8 3 82 9 2 9

7 26 33 82 9

18

18

18 18

18 1821 1227 1633113

⋅ ++=

− ⋅ −

⋅ + ⋅=

⋅ − ⋅+

=−

=

=


( )( )

3344

1 18 8

18

8

5 8 56

8 5 86

40 16392 33 13

213

xx

x

x

x

x

x

⋅=

− ⋅ −

=⋅ − ⋅

=−

=

⋅ ⋅=

⋅

=

10. The LCD of 10, 1, and 5y is 10y.

( )331010

2 25 5

310

10 22

10

10 10 2

43 20

423

4

y yy

y

y

y y

y y

y

⋅ ++=

⋅

⋅ + ⋅=

+=

=

11. 1 1 1 1 1 15 3 4 5 3 4

1 15 121 12 1 55 12 12 512 560 601760

⋅+ ⋅ = +

⋅

= +

⋅ ⋅= +

⋅ ⋅

= +

=

12.

1 1 1 1 1 12 6 3 2 6 3

1 12 181 9 12 9 189 1

18 1810182 52 959

⋅+ ⋅ = +

⋅

= +

⋅= +

⋅

= +

=

⋅=

⋅

=

13. 5 1 1 5 3 1 5 3 1 5 56 3 4 6 1 4 2 3 1 2 2 2 2 2 8

⋅ ⋅÷ ⋅ = ⋅ ⋅ = = =

⋅ ⋅ ⋅ ⋅ ⋅ ⋅

14. 7 1 1 7 4 1 7 4 1 18 4 7 8 1 7 4 2 1 7 2

⋅ ⋅÷ ⋅ = ⋅ ⋅ = =

⋅ ⋅ ⋅

15. 2

2 1 12 43 9

4 11 94 9 11 9 936 19 9

359

⎛ ⎞− = −⎜ ⎟⎝ ⎠

= −

⋅= −

⋅

= −

=

16. 2

2 1 13 92 4

9 11 49 4 11 4 436 14 4

354

⎛ ⎞− = −⎜ ⎟⎝ ⎠

= −

⋅= −

⋅

= −

=



182

17.

2 4 1 9 2 4 1 10 9 39 9 3 10 9 3 10 10 3

6 10 279 30 306 179 306 179 30

6 173 3 5 6

173 3 51745

+⎛ ⎞⎛ ⎞ ⎛ ⎞⎛ ⎞+ − = ⋅ − ⋅⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠

⎛ ⎞⎛ ⎞= −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

−⎛ ⎞⎛ ⎞= ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

⋅= −

⋅⋅

= −⋅ ⋅ ⋅

= −⋅ ⋅

= −

18. 1 1 1 1 1 2 1 1 2 15 10 5 10 5 2 10 5 2 10

2 1 2 110 10 10 101 3

10 101 3

10 103

100

⎛ ⎞⎛ ⎞ ⎛ ⎞⎛ ⎞− + = ⋅ − ⋅ +⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠

⎛ ⎞⎛ ⎞= − +⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞= ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⋅

=⋅

=

19.

7 1 3 7 1 4 38 2 11 8 2 4 11

7 4 38 8 11

3 38 113 118 33 118 3

118

⎛ ⎞ ⎛ ⎞− ÷ = − ⋅ ÷⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞= − ÷⎜ ⎟⎝ ⎠

= ÷

= ⋅

⋅=

⋅

=

20. 2 7 4 9 43 3 9 3 9

3 41 93 91 43 91 4274

⎛ ⎞− − ÷ = − ÷⎜ ⎟⎝ ⎠

= − ÷

= − ⋅

⋅= −

⋅

= −

21.

1 1 1 5 1 42 2 2 24 5 4 5 5 4

5 42 220 2092 220

2 9 21 20

2 9 21 2 109 2 10

10 1 109 20

10 102910

⋅ ⋅⎛ ⎞ ⎛ ⎞⋅ + + = ⋅ + +⎜ ⎟ ⎜ ⎟⋅ ⋅⎝ ⎠ ⎝ ⎠⎛ ⎞= + +⎜ ⎟⎝ ⎠⎛ ⎞= +⎜ ⎟⎝ ⎠

= ⋅ +

⋅= +

⋅ ⋅⋅

= +⋅

= +

=

22.

2 1 2 5 2 15 1 15 2 5 1 2 2

2 10 1 15 2 22 9 15 22 9 15 29 55 545

⋅⎛ ⎞ ⎛ ⎞⋅ − − = ⋅ − −⎜ ⎟ ⎜ ⎟⋅⎝ ⎠ ⎝ ⎠⎛ ⎞= ⋅ − −⎜ ⎟⎝ ⎠⎛ ⎞= ⋅ −⎜ ⎟⎝ ⎠⋅

= −⋅

= −

=

23.

2 2

2

2

3 3 1 3 3 3 14 4 12 4 4 3 12

3 9 14 12 123 84 12

9 816 129 12

16 89 3 44 4 82732

⋅⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞÷ − = ÷ −⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⋅⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞= ÷ −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞= ÷⎜ ⎟⎝ ⎠

= ÷

= ⋅

⋅ ⋅=

⋅ ⋅

=



183

24.

2 2

2

2

8 2 8 2 3 229 3 9 1 3 3

8 6 29 3 38 49 3

64 481 364 381 416 4 327 3 41627

⋅⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞÷ − = ÷ −⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⋅⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞= ÷ −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞= ÷⎜ ⎟⎝ ⎠

= ÷

= ⋅

⋅ ⋅=

⋅ ⋅

=

25. 2 2

2

2

2 3 2 2 35 10 5 2 10

4 310 101

101 1

10 101

100

⎛ ⎞ ⎛ ⎞− = ⋅ −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞= −⎜ ⎟⎝ ⎠

⎛ ⎞= ⎜ ⎟⎝ ⎠

= ⋅

=

26. 3 3

3

3

3 4 3 3 4 22 3 2 3 3 2

9 86 6161 1 16 6 61

216

⋅ ⋅⎛ ⎞ ⎛ ⎞− = −⎜ ⎟ ⎜ ⎟⋅ ⋅⎝ ⎠ ⎝ ⎠

⎛ ⎞= −⎜ ⎟⎝ ⎠

⎛ ⎞= ⎜ ⎟⎝ ⎠⎛ ⎞ ⎛ ⎞ ⎛ ⎞= ⋅ ⋅⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

=

27.

2 2

2

2

3 1 1 1 3 2 1 1 4 14 8 2 8 4 2 8 2 4 8

6 1 4 18 8 8 87 58 8

49 564 849 5 864 8 849 4064 64964

⋅ ⋅⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ − + = + − +⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⋅ ⋅⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞= + − +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞= −⎜ ⎟⎝ ⎠

= −

⋅= −

⋅

= −

=

28.

3 2 3 2

3 2

3 2

3 2

1 1 2 3 1 1 2 2 36 3 5 4 6 3 2 5 4

1 2 66 6 203 66 201 32 10

1 98 1001 25 9 28 25 100 225 18

200 20043

200

⋅ ⋅⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ + ⋅ = + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⋅ ⋅⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞= + +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

= +

⋅ ⋅= +

⋅ ⋅

= +

=

29. 2 55 55 6526

2 6 51 6 612 56 676

y z ⎛ ⎞− = −⎜ ⎟⎝ ⎠

= −

= ⋅ −

= −

=

30. 5 1 5 1 62 2 26 3 3 3 3

z x ⎛ ⎞ ⎛ ⎞− = − − = + = =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠



184

31. 13

56

1 5 1 6 1 3 2 23 6 3 5 3 5 5

xz

− ⋅ ⋅= = − ÷ = − ⋅ = − = −

⋅

32. ( )

( )( )

2 15 3

562 15 3

56

30

3012 ( 10)

25225

y xz

+ −+=

⎡ ⎤+ −⎣ ⎦=

+ −=

=

33.

22 1 2 5

3 5 61 1 2 53 3 5 2 3

1 19 31 1 39 3 31 39 9

2929

x yz ⎛ ⎞ ⎛ ⎞⎛ ⎞− = − −⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠

⋅⎛ ⎞⎛ ⎞= − − −⎜ ⎟⎜ ⎟ ⋅ ⋅⎝ ⎠⎝ ⎠

= −

= − ⋅

= −

−=

= −

34. 1 5(1 )(1 ) 1 13 6

3 1 6 53 3 6 62 113 62 11

3 2 3119

x z ⎡ ⎤⎛ ⎞ ⎡ ⎤+ + = + − +⎜ ⎟⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎣ ⎦⎣ ⎦⎛ ⎞⎛ ⎞= − +⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞= ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⋅

=⋅ ⋅

=

35. 5241

12

5 1 5 12 5 12 524 12 24 1 12 2 1 2

a a a a a⋅ ⋅= ÷ = ⋅ = =

⋅ ⋅

36. 7

101425

7 1410 257 25

10 147 5 5

5 2 7 25

2 254

zz

z

z

z

z

= ÷

= ⋅

⋅ ⋅=

⋅ ⋅ ⋅ ⋅

=⋅ ⋅

=

37. 3 33 1 27 1 28 4 7 7

2 2 8 8 8 4 2 2⋅⎛ ⎞ ⎛ ⎞+ = + = = =⎜ ⎟ ⎜ ⎟ ⋅⎝ ⎠ ⎝ ⎠

38. 5 1 1 1 5 2 1 121 2 7 3 21 1 7 3

5 2 1 121 1 7 310 121 211121

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞÷ + ⋅ = ⋅ + ⋅⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⋅ ⋅= +

⋅ ⋅

= +

=

39. 21 1 1 1 1 5 1 4 5 4 9

2 5 4 5 4 5 5 4 20 20 20⋅ ⋅⎛ ⎞− + = + = + = + =⎜ ⎟ ⋅ ⋅⎝ ⎠

40. 23 3 9 3 9 3 2 9 6 15

4 8 16 8 16 8 2 16 16 16⋅⎛ ⎞− + = + = + = + =⎜ ⎟ ⋅⎝ ⎠

41. ( )( )

1166

4 43 3

1643

6 22

1 6 1

6 2 6

6 1 612 16 813

2132

++=

− −

⋅ + ⋅=

⋅ − ⋅+

=−

=−

= −

42. ( )( )

11 122 2

1 115 55

10 33 10 3 10 30 5 2540 2 424 10 4 1010 4

−− ⋅ − ⋅ −= = = =

++ ⋅ + ⋅+

43. 2 2 22 5 2 3 3 3 91

5 5 5 5 5 5 25⎛ ⎞ ⎛ ⎞ ⎛ ⎞− = − = = ⋅ =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠



185

44. 2 21 3 1 1 3 3

2 4 2 2 4 41 94 161 4 94 4 164 9

16 165

16

⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞⎛ ⎞− − = − − −⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠

= −

⋅= −

⋅

= −

= −

45. 3 1 1 3 4 1 414 8 2 4 4 8 8

1 54 8

1 54 85

32

⎛ ⎞⎛ ⎞ ⎛ ⎞⎛ ⎞− + = − +⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠

⎛ ⎞⎛ ⎞= −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

⋅= −

⋅

= −

46. 1 3 1 1 2 3 1 51

10 20 5 10 2 20 5 52 3 1 5

20 20 5 55 420 5

5 45 4 515

⋅⎛ ⎞⎛ ⎞ ⎛ ⎞⎛ ⎞+ − = + −⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⋅⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞= + −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞= −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

⋅= −

⋅ ⋅

= −

47. 4 4

4

2 7 99 9 9

( 1)( 1)( 1)( 1)( 1)1

⎛ ⎞ ⎛ ⎞− − = −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

= −= − − − −=

48. 2 2

2

2

5 2 5 2 39 3 9 3 3

5 69 9

191 19 9

181

⋅⎛ ⎞ ⎛ ⎞− = −⎜ ⎟ ⎜ ⎟⋅⎝ ⎠ ⎝ ⎠

⎛ ⎞= −⎜ ⎟⎝ ⎠

⎛ ⎞= −⎜ ⎟⎝ ⎠⎛ ⎞⎛ ⎞= − −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

=

49.

( )( )

51513 63 6

3 1 3 14 2 4 2

513 63 14 2

12

12

12 12

12 124 109 66

152 33 525

−−=

+ +

⋅ − ⋅=

⋅ + ⋅−

=+

−=

⋅= −

⋅

= −

50. ( )( )

7 17 110 210 2

34 345 4 5 4

7 110 2

345 4

20

20

20 20

20 2014 1016 152431

++=

+ +

⋅ + ⋅=

⋅ + ⋅+

=+

=

51.

3 6 3 6 3 5 3 64 5 4 5 4 6 4 5

3 5 3 64 6 4 53 5 3 2 3

4 2 3 2 2 55 98 105 5 9 48 5 10 425 3640 40

1140

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞÷ − ⋅ = ⋅ − ⋅⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⋅ ⋅= −

⋅ ⋅⋅ ⋅ ⋅

= −⋅ ⋅ ⋅ ⋅

= −

⋅ ⋅= −

⋅ ⋅

= −

= −



186

52.

1 2 1 2 1 2 1 72 7 2 7 2 7 2 2

1 2 1 72 7 2 22 7

14 42 2 7 7

14 2 4 74 4928 28

4528

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⋅ − ÷ = ⋅ − ⋅⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⋅ ⋅= −

⋅ ⋅

= −

⋅ ⋅= −

⋅ ⋅

= −

= −

53. ( )( )

331 13 3

313

3 22

5 3 5

3 3 2

3 5 36

15 16

16

xx

x

x

x

++=

+ +

⋅ + ⋅=

⋅ + ⋅+

=++

=

54. ( )( )

443 38 8

438

8 11

2 8 2

8 1 8

8 2 88 216 38 2

19

xx

x

x

x

⋅ −−=

+ ⋅ +

⋅ − ⋅=

⋅ + ⋅−

=+−

=

55. 1 3 2 1 6 1 7 13 or 32 1 2 2 2 2 2 2

+ = ⋅ + = + =

56. 2 2 3 2 6 2 8 22 or 23 1 3 3 3 3 3 3

+ = ⋅ + = + =

57. 5 9 6 5 54 5 49 19 or 86 1 6 6 6 6 6 6

− = ⋅ − = − =

58. 1 4 5 1 20 1 19 44 or 35 1 5 5 5 5 5 5

− = ⋅ − = − =

59. no; answers may vary

60. yes; answers may vary

61. ( )313 31 1

2 42 4 2 44 4 4 2 3 5

2 4(2) 4 2 8 8

++ ⋅ + ⋅ += = = =

⋅

62.

( )3 93 95 105 10

3 95 10

10

2 10(2)10 10

10 26 920

15205 35 434

++=

⋅ + ⋅=

⋅+

=

=

⋅=

⋅

=

63. ( )1 21 2 1 2

4 144 14 4 1428 28 28

2 28(2) 28 27 456

1156

++ ⋅ + ⋅+ =

⋅+

=

=

64. ( )5 75 7

6 96 9

5 76 9

18

2 18(2)18 18

18 215 14

362936

++=

⋅ + ⋅=

⋅+

=

=

65. The average of a and b should be halfway between a and b.

66. False; the average cannot be greater than the greatest number.

67. False; the average cannot be less than the smallest number.

68. True

69. False, if the absolute value of the negative fraction is greater than the absolute value of the positive fraction, the sum is negative.

70. True



187

71. True

72. addition; answers may vary

73. No; answers may vary.

74. Subtraction, multiplication, addition, division

75. Addition, multiplication, subtraction, division

76. Division, multiplication, subtraction, addition

77. Subtraction, multiplication, division, addition

78.

( )( )

( )

34

47

34

47

34

47

22

28 2

28

28 2 28

2856 21

167716

xy

++=

−

+=

−

⋅ + ⋅=

⋅ −

+=

−

= −

79.

3 44 44 7

4 3 41 4 712 44 7

12 7 4 44 7 7 484 1628 2868284 174 7

177

x y ⎛ ⎞+ = ⋅ + −⎜ ⎟⎝ ⎠⎛ ⎞= ⋅ + −⎜ ⎟⎝ ⎠

= −

⋅ ⋅= −

⋅ ⋅

= −

=

⋅=

⋅

=

80. 2

2 3 47 74 7

9 416 19 4 16

16 1 169 64

16 165516

x y ⎛ ⎞ ⎛ ⎞+ = + −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

= −

⋅= −

⋅

= −

= −

81.

( )( )

( )

9 914 14

3 44 7

914

3 44 7

3 44 7

28

2818

28 2818

21 16185

x y=

+ + −

=−

=⋅ − ⋅

=−

=

Section 4.7

Practice Problems

1.

112345 2 3 4 50

4q 2!Ωq

2. 2 11 5 11 5 11 11 21 or 13 15 3 15 3 5 3 9 9

⋅⋅ = ⋅ = =

⋅ ⋅

3. 5 5 18 5 18 5 6 3 1518 or 156 6 1 6 1 6 1 1

⋅ ⋅ ⋅⋅ = ⋅ = = =

⋅ ⋅

4. 1 3 16 11 16 11 4 4 11 44 43 2 or 85 4 5 4 5 4 5 4 5 5

⋅ ⋅ ⋅⋅ = ⋅ = = =

⋅ ⋅

Estimate: 135

rounds to 3, 324

rounds to 3.

3 ⋅ 3 = 9, so the answer is reasonable.

5. 7 3 97 3 97 97 23 6 or 19

15 1 15 1 5 3 5 5⋅

⋅ = ⋅ = =⋅ ⋅

Estimate: 76

15 rounds to 6 and 3 ⋅ 6 = 18, so the

answer is reasonable.



188

6. 4 4 7 4 1 4 1 479 9 1 9 7 9 7 63

⋅÷ = ÷ = ⋅ = =

⋅

7. 8 4 8 193

15 5 15 58 5

15 198 5

15 198 5

5 3 198

57

÷ = ÷

= ⋅

⋅=

⋅⋅

=⋅ ⋅

=

8. 2 3 23 313 27 14 7 14

23 147 3123 7 2

7 3146 15 or 131 31

÷ = ÷

= ⋅

⋅ ⋅=

⋅

=

9. 126245

+

52

301243017630

+

Estimate: 126

rounds to 2, 245

rounds to 4, and

2 + 4 = 6, so the answer is reasonable.

10. 53

14627

+

53

141221417514

+

3 35 1 614 14

= + =

11. 12637125

+

12303357235371735

+

2 217 1 1835 35

= + =

12. 7329516

18−

143218516

18916

18

−

1162

=

13. 79

15345

−

79

1594

15−

2281594

1513415

−

14. 252109

−

924921097149

−

15. 1234519

12−

323

12519

12−

152212519

1210312

−

536

=

The girth of the largest known American beech

tree is 536

feet larger than the girth of the largest

known sugar maple tree.

16. 1 2244 3 447 7

44 71 222 22 71 22

14 or 141

÷ = ÷

= ⋅

⋅ ⋅=

⋅

=

Therefore, 14 dresses can be made from 44 yards.

17. 3 7 9 3 6697 7 7

⋅ +− = − = −

18. 10 11 5 10 65511 11 11

⋅ +− = − = −



189

19. 37 548 8

− = − 4

8 3732

5

20. 46 195 5

− = − 9

5 4645

1

21. 3 3 11 182 34 5 4 5

11 184 5

11 2 92 2 599 9 or 910 10

⎛ ⎞ ⎛ ⎞⋅ − = ⋅ −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⋅= −

⋅⋅ ⋅

= −⋅ ⋅

= − −

22. 2 1 30 54 17 4 7 4

30 47 5

30 47 5

5 6 47 5

24 3 or 37 7

− ÷ = − ÷

= − ⋅

⋅= −

⋅⋅ ⋅

= −⋅

= − −

23. 3124263

−

912

1286

1216

12

−

2 3 16 12 63 4 12− = −

24. 297

113014

+

49

14113014153914

+

14014

=

2 11 19 30 407 14 14

− − = −


1. 5 28025

11 11=

2. 14 10196715 15

=

3. 31 377610735 35

=

4. 17 392318621 21

=

5. 365 12614 14

=

6. 290 42213 13

=

7. 2769 39230 10

=

8. 3941 1423117 17

=


1. The number 354

is called a mixed number.

2. For 35 ,4

the 5 is called the whole number part

and 34

is called the fraction part.

3. To estimate operations on mixed numbers, we round mixed numbers to the nearest whole number.

4. The mixed number 528

written as an improper

fraction is 21.8

Exercise Set 4.7

1.

2s a √

112345 2 3 4 50



190

2. 1~ 3~~

112345 2 3 4 50

3. 3R 1aa

112345 2 3 4 50

4. 4a ≈Ï

112345 2 3 4 50

5. 11212

rounds to 3.

114

rounds to 1.

3 ⋅ 1 = 3 The best estimate is b.

6. 156

rounds to 5.

537

rounds to 4.

5 ⋅ 4 = 20 The best estimate is d.

7. 212

11 rounds to 12.

9310

rounds to 4.

12 ÷ 4 = 3 The best estimate is a.

8. 320

14 rounds to 20.

8411

rounds to 5.

20 ÷ 5 = 4 The best estimate is c.

9. 2 1 8 1 823 7 3 7 21⋅ = ⋅ =

10. 5 1 5 21 5 3 7 7 14 or 29 5 9 5 3 3 5 3 3

⋅ ⋅⋅ = ⋅ = =

⋅ ⋅

11. 3 7 8 7 5 7 5 35 37 1 or 45 1 5 1 8 1 8 8 8

⋅÷ = ÷ = ⋅ = =

⋅

12. 2 9 5 9 3 9 3 27 29 1 or 53 1 3 1 5 1 5 5 5

⋅÷ = ÷ = ⋅ = =

⋅

13. Exact: 1 1 11 7 77 72 3 or 75 2 5 2 10 10⋅ = ⋅ =

Estimate: 125

rounds to 2, 132

rounds to 4.

2 ⋅ 4 = 8 so the answer is reasonable.

14. Exact: 1 1 9 57 513 12 7 or 164 8 4 8 32 32⋅ = ⋅ =

Estimate: 124

rounds to 2, 178

rounds to 7.

2 ⋅ 7 = 14

15. Exact: 4 2 19 44 836 313 6 or 235 7 5 7 35 35⋅ = ⋅ =

Estimate: 435

rounds to 4, 267

rounds to 6.

4 ⋅ 6 = 24

16. Exact: 5 3 35 38 7 5 2 19 133 15 7 or 446 5 6 5 2 3 5 3 3

⋅ ⋅ ⋅⋅ = ⋅ = =

⋅ ⋅

Estimate: 556

rounds to 6, 375

rounds to 8.

6 ⋅ 8 = 48

17. 1 5 5 25 15 2 or 122 1 2 2 2

⋅ = ⋅ =

18. 1 6 10 2 3 10 206 3 203 1 3 1 3 1

⋅ ⋅⋅ = ⋅ = = =

⋅

19. 2 1 11 3 11 3 11 13 1 or 53 2 3 2 3 2 2 2

⋅⋅ = ⋅ = =

⋅

20. 4 5 14 21 2 7 21 147 72 2 or 75 8 5 8 5 2 4 20 20

⋅ ⋅⋅ = ⋅ = =

⋅ ⋅

21. 2 1 8 7 56 22 or 183 7 3 1 3 3÷ = ⋅ =

22. 5 1 5 21 5 5 5 5 2549 5 9 5 9 21 9 21 189

⋅÷ = ÷ = ⋅ = =

⋅



191

23. 738

rounds to 4.

125

rounds to 2.

4 + 2 = 6 The best estimate is a.

24. 738

rounds to 4.

125

rounds to 2.

4 − 2 = 2 The best estimate is d.

25. 183

rounds to 8.

112

rounds to 2.

8 + 2 = 10 The best estimate is b.

26. 183

rounds to 8.

112

rounds to 2.

8 − 2 = 6 The best estimate is c.

27. Exact: 74

1212

128 26 6

12 3

+

=

Estimate: 74

12 rounds to 5,

1212

rounds to 2.

5 + 2 = 7 so the answer is reasonable.

28. Exact: 47

1123

11610

11

+

Estimate: 47

11 rounds to 7,

2311

rounds to 3.


29. Exact: 310

14437

+

310

1483

14111314

+

Estimate: 310

14 rounds to 10,

437

rounds to 4.


30. Exact: 512

12146

+

512

1224

12716

12

+

Estimate: 512

12 rounds to 12,

146

rounds to 4.


31. 1952825

+

5925282571725

+

32. 26

137826

+

46

267826111426

+

33. 312

1410

52512

+

181284

10352584534784

+

34. 289

3210921

+

14863

3230963444963

+



192

35. 4157

11914

+

815

141191419 5 524 24 1 2514 14 14

+

= + =

36. 323588

15+

923

1588

1517 2 231 31 1 3215 15 15

+

= + =

37. 538126374

+

1532442

241872437 13 1312 12 1 1324 24 24

+

= + =

38. 143295136

+

104301293053

3027 916 1630 10

+

=

39. Exact: 74

1012

10−

6 32 2

10 5=

Estimate: 74

10 rounds to 5,

1210

rounds to 2.

5 − 2 = 3 so the answer is reasonable.

40. Exact: 479239249

−

Estimate: 479

rounds to 7, 239

rounds to 3.


41. Exact: 131014437

−

13101483

14−

57

14

Estimate: 131014

rounds to 11, 437

rounds to 4.


42. Exact: 512

12146

−

512

1224

12−

3 18 8

12 4=

Estimate: 512

12 rounds to 12,

146

rounds to 4.


43. 1956825

−

59256825

−

308256825

−

2425

44. 25

137426

−

45267426

−

3042674262326

−



193

45. 6429

−

959429539

−

46. 871

10−

1071071

1036

10

−

47. 1636547

12−

263

12547

12−

146212547

129 315 15

12 4

−

=

48. 286

15327

10−

486

30927

30−

3485309273025 558 5830 6

−

=

49. 32411443 3 1 44

+

= + =

50. 55832887 7 1 88

+

= + =

51. 4157

11914

−

815

1411914

−

2214141191411514

−

52. 323588

15−

923

1588

15115

15

−

53. 1 28 2 56 23 2 69 9 1 9 9⋅ = ⋅ = =

54. 1 9 3 27 14 3 132 2 1 2 2⋅ = ⋅ = =

55. 2 1 5 11 5 5 251 23 5 3 5 3 11 33÷ = ÷ = ⋅ =

56.

1 1 26 135 35 4 5 4

26 45 132 13 45 13

2 45

85315

÷ = ÷

= ⋅

⋅ ⋅=

⋅⋅

=

=

=

57. 4 5 8 5 1322 13 22 13 359 18 18 18 18+ = + =

58. 3 2 3 10 28 10 1815 5 15 5 14 5 925 5 25 25 25 25 25

− = − = − =

59. 2 1 4 1 3 15 3 5 3 2 23 6 6 6 6 2− = − = =

60. 3 13 6 13 22 13 95 2 5 2 4 2 28 16 16 16 16 16 16− = − = − =

61. 1155320

10237

15+

615

3092030437

30197230

+



194

62. 377

151202

+

67

1415

72014134214

+

63. 467

11514

−

86

1411514

−

22514115141114

−

64. 547

12192324

−

104724192324

−

34462419232415 523 2324 8

−

=

65. 2 3 30 134 17 10 7 10

30 137 103 10 13

7 10397457

⋅ = ⋅

⋅=

⋅⋅ ⋅

=⋅

=

=

66. 2 3 20 11 20 11 4 5 11 55 16 2 183 4 3 4 3 4 3 4 3 3

⋅ ⋅ ⋅⋅ = ⋅ = = = =

⋅ ⋅

67. 26

11310433

+

6633

310433161333

+

68. 73

16162398

+

73

1686

1669

1621 5 518 18 1 1916 16 16

+

= + =

69. 257

− decreased by a number translates as

25 .7x− −

70. The sum of 384

and a number translates as

38 .4x+

71. Multiply 91

10 by a number translates as

91 .10

x⋅

72. Divide a number by 1611

− translates as

16 .11

x ⎛ ⎞÷ −⎜ ⎟⎝ ⎠

73. Subtract the standard gauge in the U.S. from the standard gauge in Spain.

965101562

−

965

10556

104 29 9

10 5

−

=

Spain’s standard gauge is 295

inches wider than

the U.S. standard gauge.

74. Subtract the standard gauge in Portugal from the standard gauge in Spain.

96510116520

−

186520116520720

−

Spain’s standard gauge is 720

inch wider than

that of Portugal.



195

75. 1114335

−

5112012320

−

2510201232013720

−

Tucson gets an average of 13720

inches more

rain than Yuma.

76. 435418

−

84285418318

−

The short crutch should be lengthened 318

inches.

77. 3 2 7 2 7 2 7 72 14 1 4 1 4 1 2 2 2

⋅ ⋅⋅ = ⋅ = = =

⋅ ⋅ ⋅ or

132

The area is 72

or 132

square yards.

78. 1 5 7 5 7 35 15 3 or 172 1 2 1 2 2 2

⋅⋅ = ⋅ = =

⋅

The area is 352

or 1172

square inches.

79. 3 1 3 5 3 5 1514 4 4 4 4 4 16

⋅⋅ = ⋅ = =

⋅

The area is 1516

square inch.

80. 1 1 9 19 3 3 19 57 14 6 or 282 3 2 3 2 3 2 2

⋅ ⋅⋅ = ⋅ = =

⋅

The area is 572

or 1282

square yards.

81. 153

5778

3+

8524

521724

329 5 520 20 1 2124 24 24

+

= + =


meters.

82. 1 13 5 65 13 5 164 4 1 4 4⋅ = ⋅ = =


yards.

83. 2 1 1 2 1 215 3 2 15 3 23 4 2 3 4 4

2 315 53 4

⎛ ⎞ ⎛ ⎞− + = − +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

= −

2153354

−

815

1295

12−

20141295

1211912

−

No; the remaining board is 11912

feet which is

112

foot short.

84. 6338

−

858338528

−

The remaining piece is 528

feet.

85. 3 51 4 51 1 5112 44 4 1 4 4 16÷ = ÷ = ⋅ =

The patient walked 51 3316 16

= miles per day.

86. 3 1 111 1 111 4 11127 1114 4 4 4 4 1 1÷ = ÷ = ⋅ = =

This will make 111 quarter-pound hamburgers.



196

87. 4 12 18 12 7 6 2 7 14 212 2 47 1 7 1 18 1 6 3 3 3

⋅ ⋅÷ = ÷ = ⋅ = = =

⋅ ⋅

The length is 243

meters.

88. 1 47 4 47 1 47 1 47 723 4 52 2 1 2 4 2 4 8 8

⋅÷ = ÷ = ⋅ = = =

⋅

The length of each side is 758

feet.

89. 922022374

15+

27260402602846095 19 7 78 8 8 1 960 12 12 12

+

= = + =

The total duration of the eclipses is 79 minutes.

12

90. 922074

15+

272602846055 116 660 12

+

=

The total duration for the eclipses in even-

numbered years is 11612

minutes.

91. 74

15223

−

74

1510215

−

223151021512 41 115 5

−

=

It will be 415

minutes longer.

92. 2239220

−

40260272601360

−

It will be 1360

minute longer.

93. 2 3 22 234 25 10 5 10

2 11 235 2 5

25325

31025

− ⋅ = − ⋅

⋅ ⋅= −

⋅ ⋅

= −

= −

94. 5 2 23 113 36 3 6 3

23 36 1123 3

2 3 112322

1122

⎛ ⎞− ÷ − = ÷⎜ ⎟⎝ ⎠

= ⋅

⋅=

⋅ ⋅

=

=

95. 1 3 1 35 19 5 198 4 8 4

1 65 198 8

7248

7248

⎛ ⎞− − = − +⎜ ⎟⎝ ⎠⎛ ⎞= − +⎜ ⎟⎝ ⎠⎛ ⎞= −⎜ ⎟⎝ ⎠

= −

96. 5 2 5 217 14 17 149 3 9 3

5 617 149 914 616 149 9

829

⎛ ⎞+ − = −⎜ ⎟⎝ ⎠

= −

= −

=



197

97. 2 3 8 931 17 31 17

15 20 60 6068 930 1760 60591360

− + = − +

= − +

= −

98. 5 1 12 5 6 2 5 30 21 2 47 2 7 2 7 2 7 7

⋅ ⋅⎛ ⎞− ⋅ − = ⋅ = = =⎜ ⎟ ⋅⎝ ⎠

99. 3 1 7 71 34 2 4 2

7 24 7

7 24 72412

⎛ ⎞ ⎛ ⎞÷ − = ÷ −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞= ⋅ −⎜ ⎟⎝ ⎠⋅

= −⋅

= −

= −

100. 7 5 7 531 26 31 268 12 8 12

21 1031 2624 24

11524

⎛ ⎞− − − = − +⎜ ⎟⎝ ⎠

= − +

= −

101. 7 5 21 2011 13 11 138 6 24 24

20 2113 1124 2444 2112 1124 24

23124

− = −

⎛ ⎞= − −⎜ ⎟⎝ ⎠⎛ ⎞= − −⎜ ⎟⎝ ⎠

= −

102. 2 3 4 3 720 30 20 30 505 10 10 10 10

⎛ ⎞ ⎛ ⎞− + − = − + − = −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

103. 3 73 1007 ( 100)

10 10 173 110 10073 1

10 10073

1000

⎛ ⎞− ÷ − = − ÷ −⎜ ⎟⎝ ⎠⎛ ⎞= − ⋅ −⎜ ⎟⎝ ⎠

⋅=

⋅

=

104. 1 3 17 194 24 8 4 8

17 84 19

17 4 24 19

341915119

− ÷ = − ÷

= − ⋅

⋅ ⋅= −

⋅

= −

= −

105. 1 1(3 ) 3 13 3x x x x⎛ ⎞= ⋅ = ⋅ =⎜ ⎟

⎝ ⎠

106. 1 1(5 ) 5 15 5

y y y y⎛ ⎞= ⋅ = ⋅ =⎜ ⎟⎝ ⎠

107. 2 3 2 3 13 2 3 2

a a a a⎛ ⎞ ⎛ ⎞= ⋅ = ⋅ =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

108. 9 10 9 10 1

10 9 10 9m m m m⎛ ⎞ ⎛ ⎞− − = ⋅ = ⋅ =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

109. a. 59 9 1 105= + =

b. 1009 9 1 10100

= + =

c. 446 6 4 1011

= + =

d. 138 8 1 913

= + =

a, b, and c are equivalent to 10.

110. a. 7 3 36 6 1 74 4 4= + =

b. 11 3 35 5 2 74 4 4= + =

c. 12 37 716 4

=

The answer is d, all of them.

111. Incorrect, to divide mixed number, first write each mixed number as an improper fraction.



198




115. Incorrect; 2 1 11 8 88 43 1 or 43 7 3 7 21 21⋅ = ⋅ =



The Bigger Picture

1. 3 2 3 2 5

17 17 17 17+

+ = =

2. 9 1 9 1 8 4

10 10 10 10 5−

− = = =

3. 2 3 2 10 3 3 20 9 293 10 3 10 10 3 30 30 30

⋅ ⋅+ = + = + =

⋅ ⋅

4. 23 11 23 11 2 23 22 124 12 24 12 2 24 24 24

⋅− = − = − =

⋅

5. 7 19 7 5 19 2 35 38 73 3318 20 8 5 20 2 40 40 40 40

⋅ ⋅+ = + = + = =

⋅ ⋅

6. 3

33 3 3 3 27

4 4 4 644⋅ ⋅

= =⋅ ⋅

7. 16437

−

71574373127

−

8. 528116354

+

1522441

241852437 13 138 8 1 924 24 24

+

= + =

9. 6 8 6 8 2 3 8 2 8 16

11 9 11 9 11 3 3 11 3 33⋅ ⋅ ⋅ ⋅

⋅ = = = =⋅ ⋅ ⋅ ⋅

10. 4 4 34 9 34 5 34 5 34 72 1 1

15 5 15 5 15 9 3 5 9 27 27⋅

÷ = ÷ = ⋅ = = =⋅ ⋅

Section 4.8

Practice Problems

1. 2 53 12

2 2 5 23 3 12 3

5 2 412 3 45 8

12 121312

y

y

y

y

y

− =

− + = +

⋅= +

⋅

= +

=

Check: 2 53 12

13 2 512 3 12

13 2 4 512 3 4 12

13 8 512 12 12

5 5 True12 12

y − =

−

⋅−

⋅

−

=

The solution is 13 .12

2. 1 2515 5 25

10

y

y

y

=

⋅ = ⋅

=

Check: 1 25

1 10 25

2 2 True

y =

⋅

=

The solution is 10.



199

3. 5 257

7 5 7 255 7 5

7 2515

35

b

b

b

b

=

⋅ = ⋅

⋅=

=

Check: 5 257

5 35 257

25 25 True

b =

⋅

=

The solution is 35.

4. 7 2

10 510 7 10 27 10 7 5

10 27 547

x

x

x

x

− =

− ⋅ = − ⋅

⋅= −

⋅

= −

Check: 7 2

10 57 4 2

10 7 528 270 5

2 2 True5 5

x− =

− ⋅−

=


−

5. 354

1 1 355 5 4

1 35 4320

x

x

x

x

= −

⋅ = ⋅−

⋅= −

⋅

= −

Check: 354

3 3520 415 320 4

3 3 True4 4

x = −

⋅− −

− −

− = −

The solution is 3 .

20−

6. The LCD of 15 and 5 is 15. 11 315 511 315 1515 511 911 911 11

911

x

x

xx

x

= −

⋅ = ⋅−

= −−

=

= −

The solution is 9 .

11−

7. The LCD of 8 and 4 is 8. 3 2

8 438 8(2)

8 438 8 8(2)

8 46 16

6 6 16 610

y

y

y

yy

y

+ =

⎛ ⎞+ =⎜ ⎟⎝ ⎠

⋅ + ⋅ =

+ =+ − = −

=

Check: 3 2

8 410 3 28 4

10 6 28 8

16 282 2 True

y+ =

+

+

=

The solution is 10.



200

8.

15 5

15 55 5

15 5( ) 55 5

5 14 14 14 4

14

x x

x x

x x

x xxx

x

− =

⎛ ⎞ ⎛ ⎞− =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞− =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

− =− =−

=− −

= −

To check, replace x with 14

− in the original

equation to see that a true statement results. The

solution is 1 .4

−

9. The LCD of 2 and 5 is 10. 3

2 5 2310 10

2 5 2310 10 10

2 5 25 2 15

5 2 2 2 153 153 153 3

5

y y

y y

y y

y yy y y y

yy

y

= +

⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎛ ⎞ ⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

= +− = − +

=

=

=

Check: 3

2 5 25 5 32 5 21 12 1 12 21 12 2 True2 2

y y= +

+

+

=

The solution is 5.

10. 9 9 3 10 27 10 27 10

10 3 10 3 3 10 30 30 30y y y y−

− = ⋅ − ⋅ = − =

Exercise Set 4.8

1. 1 13 3

1 1 1 13 3 3 3

23

x

x

x

+ = −

+ − = − −

= −

Check: 1 13 3

2 1 13 3 32 1 13 3

1 1 True3 3

x + = −

− + −

− +−

− = −


−

2. 1 79 9

1 1 7 19 9 9 9

89

x

x

x

+ = −

+ − = − −

= −

Check: 1 79 9

8 1 79 9 98 1 79 9

7 7 True9 9

x + = −

− + −

− +−

− = −


−

3. 3 2

13 133 3 2 3

13 13 13 132 313

113

y

y

y

y

− = −

− + = − +

− +=

=



201

Check: 3 2

13 131 3 2

13 13 131 3 213 13

2 2 True13 13

y − = −

− −

−−

− = −

The solution is 1 .

13

4. 5 4

14 145 5 4 5

14 14 14 144 5149

14

z

z

z

z

− =

− + = +

+=

=

Check: 5 4

14 149 5 4

14 14 149 5 414 14

4 4 True14 14

z − =

−

−

=

The solution is 9 .

14

5. 1 1 23 25 5 5

1 35 5

1 1 3 15 5 5 5

45

x x

x

x

x

− − = +

− =

− + = +

=

Check: 1 1 23 25 5 5

4 1 4 1 23 25 5 5 5 5

12 1 8 1 25 5 5 5 5

3 3 True5 5

x x− − = +

⋅ − − ⋅ +

− − +

=


6. 1 2 55 4

11 11 111 311 11

1 1 3 111 11 11 11

411

x x

x

x

x

+ − = −

+ = −

+ − = − −

= −

Check: 1 2 55 4

11 11 114 1 4 2 55 4

11 11 11 11 1120 1 16 2 511 11 11 11 11

3 3 True11 11

x x+ − = −

⎛ ⎞ ⎛ ⎞− + − − −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

− + + −

− = −

The solution is 4 .

11−

7. 1 5

12 61 1 5 1

12 12 6 125 2 16 2 1210 112 121112

x

x

x

x

x

− =

− + = +

⋅= +

⋅

= +

=

Check: 1 5

12 611 1 512 12 6

10 512 6

5 5 True6 6

x − =

−

=

The solution is 11.12

8. 8 19 3

8 8 1 89 9 3 9

1 3 83 3 93 89 9119

y

y

y

y

y

− =

− + = +

⋅= +

⋅

= +

=



202

Check: 8 19 3

11 8 19 9 3

3 19 31 1 True3 3

y − =

−

=


9. 2 35 10

2 2 3 25 5 10 5

3 2 210 5 23 4

10 107

10

y

y

y

y

y

+ = −

+ − = − −

⋅= − −

⋅

= − −

= −

Check: 2 35 10

2 7 35 10 10

2 2 7 35 2 10 10

4 7 310 10 10

3 3 True10 10

y+ = −

⎛ ⎞+ − −⎜ ⎟⎝ ⎠⎛ ⎞⋅ + − −⎜ ⎟⎝ ⎠⎛ ⎞+ − −⎜ ⎟⎝ ⎠

− = −

The solution is 7 .

10−

10. 1 32 8

1 1 1 32 2 2 8

1 4 32 4 84 38 878

a

a

a

a

a

+ = −

− + + = − −

⋅= − −

⋅

= − −

= −

Check: 1 32 8

1 7 32 8 8

4 7 38 8 8

3 3 True8 8

a+ = −

⎛ ⎞+ − −⎜ ⎟⎝ ⎠

− −

− = −


−

11.

1 37 616 4

1 316 4

1 1 3 116 16 4 16

3 4 14 4 1612 116 161116

z z

z

z

z

z

z

+ − =

+ =

+ − = −

⋅= −

⋅

= −

=

Check: 1 37 6

16 411 1 11 37 616 16 16 4

77 1 66 316 16 16 4

12 316 4

3 3 True4 4

z z+ − =

⎛ ⎞ ⎛ ⎞+ −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

+ −

=


12.

2 119 87 14

2 117 14

2 2 11 27 7 14 7

11 2 214 7 211 414 141514

x x

x

x

x

x

x

− − =

− =

− + = +

⋅= +

⋅

= +

=



203

Check: 2 119 87 14

15 2 15 119 814 7 14 14

135 4 120 1114 14 14 14

11 11 True14 14

x x− − =

⎛ ⎞ ⎛ ⎞− −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

− −

=


13. 2 59 6

2 5 5 59 6 6 6

2 2 5 39 2 6 3

4 1518 18

1118

x

x

x

x

x

− = −

− + = − +

− ⋅ + ⋅ =

− + =

=

Check: 2 59 62 11 59 18 6

2 2 11 5 39 2 18 6 3

4 11 1518 18 184 4 True

18 18

x− = −

− −

− ⋅ − ⋅

− −

− =−


14. 1 74 10

1 7 7 74 10 10 10

1 5 7 24 5 10 2

5 1420 20

920

y

y

y

y

y

− = −

− + = − +

⋅ ⋅− + =

⋅ ⋅

− + =

=

Check: 1 74 101 9 74 20 10

1 5 9 7 24 5 20 10 2

5 9 1420 20 205 5 True20 20

y− = −

− −

⋅ ⋅− −

⋅ ⋅

− −

− = −


15. 7 21 17 27 7

27

x

x

x

=

⋅ = ⋅

=

16. 5 41 1( 5 ) 45 5

45

x

x

x

− =

− ⋅ − = − ⋅

= −

17. 1 3414 4 34

12

x

x

x

=

⋅ = ⋅

=

18. 1 6313 3 63

18

x

x

x

=

⋅ = ⋅

=

19. 2 69

9 2 9 62 9 2

9 62

9 2 32

27

y

y

y

y

y

= −

⋅ = ⋅−

⋅= −

⋅ ⋅= −

= −



204

20. 4 87

7 4 7 84 7 4

564

14

x

x

x

x

= −

⋅ = ⋅−

= −

= −

21. 4 39 2

9 4 9 34 9 4 2

278

z

z

z

− = −

− ⋅ = − ⋅−

=

22. 11 210 7

10 11 10 211 10 11 7

2077

x

x

x

− = −

− ⋅− = − ⋅−

=

23. 173

1 1 177 7 3

121

a

a

a

=

⋅ = ⋅

=

24. 52

121 1 522 2 12

524

z

z

z

= −

⋅ = ⋅−

= −

25. 63

111 1 633 3 11

63 112

11

x

x

x

x

− = −

− ⋅− = − ⋅−

=⋅

=

26. 12425

1 1 1244 4 25

124 25325

z

z

z

z

− = −

− ⋅− = − ⋅−

=⋅

=

27. 5 39 18

5 318 189 1810 310 310 10

310

x

x

xx

x

= −

⎛ ⎞ ⎛ ⎞= −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

= −−

=

= −

28. 3 75 20

3 720 205 2012 712 712 12

712

y

y

yy

y

= −

⎛ ⎞ ⎛ ⎞= −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

= −−

=

= −

29. 72

3 373 2 3

3 3

3 3 2 73

6 76 6 7 6

1

x

x

x

xx

x

+ =

⎛ ⎞+ = ⋅⎜ ⎟⎝ ⎠

⋅ + ⋅ =

+ =+ − = −

=

30. 71

5 575 1 5

5 5

5 5 1 75

5 75 5 7 5

12

x

x

x

xx

x

− =

⎛ ⎞− = ⋅⎜ ⎟⎝ ⎠

⋅ − ⋅ =

− =− + = +

=



205

31.

5 15 10

510 10(1)5 10

510 10 105 10

2 5 102 5 5 10 5

2 152 152 2

152

x

x

x

xx

xx

x

− =

⎛ ⎞− =⎜ ⎟⎝ ⎠

⋅ − ⋅ =

− =− + = +

=

=

=

32. 63

3 3( 6)3

3 3( ) 183

3 182 182 182 2

9

x x

x x

x x

x xxx

x

− = −

⎛ ⎞− = −⎜ ⎟⎝ ⎠

⎛ ⎞ − = −⎜ ⎟⎝ ⎠

− = −− = −− −

=− −

=

33. 1 32 5 10

1 310 102 5 10

1 310 102 5

5 61

x

x

x

xx

− =

⎛ ⎞− = ⋅⎜ ⎟⎝ ⎠

⋅ − ⋅ =

− =− =

34. 2 13 4 12

2 112 123 4 12

2 112 123 4

8 35

x

x

x

xx

− =

⎛ ⎞ ⎛ ⎞− =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞− =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

− ==

35.

23 5

15 15 23 5

5 15 15 25

5 3 305 3 3 3 30

2 302 302 2

15

x x

x x

xx

x xx x x x

xx

x

= −

⎛ ⎞ ⎛ ⎞= −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

= ⋅ − ⋅

= −− = − −

= −−

=

= −

36.

52 7 2

514 142 7 2

57 14 147 2

7 2 357 2 2 2 35

5 355 355 5

7

a a

a a

aa

a aa a a a

aa

a

= +

⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

= ⋅ + ⋅

= +− = − +

=

=

=

37. 4 3 4 7

7 3 7 3 3 73 2821 213 28

21

x x

x

x

− = ⋅ − ⋅

= −

−=

38. 5 5 8 9 40 9 40 99 8 9 8 8 9 72 72 72

y y y y− +− + = − ⋅ + ⋅ = − + =

39. 5 2 10 105

2 2 1 2 2 2 2y y y y ++ = + ⋅ = + =

40. 7 2 3 7 6 7 6 723 1 3 3 3 3 3x x x x+

+ = ⋅ + = + =



206

41. 3 3 3 510 6 10 3 6 5

9 530 3014302 72 15

715

x x x x

x x

x

x

x

+ = ⋅ + ⋅

= +

=

⋅ ⋅=

⋅

=

42. 9 5 9 3 5 4 27 20 78 6 8 3 6 4 24 24 24x x x x x x x− = ⋅ − ⋅ = − =

43. 3 18 2

8 3 8 13 8 3 2

8643

x

x

x

x

=

⋅ = ⋅

=

=

44. 2 35 10

5 2 5 32 5 2 10

152034

y

y

y

y

=

⋅ = ⋅

=

=

45.

2 43 5 15

2 415 153 5 15

215 15 43 5

10 3 410 3 10 4 10

3 63 63 3

2

x

x

x

xx

xx

x

− =

⎛ ⎞− = ⋅⎜ ⎟⎝ ⎠

⋅ − ⋅ =

− =− − = −

− = −− −

=− −

=

46.

4 215 4 20

4 2120 205 4 20

420 20 215 4

16 5 2116 5 16 21 16

5 55 55 5

1

x

x

x

xx

xx

x

+ =

⎛ ⎞+ = ⋅⎜ ⎟⎝ ⎠

⋅ + ⋅ =

+ =+ − = −

=

=

=

47. 9 27

14 2014 9 14 279 14 9 20

2 7 9 39 2 102110

z

z

z

z

=

⋅ = ⋅

⋅ ⋅ ⋅=

⋅ ⋅

=

48. 5 5

16 616 5 16 55 16 5 6

2 8 55 2 383

a

a

a

a

=

⋅ = ⋅

⋅ ⋅=

⋅ ⋅

=

49. 43 57487

1 1 488 8 7

1 42 4 71

14

m m

m

m

m

m

− − =

− =

− ⋅− = − ⋅

⋅= −

⋅ ⋅

= −

50. 330 34

2034

201 1 344 4 20

380

n n

n

n

n

− =

− =

− ⋅− = − ⋅

= −



207

51. 11

4 414 1 4

4 4

4 4 1 14

4 14 4 1 4

3

x

x

x

xx

x

+ =

⎛ ⎞ ⎛ ⎞+ =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⋅ + ⋅ =

+ =+ − = −

= −

52. 12

7 717 2 7

7 7

7 7 2 17

14 114 14 1 14

15

y

y

y

yy

y

− =

⎛ ⎞ ⎛ ⎞− =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⋅ − ⋅ =

− =− + = +

=

53. 5 2 5 2 3 5 6 19 3 9 3 3 9 9 9− = − ⋅ = − = −

54. 8 1 8 2 1 11 16 11 511 2 11 2 2 11 22 22 22

− = ⋅ − ⋅ = − =

55. 1 10515 5 105

50

y

y

y

=

⋅ = ⋅

=

56. 1 2414 4 ( 2)4

8

x

x

x

= −

⋅ = ⋅ −

= −

57. 5 157 49

7 5 7 155 7 5 49

7 155 497 3 55 7 737

y

y

y

y

y

= −

⋅ = ⋅−

⋅= −

⋅⋅ ⋅

= −⋅ ⋅

= −

58. 3 94 2

4 3 4 93 4 3 2

4 93 22 2 3 3

3 26

x

x

x

x

x

− =

− ⋅− = − ⋅

⋅= −

⋅⋅ ⋅ ⋅

= −⋅

= −

59. 22

2 2( 2)2

2 2 42

2 444

1 14

x x

x x

x x

x xxx

x

− = −

⎛ ⎞− = −⎜ ⎟⎝ ⎠

⋅ − ⋅ = −

− = −− = −− −

=− −

=

60. 43

3 3( 4 )3

3( 4) 312 3

3 12 3 32 122 122 2

6

y y

y y

y yy y

y y y yyy

y

= − +

⎛ ⎞ = − +⎜ ⎟⎝ ⎠

= − + ⋅= − +

− = − + −− = −− −

=− −

=

61.

5 3 98 16 165 68 165 38 8

8 5 8 35 8 5 8

8 35 835

y

y

y

y

y

y

− = −

− = −

− = −

− ⋅− = − ⋅−

⋅=

⋅

=



208

62. 7 5 49 18 187 99 18

9 7 9 97 9 7 18

9 97 9 29

14

x

x

x

x

x

− = − −

− = −

− ⋅− = − ⋅−

⋅=

⋅ ⋅

=

63. 117 253183

1 1 188 8 3

124

x x

x

x

x

− =

− =

− ⋅− = − ⋅

= −

64. 427 309439

1 1 433 3 9

427

x x

x

x

x

− =

− =

− ⋅− = − ⋅

= −

65.

7 1 26 4 37 1 212 126 4 3

1 214 12 124 3

14 3 814 514 514 14

514

x

x

x

xxx

x

= −

⎛ ⎞⋅ = −⎜ ⎟⎝ ⎠

= ⋅ − ⋅

= −= −−

=

= −

66.

5 1 74 2 10

5 1 720 204 2 10

1 725 20 202 10

25 10 1425 425 425 25

425

y

y

y

yyy

y

= −

⎛ ⎞ ⎛ ⎞= −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

= ⋅ − ⋅

= −= −−

=

= −

67.

24 12 3

212 124 12 3

23 12 1212 3

3 83 8

2 82 82 2

4

b b

b b

bb

b bb b b bbb

b

= +

⎛ ⎞⋅ = +⎜ ⎟⎝ ⎠

= ⋅ + ⋅

= +− = + −

=

=

=

68.

16 3 2

16 66 3 2

16 63 2

2 32 2 3 2

33

1 13

a a

a a

aa

a aa a a a

aa

a

= +

⎛ ⎞⋅ = +⎜ ⎟⎝ ⎠

= ⋅ + ⋅

= +− = + −− =−

=− −

= −

69. 2 83 2

6 2 6 83 2

6 6 2 6 6 83 22 12 3 48

2 12 2 3 48 212 48

12 48 48 4836

x x

x x

x x

x xx x x x

xxx

+ = +

⎛ ⎞ ⎛ ⎞+ = +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⋅ + ⋅ = ⋅ + ⋅

+ = ++ − = + −

= +− = + −− =



209

70.

2 45 3

15 2 15 45 3

15 15 2 15 15 45 33 30 5 60

3 5 30 5 5 602 30 60

2 30 30 60 302 302 302 2

15

y y

y y

y y

y yy y y y

yy

yy

y

− = −

⎛ ⎞ ⎛ ⎞− = −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⋅ − ⋅ = ⋅ − ⋅

− = −− − = − −− − = −

− − + = − +− = −− −

=− −

=

71. 57,236 rounded to the nearest hundred is 57,200.

72. 576 rounded to the nearest hundred is 600.

73. 327 rounded to the nearest ten is 330.



76.

14 311 8 2

14 388 8811 8 2

14 388 88 4411 8

112 33 44112 33 33 44 33

112 11112 1111 11

11211

x x

x x

x x

x xx x x x

xx

x

+ =

⎛ ⎞+ = ⋅⎜ ⎟⎝ ⎠

⋅ + ⋅ =

+ =+ − = −

=

=

=

77.

19 353 2353 1431 2719 353 231431 143153 1431 27

353 2327 19 1431 14311431 27

513 353 1219513 1219 353 1219 1219

706 353706 353

353 3532

x

x

x

xxxx

x

= +

⎛ ⎞⋅ = +⎜ ⎟⎝ ⎠

⋅ = ⋅ + ⋅

= +− = + −− =−

=

− =

78. 5 3 5 4 5 4 5 4 5

12 4 12 3 12 3 4 3 3 9⋅ ⋅

÷ = ⋅ = = =⋅ ⋅ ⋅

The length is 5 inch.9

Chapter 4 Vocabulary Check

1. Two numbers are reciprocals of each other if their product is 1.

2. A composite number is a natural number greater than 1 that is not prime.


4. An improper fraction is a fraction whose numerator is greater than or equal to its denominator.

5. A prime number is a natural number greater than 1 whose only factors are 1 and itself.

6. A fraction is in simplest form when the numerator and the denominator have no factors in common other than 1.

7. A proper fraction is one whose numerator is less than its denominator.

8. A mixed number contains a whole number part and a fraction part.


the 7 is called the numerator

and the 9 is called the denominator.

10. The prime factorization of a number is the factorization in which all the factors are prime numbers.

11. The fraction 30

is undefined.

12. The fraction 0 0.5=

13. Fractions that have the same denominator are called like fractions.

14. The LCM of the denominators in a list of fractions is called the least common denominator.



210

15. A fraction whose numerator or denominator or both numerator and denominator contain fractions is called a complex fraction.

16. In ,a cb d= a ⋅ d and b ⋅ c are called cross products.

Chapter 4 Review



3. Each part is 13

of a whole and there are 7 parts shaded, or 2 wholes and 1 more part: 7 1 or 23 3

4. Each part is 14

of a whole and there are 13 parts shaded, or 3 wholes and 1 more part: 13 1 or 34 4

5. successful free throws 11total free throws 12

→→

1112

of the free throws were made.

6. a. 131 − 23 = 108 108 cars are not blue.

b. not blue 108total 131

→→

108131

of the cars on the lot are not blue.

7. 3 13= −

−

8. 20 120−

=−

9. 0 01=

−

10. 4 is undefined0

11.

·

10

12. ¢

1 20



211

13. @

1 2 30

14.

Y

1 20

15. 3

4 1512

3

15 334 4=

16. 3

13 39390

39 313

=

17. 1 5 2 1 10 1 1125 5 5 5

⋅ + += = =

18. 8 9 3 8 27 8 3539 9 9 9

⋅ + += = =

19. 12 4 3 428 7 3 7

⋅= =

⋅

20. 15 3 5 527 3 9 9

⋅= =

⋅

21. 225 1 25 1

3 25 375x x

x x xx⋅ ⋅

− = − = −⋅ ⋅ ⋅

22. 3 236 1 36

72 2 36 2y y y y yy y

⋅ ⋅ ⋅ ⋅− = − = −

⋅ ⋅

23. 29 29 2932 32 32ab a babc a b c c

⋅ ⋅= =

⋅ ⋅ ⋅

24. 18 18 1823 23 23xyz x y z zxy x y

⋅ ⋅ ⋅= =

⋅ ⋅

25. 2

3 245 9 5 5

9 327 3x y x x y x

x y y yxy y⋅ ⋅ ⋅ ⋅

= =⋅ ⋅ ⋅ ⋅ ⋅



212

26. 2

3 242 6 7 7

6 530 5ab c a b b c b

a b c c cabc c⋅ ⋅ ⋅ ⋅ ⋅

= =⋅ ⋅ ⋅ ⋅ ⋅ ⋅

27. 8 inches 8 4 2 2

12 inches 12 4 3 3⋅

= = =⋅

8 inches represents 23

of a foot.

28. 15 − 6 = 9 cars are not white. 9 non-white cars 9 3 3 3

15 total cars 15 3 5 5⋅

= = =⋅

35

of the cars are not white.



31. 3 1 3 1 35 2 5 2 10

⋅⋅ = =

⋅

32. 6 5 6 5 6 5 5 57 12 7 12 7 6 2 7 2 14

⋅ ⋅− ⋅ = − = − = − = −

⋅ ⋅ ⋅ ⋅

33. 3 3

2

24 15 24 155 8 5 8

8 3 5 35 83 3

9

x xx x

xx x x

x x

x

⋅⎛ ⎞− ⋅ − =⎜ ⎟

⋅⎝ ⎠⋅ ⋅ ⋅ ⋅

=⋅ ⋅ ⋅ ⋅⋅

=⋅

=

34. 3 3

2 227 7 27 7 3 9 7

21 3 7 9 2 218 21 18y y y y y y

y yy y⋅ ⋅ ⋅ ⋅ ⋅ ⋅

⋅ = = =⋅ ⋅ ⋅ ⋅ ⋅⋅

35. 31 1 1 1 1 1 1 1

3 3 3 3 3 3 3 27⋅ ⋅⎛ ⎞ ⎛ ⎞⎛ ⎞⎛ ⎞− = − − − = − = −⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⋅ ⋅⎝ ⎠ ⎝ ⎠⎝ ⎠⎝ ⎠

36. 25 5 5 5 5 25

12 12 12 12 12 144⋅⎛ ⎞ ⎛ ⎞⎛ ⎞− = − − = =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⋅⎝ ⎠ ⎝ ⎠⎝ ⎠

37. 3 3 3 8 3 8 3 4 2 2 24 8 4 3 4 3 1 4 3 1

⋅ ⋅ ⋅− ÷ = − ⋅ = − = − = − = −

⋅ ⋅ ⋅

38. 21 7 21 54 5 4 7

21 54 77 3 54 7

3 54

154

a a aa

aaaa

÷ = ⋅

⋅=

⋅⋅ ⋅ ⋅

=⋅ ⋅⋅

=

=

39. 9 1 9 3 9 3 272 3 2 1 2 1 2

⋅⎛ ⎞− ÷ − = − ⋅ − = =⎜ ⎟ ⋅⎝ ⎠

40. 5 5 2 5 1 5 1 523 3 1 3 2 3 2 6

yyy y y

⋅− ÷ = − ÷ = − ⋅ = − = −

⋅

41.

9 37 49 47 39 47 33 3 47 3

3 47

127

x y÷ = ÷

= ⋅

⋅=

⋅⋅ ⋅

=⋅⋅

=

=

42. 9 7 9 7 9 637

10 1 10 1 10 10ab ⋅

= − ⋅ = − ⋅ = − = −⋅

43. 11 7 11 7 77area length width6 8 6 8 48

⋅= ⋅ = ⋅ = =

⋅

The area is 7748

square feet.

44. 2 2 2 2 4area side side3 3 3 3 9

⋅= ⋅ = ⋅ = =

⋅

The area is 49

square meter.

45. 7 3 7 3 1011 11 11 11

++ = =

46. 4 2 4 2 6 2 3 29 9 9 9 3 3 3

+ ⋅+ = = = =

⋅



213

47. 1 5 1 5 4 1 4 1

12 12 12 12 3 4 3− − ⋅

− = = = − = −⋅

48. 11 11 12 3 4 415 15 15 15 3 5 5x x x x x x x+ ⋅ ⋅+ = = = =

⋅

49. 4 3 4 321 21 21y y −− =

50. 4 3 2 4 3 2 1 1

15 15 15 15 15 15− − −

− − = = = −

51. 3 = 3 x = x LCD = 3 ⋅ x = 3x

52. 4 = 2 ⋅ 2 8 = 2 ⋅ 2 ⋅ 2 12 = 2 ⋅ 2 ⋅ 3 LCD = 2 ⋅ 2 ⋅ 2 ⋅ 3 = 24

53. 2 2 10 2 10 203 3 10 3 10 30

⋅= ⋅ = =

⋅

54. 5 5 7 5 7 358 8 7 8 7 56

⋅= ⋅ = =

⋅

55. 7 7 7 7 7 496 6 7 6 7 42a a a a⋅= ⋅ = =

⋅

56. 9 9 5 9 5 454 4 5 4 5 20b b b b⋅= ⋅ = =

⋅

57. 4 4 10 4 10 40

5 5 10 5 10 50x x x x⋅

= ⋅ = =⋅

58. 5 5 2 5 2 10

9 9 2 9 2 18y y y y⋅

= ⋅ = =⋅

59. 3 2 1 3 2 1 6 2 3 38 8 8 8 8 2 4 4

+ + ⋅+ + = = = =

⋅

He did 34

of his homework that evening.

60. 9 3 9 3 9 3 9 3

16 16 16 16 1624163 82 832

+ + ++ + + =

=

⋅=

⋅

=

The perimeter is 32

miles.

61. The LCD of 18 and 9 is 18. 7 2 7 2 2 7 4 11

18 9 18 9 2 18 18 18⋅

+ = + = + =⋅

62. The LCD of 13 and 26 is 26. 4 1 4 2 1 8 1 7

13 26 13 2 26 26 26 26⋅

− = − = − =⋅

63. The LCD of 3 and 4 is 12. 1 1 1 4 1 3 4 3 13 4 3 4 4 3 12 12 12

⋅ ⋅− + = − + = − + = −

⋅ ⋅

64. The LCD of 3 and 4 is 12. 2 1 2 4 1 3 8 3 53 4 3 4 4 3 12 12 12

⋅ ⋅− + = − + = − + = −

⋅ ⋅

65. The LCD of 11 and 55 is 55. 5 2 5 5 2 25 2 25 211 55 11 5 55 55 55 55x x x x⋅ ++ = + = + =

⋅

66. The LCD of 15 and 5 is 15. 4 4 3 4 3 4 3

15 5 15 5 3 15 15 15b b b b⋅ +

+ = + = + =⋅

67. The LCD of 12 and 9 is 36. 5 2 5 3 2 4 15 8 712 9 12 3 9 4 36 36 36y y y y y y y⋅ ⋅− = − = − =

⋅ ⋅

68. The LCD of 18 and 9 is 18. 7 2 7 2 2 7 4 1118 9 18 9 2 18 18 18x x x x x x x⋅+ = + = + =

⋅

69. The LCD of 9 and y is 9y. 4 5 4 5 9 4 45 4 459 9 9 9 9 9

y y yy y y y y y

⋅ ⋅ ++ = + = + =

⋅ ⋅

70. The LCD of 14 and 7 is 14. 9 3 9 3 2 9 6 15

14 7 14 7 2 14 14 14⋅

− − = − − = − − = −⋅



214

71. The LCD of 25, 75, and 50 is 150. 4 23 7 4 6 23 2 7 325 75 50 25 6 75 2 50 3

24 46 21150 150 15091

150

⋅ ⋅ ⋅+ + = + +

⋅ ⋅ ⋅

= + +

=

72. The LCD of 3, 9, and 6 is 18. 2 2 1 2 6 2 2 1 33 9 6 3 6 9 2 6 3

12 4 318 18 185

18

⋅ ⋅ ⋅− − = − −

⋅ ⋅ ⋅

= − −

=

73. The LCD of 9 and 6 is 18. 2 5 2 5 2 2 5 3 2 2 5 39 6 9 6 9 2 6 3 9 2 6 3

4 15 4 1518 18 18 1838182 192 9

199

⋅ ⋅ ⋅ ⋅+ + + = + + +

⋅ ⋅ ⋅ ⋅

= + + +

=

⋅=

⋅

=


meters.

74. The LCD of 5 and 10 is 10. 1 3 7 1 2 3 2 75 5 10 5 2 5 2 10

2 6 710 10 1015103 52 532

⋅ ⋅+ + = + +

⋅ ⋅

= + +

=

⋅=

⋅

=

The perimeter is 32

feet.

75. The LCD of 25 and 50 is 50. 9 3 9 2 3 18 3 2125 50 25 2 50 50 50 50

⋅+ = + = + =

⋅

2150

of the donors have type A blood.

76. The LCD of 12 and 3 is 12. 2 5 2 4 5 8 5 3 1 3 13 12 3 4 12 12 12 12 4 3 4

⋅ ⋅− = − = − = = =

⋅ ⋅

The difference in length is 14

yard.

77. 257

10

2 75 10

2 105 7

2 105 7

2 5 25 7

47

x x

x

x

x

x

= ÷

= ⋅

⋅=

⋅⋅ ⋅ ⋅

=⋅

=

78. 37

117

3 11 3 7 3 7 37 7 7 11 7 11 11

yy y y y⋅

= ÷ = ⋅ = =⋅

79. ( )( )

2 12 15 25 2

3 7 3 74 10 4 10

2 15 23 74 10

20

20

20 20

20 208 10

15 142

12

−−=

− −

⋅ − ⋅=

⋅ − ⋅−

=−

−=

= −

80. ( )( )5 15 16 46 4

1 112 12

5 16 4

12

12

12 12

110 3

17

17

y y

y

y

y y

y y

y

y

− −

−−=

⋅ − ⋅=

−−

=−

=−

= −



215

81.

( )

( )

12

2 43 5

12

2 43 5

2 43 5

30

3015

30 3015

20 24154

xy z

=+ − +

⋅=

− +

=− + ⋅

=− +

=

82.

( )

( )

1 22 3

451 22 3

45

1 22 3

30

30

30 30

6 415 20

24524

x yz

+ −+=

−=

⋅

⋅ − ⋅=

⋅−

=

= −

83. 5 1 4 5 2 4 5 2 4 8

13 2 5 13 1 5 13 1 5 13⋅ ⋅

÷ ⋅ = ⋅ ⋅ = =⋅ ⋅

84. 22 1 2 1

27 3 27 92 1 3

27 9 32 3

27 27127

⎛ ⎞− = −⎜ ⎟⎝ ⎠

⋅= −

⋅

= −

= −

85. 9 1 2 1 9 1 2 1

10 3 5 11 10 3 5 113 2

10 553 11 2 2

10 11 55 233 4

110 11029

110

⋅ ⋅⋅ − ⋅ = −

⋅ ⋅

= −

⋅ ⋅= −

⋅ ⋅

= −

=

86. 2 1 3 2 1 2 37 5 10 7 5 2 10

2 2 37 10 102 57 102 17 217

⋅⎛ ⎞ ⎛ ⎞− ⋅ + = − ⋅ +⎜ ⎟ ⎜ ⎟⋅⎝ ⎠ ⎝ ⎠⎛ ⎞= − ⋅ +⎜ ⎟⎝ ⎠

= − ⋅

= − ⋅

= −

87. 37859613

12+

97242092423

2431 7 719 19 1 2024 24 24

+

= + =

88. 185

rounds to 8.

3511

rounds to 5.

An estimate is 8 + 5 = 13. 18535

11−

1185515555

−

667551555551255

−

89. 518

rounds to 2.

135

rounds to 3.

An estimate is 2 ⋅ 3 = 6. 5 1 13 16 13 16 13 8 2 26 11 3 58 5 8 5 8 5 8 5 5 5

⋅ ⋅ ⋅⋅ = ⋅ = = = =

⋅ ⋅

90. 3 2 27 9 27 7 9 3 7 21 16 1 54 7 4 7 4 9 4 9 4 4

⋅ ⋅÷ = ÷ = ⋅ = = =

⋅

91. 1 341 31 341 2 11 31 2341 15 222 1 2 1 31 1 31

⋅ ⋅÷ = ÷ = ⋅ = =

⋅

We would expect 22 miles on one gallon.



216

92. 1 22 5 22 5 110 27 5 363 3 1 3 1 3 3

⋅⋅ = ⋅ = = =

⋅

There are 110

3 or

2363

grams of fat in a 5-ounce

hamburger patty.

93. 7 3 4 118 10 8 88 8 8 2− = =

1 17 2 17 1 17 18 2 42 2 1 2 2 4 4÷ = ÷ = ⋅ = =

Each measurement is 14 inches.4

94. 3 3 13 3 13 3 2 13 6 71

10 5 10 5 10 5 2 10 10 10⋅

− = − = − = − =⋅

The unknown measurement is 7

10 yard.

95. 1 3 2 3 512 15 12 15 277 14 14 14 14

⎛ ⎞ ⎛ ⎞− + − = − + − = −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

96. 1 7 16 273 25 10 5 10

16 105 27

16 5 25 27

3227

5127

⎛ ⎞ ⎛ ⎞− ÷ − = − ÷ −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞= − ⋅ −⎜ ⎟⎝ ⎠

⋅ ⋅=

⋅

=

=

97. 1 3 9 7 63 152 1 34 4 4 4 16 16

− ⋅ = − ⋅ = − = −

98. 7 7 7 723 24 24 238 10 10 8

⎛ ⎞− = − −⎜ ⎟⎝ ⎠

7 28 6824 24 2310 40 407 35 3523 23 238 40 40

3340

− − −

7 7 3323 248 10 40− = −

99. 2 13 6

2 2 1 23 3 6 3

1 2 26 3 21 46 656

a

a

a

a

a

− =

− + = +

⋅= +

⋅

= +

=

100. 1 79 85 10

1 75 10

1 1 7 15 5 10 5

7 210 109

10

x x

x

x

x

x

+ − = −

+ = −

+ − = − −

= − −

= −

101. 3 65

5 3 5 63 5 3

10

x

x

x

− =

− ⋅− = − ⋅

= −

102. 2 49 3

9 2 9 42 9 2 3

9 42 33 3 2 2

2 3616

y

y

y

y

y

y

= −

⋅ = ⋅−

⋅= −

⋅⋅ ⋅ ⋅

= −⋅

= −

= −

103. 63

7 767 3 7

7 7

7 7 3 67

21 621 21 6 21

15

x

x

x

xx

x

− = −

⎛ ⎞ ⎛ ⎞− = −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⋅ − ⋅ = −

− = −− + = − +

=



217

104. 112

5 5115 2 5

5 5

5 5 2 115

10 1110 10 11 10

1

y

y

y

yy

y

+ =

⎛ ⎞ ⎛ ⎞+ =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⋅ + ⋅ =

+ =+ − = −

=

105.

1 176 4 12

1 1712 126 4 12

112 12 176 4

2 3 172 3 2 17 2

3 153 153 3

5

x

x

x

xxxx

x

+ =

⎛ ⎞+ = ⋅⎜ ⎟⎝ ⎠

⋅ + ⋅ =

+ =+ − = −

=

=

=

106.

5 15 4 2 20

5 120 205 4 2 20

5 120 20 20 205 4 2 20

4 25 10 14 25 10 10 1 10

25 6 125 6 25 1 25

6 246 246 6

4

x x

x x

x x

x xx x x x

xx

xx

x

− = −

⎛ ⎞ ⎛ ⎞− = −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⋅ − ⋅ = ⋅ − ⋅

− = −− − = − −− − = −

− − + = − +− =−

=− −

= −

107. 6 5 6 5 3 2 5 1

15 8 15 8 5 3 2 4 4⋅ ⋅ ⋅

⋅ = = =⋅ ⋅ ⋅ ⋅

108. 2 3 2 3

3 3

2 3

3

2

5 10 510

510

55 2

2

x x x yy yy x

x yy xx x y y y

y x x xyx

÷ = ⋅

⋅=

⋅⋅ ⋅ ⋅ ⋅ ⋅

=⋅ ⋅ ⋅ ⋅ ⋅

=

109. 3 1 3 1 2 1 2 1

10 10 10 10 5 2 5− ⋅

− = = = =⋅

110. 7 2 7 2 7 2 7

8 3 8 3 2 4 3 12x x x x⋅ ⋅

⋅ − = − = − = −⋅ ⋅ ⋅ ⋅

111. 2 2 4 33 4 3 4 4 3

8 312 128 3

121112

x x x x

x x

x x

x

+ = ⋅ + ⋅

= +

+=

=

112. 5 2 5 5 2 25 2 23

11 55 11 5 55 55 55 55− + = − ⋅ + = − + = −

113. 3 1 8 115 4 5 4

8 45 18 45 132 2or 65 5

− ÷ = − ÷

= − ⋅

⋅= −

⋅

= − −

114. 728

rounds to 3.

192

rounds to 10.

An estimate is 3 + 10 = 13. 728192

+

728498

11 3 311 11 1 128 8 8

+

= + =



218

115. 1127

rounds to 12.

395

rounds to 10.

An estimate is 12 − 10 = 2. 1127395

−

512

3521935

−

4011352193519235

−

116. ( )( )

3344

1 18 8

3418

8 22

1 8 1

8 2 8

8 1 816 68 1

22 1 or 37 7

++=

− −

⋅ + ⋅=

⋅ − ⋅+

=−

=

117.

3 2 4 3 2 3 48 3 9 8 3 3 9

3 6 48 9 93 28 93 28 9

3 22 4 3 31

12

⋅⎛ ⎞ ⎛ ⎞− ⋅ − = − −⎜ ⎟ ⎜ ⎟⋅⎝ ⎠ ⎝ ⎠⎛ ⎞= − −⎜ ⎟⎝ ⎠⎛ ⎞= − ⎜ ⎟⎝ ⎠⋅

= −⋅⋅

= −⋅ ⋅ ⋅

= −

118.

2 1311 107 14

2 137 14

2 2 13 27 7 14 7

13 2 214 7 213 414 149

14

x x

x

x

x

x

x

− − = −

− = −

− + = − +

⋅= − +

⋅

= − +

= −

119. 3 45 15

5 3 5 43 5 3 15

5 43 155 4

3 5 349

x

x

x

x

x

− =

− ⋅− = − ⋅

⋅= −

⋅⋅

= −⋅ ⋅

= −

120. 5 3

12 6 45 312 12

12 6 4512 12 9

12 610 9

10 10 9 1019

x

x

x

xx

x

+ = −

⎛ ⎞ ⎛ ⎞+ = −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⋅ + ⋅ = −

+ = −+ − = − −

= −

121. 1 50 1150 52 1 2

50 2 111 2 2

100 112 2

892

1442

− = −

⋅= −

⋅

= −

=

=

The length of the remaining piece is 1442

yards.

122. 1 4 11 81 11 81 81 15 7 or 402 11 2 11 2 11 2 2

⋅⋅ = ⋅ = =

⋅

The area is 812

or 1402

square feet.

Chapter 4 Test

1. 7 of the 16 equal parts are shaded: 7 .

16

2. 2 3 7 2 21 2 2373 3 3 3

⋅ + += = =



219

3. 18

4 7543532

3

75 3184 4=

4. 24 6 4 4210 6 7 5 35

⋅= =

⋅ ⋅

5. 42 3 14 370 5 14 5x x x⋅ ⋅

− = − = −⋅

6. 5 ⋅ 11 = 55 7 ⋅ 8 = 56 Since 55 ≠ 56, the fractions are not equivalent.

7. 6 ⋅ 63 = 378 27 ⋅ 14 = 378 Since 378 = 378, the fractions are equivalent.

8. 284 2 42 2 2 21 2 2 3 7 2 3 7= ⋅ = ⋅ ⋅ = ⋅ ⋅ ⋅ = ⋅ ⋅

9. 2495 3 165 3 3 55 3 3 5 11 3 5 11= ⋅ = ⋅ ⋅ = ⋅ ⋅ ⋅ = ⋅ ⋅

10. 4 3 4 4 4 4 44 4 4 3 4 3 3

⋅÷ = ⋅ = =

⋅

11. 4 4 4 4 43 4 3 4 3

⋅− ⋅ = − = −

⋅

12. 7 7 89 9 9 9x x x x x++ = =

13. The LCD of 7 and x is 7x. 1 3 1 3 7 21 217 7 7 7 7 7

x x xx x x x x x

−− = ⋅ − ⋅ = − =

14. 3 3

21

xy z xy z x y y y z y y yz xy z xy x y z

⋅ ⋅ ⋅ ⋅ ⋅ ⋅⋅ = = = =

⋅ ⋅ ⋅

15. 2 8 2 8 163 15 3 15 45

⋅− ⋅ − = =

⋅

16. 9 2 9 2 2 9 4 9 410 5 10 5 2 10 10 10a a a a⋅ ++ = + = + =

⋅

17. 8 2 8 2 10 2 5 2

15 15 15 15 3 5 3y y y y y y− − − ⋅

− − = = = − = −⋅ ⋅

18. 3 3 23 16 3 16 3 8 2 1 18 8 2 36 8 6a a a

a a a a aa a a⋅ ⋅ ⋅ ⋅

⋅ = = = =⋅ ⋅ ⋅ ⋅ ⋅ ⋅⋅

19.

11 3 5 11 2 3 3 512 8 24 12 2 8 3 24

22 9 524 24 2422 9 5

2418243 64 634

⋅ ⋅− + = − +

⋅ ⋅

= − +

− +=

=

⋅=

⋅

=

20. 738275324

+

35340167403024081 1 112 12 2 1440 40 40

+

= + =

21. 1932

11−

11181132

11816

11

−

22. 16 3 16 123 12 3 3

16 123 3

16 3 43 3

64 1or 213 3

− ÷− = − ⋅−

⋅=

⋅⋅ ⋅

=⋅

=



220

23. 1 3 10 273 63 4 3 4

10 273 4

2 5 3 93 2 2

5 92

45 1 or 222 2

⋅ = ⋅

⋅=

⋅⋅ ⋅ ⋅

=⋅ ⋅⋅

=

=

24.

2 1 2 6 167 6 7 1 6

2 6 6 17 1 6 62 36 17 6 62 357 62 357 62 7 57 2 35 2 or 13 3

⎛ ⎞ ⎛ ⎞− ⋅ − = − ⋅ −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⋅⎛ ⎞= − ⋅ −⎜ ⎟⋅⎝ ⎠⎛ ⎞= − ⋅ −⎜ ⎟⎝ ⎠

= − ⋅

⋅= −

⋅⋅ ⋅

= −⋅ ⋅

= − −

25. 1 2 3 1 3 3 1 3 3 92 3 4 2 2 4 2 2 4 16

⋅ ⋅÷ ⋅ = ⋅ ⋅ = =

⋅ ⋅

26.

2 2

2

2

3 2 5 3 2 2 54 3 6 4 3 2 6

3 4 54 6 63 94 6

9 916 69 6

16 99 2 32 8 938

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞− ÷ + = − ÷ ⋅ +⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞= − ÷ +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞= − ÷⎜ ⎟⎝ ⎠

= ÷

= ⋅

⋅ ⋅=

⋅ ⋅

=

27.

5 4 7 5 2 4 4 73 36 3 12 6 2 3 4 12

10 16 7 312 12 12

33 312 133 112 333 112 311 3 112 3

1112

⋅ ⋅⎛ ⎞ ⎛ ⎞+ + ÷ = + + ÷⎜ ⎟ ⎜ ⎟⋅ ⋅⎝ ⎠ ⎝ ⎠⎛ ⎞= + + ÷⎜ ⎟⎝ ⎠

= ÷

= ⋅

⋅=

⋅⋅ ⋅

=⋅

=

28. 2

5 27

2021

2

5 207 21

5 217 205 3 7

7 4 53

4

x

x

x x

xx

xx x

x

= ÷

= ⋅

⋅ ⋅ ⋅=

⋅ ⋅ ⋅ ⋅

=

29. ( )( )

33 377 7

1 112 22

14 55 14 5 14 70 6 7628 7 212 14 2 1414 2

++ ⋅ + ⋅ += = = =

−− ⋅ − ⋅−

30. 3 38 4

8 3 8 33 8 3 4

8 33 44 2 33 4

2

x

x

x

x

x

− =

− ⋅− = − ⋅

⋅= −

⋅⋅ ⋅

= −⋅

= −

31. 24

5 5245 5

5 5

5 5 245

5 246 246 246 6

4

x x

x x

x x

x xxx

x

+ = −

⎛ ⎞ ⎛ ⎞+ = −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⋅ + ⋅ = −

+ = −= −−

=

= −



221

32.

2 53 4 12 2

2 512 123 4 12 2

2 512 12 12 123 4 12 2

8 3 5 68 3 6 5 6 6

8 3 58 8 3 5 8

3 33 33 3

1

x x

x x

x x

x xx x x x

xxxx

x

+ = +

⎛ ⎞ ⎛ ⎞+ = +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⋅ + ⋅ = ⋅ + ⋅

+ = ++ − = + −

− =− − = −

− = −− −

=− −

=

33. 1 5 1 5 1 55 52 1 2 1 2 2

x ⋅⎛ ⎞− = − − = ⋅ = =⎜ ⎟ ⋅⎝ ⎠

34.

1 732 81 312 81 82 311 82 311 2 42 31431

x y÷ = ÷

= ÷

= ⋅

⋅=

⋅⋅ ⋅

=⋅

=

35. 1 2 66 6 52 4 43 3 32 2 24 4 4

334

− − −

The remaining piece is 334

feet.

36. Housing: 825

Food: 750

8 7 8 2 7 16 7 16 7 2325 50 25 2 50 50 50 50 50

⋅ ++ = + = + = =

⋅

2350

of spending goes for housing and food

combined.

37. Education: 150

Transportation: 15

Clothing: 125

1 1 1 1 1 10 1 250 5 25 50 5 10 25 2

1 10 250 50 501350

+ + = + ⋅ + ⋅

= + +

=

1350

of spending goes for education,

transportation, and clothing.

38. 3 3 47,000 3 50 94047,000 2820

50 50 1 50 1⋅ ⋅

⋅ = ⋅ = =⋅

Expect to spend $2820 on health care.

39. 2 2perimeter 1 13 3

3 2 3 23 3 3 3103133

= + + +

= + + +

=

=

2 2area length width 13 3

= ⋅ = ⋅ =


feet and the area is

23

square feet.

40. 3 258 43 258 4 43 6 4258 10 244 1 4 1 43 1 43

⋅ ⋅÷ = ÷ = ⋅ = =

⋅

Expect to travel 24 miles on 1 gallon of gas.

Cumulative Review Chapters 1−4

1. 546 in words is five hundred forty-six.

2. 115 in words is one hundred fifteen.

3. 27,034 in words is twenty-seven thousand, thirty-four.

4. 6573 in words is six thousand five hundred seventy-three.



222

5. 46713759

+

6. 58744

631+

7. 54329

514−

Check: 51429

543+

8. 99562

933−

Check: 93362

995+

9. 278,362 rounded to the nearest thousand is 278,000.


11. 480012

9 60048,00057,600

×

Therefore, 12 DVDs can hold 57,600 megabytes of information.

12. 4353

1305×

He travels 1305 miles in 3 days.

13. 7089

8 56,717560 7

07164

7772

5

56,717 ÷ 8 = 7089 R 5 Check: 7089 × 8 + 5 = 56,712 + 5 = 56,717

14. 379

12 455836958411810810

4558 ÷ 12 = 379 R 10 Check: 379 × 12 + 10 = 4548 + 10 = 4558

15. 37 7 7 7⋅ ⋅ =

16. 27 7 7⋅ =

17. 4 33 3 3 3 9 9 9 3 9⋅ ⋅ ⋅ ⋅ ⋅ ⋅ = ⋅

18. 4 29 9 9 9 5 5 9 5⋅ ⋅ ⋅ ⋅ ⋅ = ⋅

19. 2(x − y) = 2(6 − 3) = 2(3) = 6

20. 8 3( 5) 8 5 3(9 5)8 5 3 440 1252

a b+ − = ⋅ + −= ⋅ + ⋅= +=

21. Let 0 represent the surface of the earth. Then 6824 feet below the surface is represented as −6824.

22. Let 0 represent a temperature of 0°F. Then 21°F below zero is represented as −21.

23. 7 3 4− + =

24. −3 + 8 = 5



223

25. 7 8 ( 5) 1 7 8 5 17 ( 8) 5 ( 1)

1 5 ( 1)4 ( 1)3

− − − − = − + −= + − + + −= − + + −= + −=

26. 6 ( 8) ( 9) 3 6 ( 8) 9 32 9 3

7 310

+ − − − + = + − + += − + += +=

27. 2( 5) ( 5)( 5) 25− = − − =

28. 42 2 2 2 2 16− = − ⋅ ⋅ ⋅ = −

29. 3(4 7) ( 2) 5 3( 3) ( 2) 59 ( 2) ( 5)11 ( 5)16

− + − − = − + − −= − + − + −= − + −= −

30. 2 2 2 2(20 5 ) (20 25) ( 5) 25− = − = − =

31. 2 6 4 8 (2 4 ) ( 6 8) 6 2y y y y y− + + = + + − + = +

32. 5x − 1 + x + 10 = (5x + x) + (−1 + 10) = 6x + 9

33. 5 2 4 7 192 12

2 2 12 214

x xx

xx

+ − = −+ = −

+ − = − −= −

34. 9 1 8 3 201 17

1 1 17 118

y yy

yy

+ − = −+ = −

+ − = − −= −

35. 17 7 3 3 21 320 7 21 6

20 7 7 21 6 720 21

20 21 21 211 or 1

x x xx x

x x x xx

xx x

− + = − + −− = −

− + = − += +

− = − +− = = −

36. 9 2 7 249 7 2 7 7 24

2 2 242 2 2 24 2

2 222 222 2

11

x xx x x x

xx

xx

x

− = −− − = − −

− = −− + = − +

= −−

=

= −

37. Two of five equal parts are shaded: 25

38. 156 2 78

2 2 39

2 2 3 13

= ⋅↓ ↓⋅ ⋅

↓ ↓ ↓⋅ ⋅ ⋅

2156 2 3 13= ⋅ ⋅

39. a. 2 9 4 2 36 2 3849 9 9 9

⋅ + += = =

b. 8 11 1 8 11 8 191

11 11 11 11⋅ + +

= = =

40. 4 5 7 4 35 4 3975 5 5 5

⋅ + += = =

41. 42 6 7 766 6 11 11x x x⋅ ⋅= =

⋅

42. 70 35 2 2

105 35 3 3y y y⋅

= =⋅ ⋅

43. 1 7 10 733 8 3 8

10 73 82 5 73 2 45 73 435 11or 212 12

⋅ = ⋅

⋅=

⋅⋅ ⋅

=⋅ ⋅⋅

=⋅

=

44. 2 2 4 2 4 8 24 or 23 3 1 3 1 3 3

⋅⋅ = ⋅ = =

⋅

45. 5 3 5 4 5 4 5 4 5 5

16 4 16 3 16 3 4 4 3 4 3 12⋅ ⋅

÷ = ⋅ = = = =⋅ ⋅ ⋅ ⋅

46. 1 3 11 28 11 5 11 5 111 5

10 5 10 5 10 21 5 2 28 56⋅

÷ = ÷ = ⋅ = =⋅ ⋅


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