chapter 6 · chapter 6: ratio, proportion, and ... the ratio of medals won by canada to the total...

288

Chapter 6

Section 6.1

Practice Problems

1. The ratio of 19 to 30 is 19 .30

2. The ratio of $16 to $12 is $16 16 4 4 4 .$12 12 4 3 3

⋅= = =

⋅

3. The ratio of 1.68 to 4.8 is 1.68 1.68 100 168 7 24 7 .4.8 4.8 100 480 20 24 20

⋅ ⋅= = = =

⋅ ⋅

4. The ratio of 223

to 13115

is

23

1315

2 2 132 13 151

8 283 158 153 284 2 3 53 7 4

10 .7

= ÷

= ÷

= ⋅

⋅ ⋅ ⋅=

⋅ ⋅

=

5. The ratio of work miles to total miles is 4800 miles 8 600 8 .

15,000 miles 25 600 25⋅

= =⋅

6. a. The ratio of the length of the shortest side to the length of the longest side is 7 meters 7 .9 meters 9

=

b. The ratio of the length of the longest side to

the perimeter is 9 meters 9 .24 meters 24

=

7. $1350 $225

6 weeks 1 week=

8. 295 miles 59 miles15 gallons 3 gallons

=

9. 3200 feet8 seconds

400

8 320032000

−

The unit rate is 400 feet1 second

or 400 feet/second

10. 78 bushels

12 trees

6.512 78.0

726060

0

−

−

The unit rate is 6.5 bushels

1 tree or 6.5 bushels/ tree.

11. $170 $34unit price or $34 per day

5 days 1 day= =

12. $2.32 $0.21unit price

11 ounces 1 ounce= ≈

$3.59 $0.22unit price16 ounces 1 ounce

= ≈

Since the 11-ounce bag has a cheaper price per ounce, it is the better buy.

Vocabulary and Readiness Check

1. A rate with a denominator of 1 is called a unit rate.

2. When a rate is written as money per item, a unit rate is called a unit price.

3. The word per translates to “division”.

4. Rates are used to compare different types of quantities.

5. To write a rate as a unit rate, divide the numerator of the rate by the denominator.

6. The quotient of two quantities is called a ratio.

7. True or false: The ratio 75

means the same as the

ratio 5 .7

false

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

ISM: Prealgebra Chapter 6: Ratio, Proportion, and Triangle Applications

289

Exercise Set 6.1

1. The ratio of 16 to 24 is 16 8 2 2 .24 8 3 3

⋅= =

⋅

2. The ratio of 25 to 150 is 25 1 25 1 .

150 6 25 6⋅

= =⋅

3. The ratio of 7.7 to 10 is 7.7 7.7 10 77 .10 10 10 100

⋅= =

⋅

4. The ratio of 8.1 to 10 is 8.1 8.1 10 81 .10 10 10 100

⋅= =

⋅

5. The ratio of 4.63 to 8.21 is 4.63 4.63 100 463.8.21 8.21 100 821

⋅= =

⋅

6. The ratio of 9.61 to 7.62 is 9.61 9.61 100 961 .7.62 7.62 100 762

⋅= =

⋅

7. The ratio of 6 ounces to 16 ounces is 6 ounces 3 2 3 .

16 ounces 8 2 8⋅

= =⋅

8. The ratio of 35 meters to 100 meters is 35 meters 7 5 7 .

100 meters 20 5 20⋅

= =⋅

9. The ratio of $32 to $100 is $32 4 8 8 .$100 4 25 25

⋅= =

⋅

10. The ratio of $46 to $102 is $46 23 2 23.$102 51 2 51

⋅= =

⋅

11. The ratio of 24 days to 14 days is 24 days 12 2 12 .14 days 7 2 7

⋅= =

⋅

12. The ratio of 120 miles to 80 miles is 120 miles 3 40 3 .80 miles 2 40 2

⋅= =

⋅

13. The ratio of 132

to 1124

is

1214

3 1 13 122 412

7 492 47 42 497 2 22 7 72 .7

= ÷

= ÷

= ⋅

⋅ ⋅=

⋅ ⋅

=


to 146

is

1316

3 1 13 43 64

10 253 6

10 63 255 2 3 2

3 5 54 .5

= ÷

= ÷

= ⋅

⋅ ⋅ ⋅=

⋅ ⋅

=


hours to 91

10 hours is

359

10

7 hours 3 97 15 101 hours

38 195 1038 105 19

19 2 2 55 19 1

4 .1

= ÷

= ÷

= ⋅

⋅ ⋅ ⋅=

⋅ ⋅

=


Chapter 6: Ratio, Proportion, and Triangle Applications ISM: Prealgebra

290


days to 526

days is

12

56

25 days 1 525 22 62 days

51 172 651 62 173 17 2 3

2 17 191

= ÷

= ÷

= ⋅

⋅ ⋅ ⋅=

⋅ ⋅

=

17. Perimeter = 125 + 130 + 125 + 130 = 510 The ratio of height to perimeter is 130 feet 10 13 13510 feet 10 51 51

⋅= =

⋅

18. The ratio of the length to the width is 94 feet 47 2 47 .50 feet 25 2 25

⋅= =

⋅

19. The ratio of the number of gallons of bottled water to the number of gallons of soft drinks is 25 gallons 5 5 5 .55 gallons 5 11 11

⋅= =

⋅

20. The ratio of the Sam’s Clubs to the Wal-Marts is 3 stores 3 .22 stores 22

=

21. The ratio of women to men is 125 women 5 25 5 .

100 men 4 25 4⋅

= =⋅

22. The ratio of the men to the total is 100 men 100 4 25 4 .

125 women 100 men 225 9 25 9⋅

= = =+ ⋅

23. The ratio of red blood cells to platelet cells is 600 cells 40 15 15 .40 cells 40 1 1

⋅= =

⋅

24. The ratio of white blood cells to red blood cells

is 1 cell 1 .

600 cells 600=

25. The ratio of the base to the height is 6 feet 1 6 1 .

18 feet 3 6 3⋅

= =⋅

26. The ratio of the longest side to the perimeter is 17 feet 17 .

(8 15 17) feet 40=

+ +

27. The ratio of calories from fat to total calories is 270 calories 30 9 9 .570 calories 30 19 19

⋅= =

⋅

28. The ratio of the amount of fat in a Quarter Pounder to the amount of fat in a Crispy Chicken

Ranch BLT sandwich is 25 grams 5 5 5 .20 grams 5 4 4

⋅= =

⋅

29. The ratio of average amount eaten by New Zealanders to the average amount eaten by

Americans is 56 pints 7 8 7 .48 pints 6 8 6

⋅= =

⋅

30. The ratio of medals won by Canada to the total

medals awarded is 24 medals 8 3 3 .256 medals 8 32 32

⋅= =

⋅

31. The rate of 5 shrubs every 15 feet is 5 shrubs 1 shrub .15 feet 3 feet

=

32. The rate of 14 lab tables for 28 students is 14 lab tables 1 lab table .28 students 2 students

=

33. The rate of 15 returns for 100 sales is 15 returns 3 returns .100 sales 20 sales

=

34. The rate of 8 phone lines for 36 employees is 8 phone lines 2 phone lines .36 employees 9 employees

=

35. The rate of 6 laser printers for 28 computers is 6 laser printers 3 laser printers .28 computers 14 computers

=

36. The rate of 4 inches of rain in 18 hours is 4 inches 2 inches .18 hours 9 hours

=



291

37. The rate of 18 gallons of pesticide for 4 acres of

crops is 18 gallons 9 gallons .

4 acres 2 acres=

38. The rate of 150 graduate students for 8 advisors is 150 graduate students 75 graduate students .

8 advisors 4 advisors=

39. 110

3 330303

30

−

−

330 calories in a 3-ounce serving is 110 calories

1 ounce

or 110 calories/ounce.

40. 25

11 2752255550

−

−

275 miles in 11 hours is 25 miles11 hours

or

25 miles/hour.

41. 75

5 3753525250

−

−

375 riders in 5 subway cars is 75 riders

1 car or

75 riders/car.

42. 3

6 18180

−

18 signs in 6 blocks is 3 signs1 block

or 3 signs/block.

43. 90

60 5400540

00−

5400 wingbeats per 60 seconds is 90 wingbeats

1 second

or 90 wingbeats/second.

44. 20

60 1200120

00−

1200 wingbeats per 60 seconds is 20 wingbeats

1 second

or 20 wingbeats/second.

45. 50,000

20 1,000,0001 00

0−

$1,000,000 paid over 20 years is $50,0001 year

or

$50,000/year.

46. 50

8000 400,000400 00

00−

400,000 library books for 8000 students is 50 library books

1 student or 50 library books/student.



292

47.

1,369,0222 2,738,044

20 7

61312

1818

00440440

−

−

−

−

−

−

2,738,044 voters for 2 senators is 1,369,022 voters

1 senator or 1,369,022 voters/senator.

48.

69,82564 4, 468,800

3 84628576

52 851 2

1 601 28

320320

0

−

−

−

−

−

4,468,800 residents for 64 parishes is 69,825 residents

1 parish or 69,825 residents/parish.

49. 300

40 12,00012 0

0−

12,000 good products to 40 defective is 300 good

1 defective or 300 good/defective.

50. 1,250,000

4 5,000,00041 0

820200

−

−

−

5,000,000 lottery tickets for 4 lottery winners is 1, 250,000 tickets

1 winner or 1,250,000 tickets/winner.

51. 65,000

20 1,300,0001 20

100100

0

−

−

$1,300,000 to build 20 houses is $65,0001 house

or

$65,000/house.

52. 2,020,000

60 121, 200,000120

1 201 20

00

−

−

$121,200,000 for 60 shows is $2,020,000

1 show or

$2,020,000/show.

53. a. 31.25

8 250.002410

82 01 6

40400

−

−

−

−

The unit rate for Charlie is 31.25 boards/hour.



293

b. 33.5

12 402.03642366 06 0

0

−

−

−

The unit rate for Suellen is 33.5 boards/hour.

c. Suellen can assemble boards faster, since 33.5 > 31.25.

54. a. 15.5

46 713.0462532302 302 30

0

−

−

−

The unit rate for Jerry is 15.5 bricks/minute.

b. 13.2

30 396.03096906 06 0

0

−

−

−

The unit rate for Bobby is 13.2 bricks/minute.

c. Jerry is the faster bricklayer, since 15.5 > 13.2.

55. a. 14.5 400 becomes 27.58 27.6

145 4000.0029011001015

85072512501160

90

≈

−

−

−

−

The unit rate for the car is ≈27.6 miles/gallon.

b. 9.25 270 becomes 29.18 29.2

925 27000.00185085008325175 0

92 582 5074 00

8 50

≈

−

−

−

−

The unit rate for the truck is ≈29.2 miles/gallon.

c. The truck has the better gas mileage, since 29.2 > 27.6.

56. a. 3.5 100 becomes 28.57 28.6

35 1000.0070300280

200175

250245

5

≈

−

−

−

−

The unit rate for Charlotte is ≈28.6 items/minute.

b. 5.5 148 becomes 26.90 26.9

55 1480.0011038033050 049 5

50

≈

−

−

−

The unit rate for Leo is ≈26.9 items/minute.

c. Charlotte is the faster scanner, since 28.6 > 26.9.



294

57. 11.50

5 57.50507

52 52 5

0

−

−

−

The unit price is $11.50 per compact disc.

58. 0.29

3 0.8762727

0

−

−

The unit price is $0.29 per apple.

59. 0.17

7 1.19749490

−

−

The unit price is $0.17 per banana.

60. 12.25

6 73.5061312

1 51 2

30300

−

−

−

−

The unit price is $12.25 per lawn chair.

61. 0.1487 0.149

8 1.190083932

7064

60564

≈

−

−

−

−

The 8-ounce size costs $0.149 per ounce.

0.1325 0.13312 1.5900

1 23936

302460600

≈

−

−

−

−

The 12-ounce size costs $0.133 per ounce. The 12-ounce size is the better buy.

62. 0.0590 0.059

32 1.89001 60

290288

20

≈

−

−

The 32-ounce size costs $0.059 per ounce. 0.0494 0.049

18 0.890072170162

8072

8

≈

−

−

−


63. 0.1056 0.106

16 1.69001 6

09080100964

≈

−

−

−




295

0.115

6 0.690609630300

−

−

−


64. 0.0575 0.058

12 0.690060

908460600

≈

−

−

−

The dozen costs $0.058 per egg. 0.07

30 2.102 10

0−

The flat costs $0.07 per egg. The dozen is the better buy.

65. 0.1908 0.191

12 2.29001 21 091 08

100964

≈

−

−

−


8 1.4900869645048

2016

4

≈

−

−

−

−


66. 0.0945 0.095

20 1.89001 80

9080100100

0

≈

−

−

−


32 3.19002 88

310288220192

28

≈

−

−

−


67. 0.0059 0.006

100 0.5900500900900

0

≈

−

−

The 100-count size costs $0.006 per napkin. 0.0051 0.005

180 0.9300900

300180120

≈

−

−

The 180-count size costs $0.005 per napkin. The 180-count size is the better buy.

68. 0.1195 0.120

20 2.39002 0

3920190180

100100

0

≈

−

−

−

−




296

0.1237 0.1248 0.9900

81916

3024

60564

≈

−

−

−

−


69. 2.3

9 20.7182 72 7

0

−

−

70. 8.6

7 60.256

42420

−

−

71. 3.7 0.555 becomes 0.15

37 5.553 71 851 85

0

−

−

72. 4.6 1.15 becomes 0.25

46 11.509 22 302 30

0

−

−

73. no; answers may vary

74. answers may vary

75. no; 6 inches 2 3 2

15 inches 5 3 5⋅

= =⋅

76. no; 1 4 2 1 942 2 2

⋅ += =

77. 10 defective to 200 good is 10 defective 1 defective .

200 good 20 good=

Yes, the machine should be repaired.

78. 3 bruised 1 bruised33 good 11 good

=

Since 1 1 ,11 10

< the shipment should not be

refused.

79. 79,543 − 79,286 = 257

13.4 257 becomes 19.17 19.2

134 2570.0013412301206

24 013 410 60

9 381 22

≈

−

−

−

−

79,895 − 79,543 = 352

15.8 352 becomes 22.27 22.3

158 3520.00316

360316

44031612401106

134

≈

−

−

−

−

80,242 − 79,895 = 347



297

16.1 347 becomes 21.55 21.6

161 3470.0032225016189 080 5

8 508 05

45

≈

−

−

−

−

Beginning Odometer Reading

Ending Odometer Reading

Miles Driven

Gallons of Gas

Used

Miles per Gallon (to the nearest

tenth)

79,286 79,543 257 13.4 19.2

79,543 79,895 352 15.8 22.3

79,895 80,242 347 16.1 21.6

80. 1.50 1.5

7759 11,674.007 7593 915 03 879 5

35 50

≈

−

The unit rate is 1.5 steps/foot.

81. 543.6 544

88300 48000000.0441500

385000353200

318000264900

531000529800

1200

≈

−

−

−

There are 544 students per school.

82. increases; answers may vary



85. no; answers may vary




298

87. a. 20 states 20 250 states 50 5

= =

b. 20 states 20 2

(50 20) states 30 3= =

−

c. no; answers may vary

Section 6.2

Practice Problems

1. a.

24 4cups cups6 1cups cups

→ ←=→ ←

b.

560 112students students25 5instructors instructors

→ ←=→ ←

2. 4 108 20

4 20 8 1080 80⋅ ⋅

=

�

�

The proportion is true.

3. 4.2 4.86 8

4.2 8 6 4.833.6 28.8⋅ ⋅

≠

�

�

The proportion is false.

4. 3 1

10 55 1

36

3 4

213 1 5 13 2 1 4

10 3 6 533 7 11 2110 3 6 5

231 23130 30

⋅ ⋅

⋅ ⋅

=

�

�

�


5. 25 25

2 25 550 550 55 510

x

xxx

x

=

⋅ = ⋅=

=

=

6. 15 602

15 2 6015 12015 12015 15

8

xxxx

x

=

⋅ = ⋅=

=

=

7. 7 28 3

47

7 4 28 7 3

1 22 3

1 3 2 32 2 3 2

34

z

z

z

z

z

=

⋅ = ⋅

= ⋅

⋅ = ⋅ ⋅

=

8. 0.6

9 1.21.2 9 0.61.2 5.41.2 5.41.2 1.2

4.5

y

yyy

y

=

= ⋅=

=

=

9. 17 8

1017 10 8

170 8170 8

8 8854

21.25

zzzz

z

z

=

⋅ = ⋅=

=

=

=

10. 4.51.8 3

4.5 3 1.813.5 1.813.5 1.81.8 1.87.5

y

yyy

y

=

⋅ ==

=

=


1. 4.2 18.4 2

= is called a proportion while 78

is called

a ratio.



299

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

products.

3. In a proportion, if cross products are equal, the proportion is true.

4. If cross products are false, the proportion is false.

Exercise Set 6.2

1. 10 diamonds 5 diamonds

6 opals 3 opals=

2. 1 raisin 8 raisins

5 cornflakes 40 cornflakes=

3. 20 students 4 students

5 microscopes 1 microscope=

4. 4 hits 1 hit

16 releasses 4 releases=

5. 6 eagles 3 eagles

58 sparrows 29 sparrows=

6. 12 errors 1.5 errors8 pages 1 page

=

7. 31

4 42 cups flour 6 cups flour

24 cookies 72 cookies=

8. 31

2 41 cups milk cup milk

10 bagels 5 bagels=

9. 22 vanilla wafers 55 vanilla wafers

1 cup cookie crumbs 2.5 cups cookie crumbs=

10. 1 cup of instant rice 1.5 cups instant rice1.5 cups cooked rice 2.25 cups cooked rice

=

11. 15 59 3

15 3 9 545 45⋅ ⋅

=

�

�


12. 8 206 15

8 15 6 20120 120⋅ ⋅

=

�

�


13. 5 48 7

5 7 8 435 32⋅ ⋅

≠

�

�


14. 7 93 5

7 5 3 935 27⋅ ⋅

≠

�

�


15. 9 2

36 89 8 36 2

72 72⋅ ⋅

=

�

�


16. 8 324 9

8 9 24 372 72⋅ ⋅

=

�

�


17. 5 6258 1000

5 1000 8 6255000 5000⋅ ⋅

=

�

�


18. 30 60050 1000

30 1000 50 60030,000 30,000⋅ ⋅

=

�

�


19. 0.8 0.20.3 0.6

0.8 0.6 0.3 0.20.48 0.06⋅ ⋅

≠

�

�




300

20. 0.7 0.30.4 0.1

0.7 0.1 0.4 0.30.07 0.12⋅ ⋅

≠

�

�


21. 4.2 58.4 10

4.2 10 8.4 542 42⋅ ⋅

=

�

�


22. 8 5.6

10 0.78 0.7 10 5.6

5.6 56⋅ ⋅

≠

�

�


23. 3 14 24 83 9

3 8 4 14 9 3 2

2 23 3

⋅ ⋅

=

�

�


24. 2 13 78 35 10

2 3 8 13 10 5 7

1 85 35

⋅ ⋅

≠

�

�


25. 102

5 92 13 4

2

2 1 2 1025 4 3 9

12 1 205 4 27

12 2020 27

⋅ ⋅

⋅

≠

�

�

�


26. 5 18 2

5 153

5 4

15 1 5 15 1 48 5 3 245 6 5 98 5 3 2

27 154 2

⋅ ⋅

⋅ ⋅

≠

�

�

�


27. 64

5 56 9

4 69 65 5

36 365 5

⋅ ⋅

=

�

�


28. 6 107 73 5

6 105 37 7

30 307 7

⋅ ⋅

=

�

�


29. 10 415 6

10 6 15 460 60⋅ ⋅

=

�

�

The proportion 10 415 6

= is true.

30. 6 98 12

6 12 8 972 72⋅ ⋅

=

�

�

The proportion 6 98 12= is true.

31. 11 54 2

11 2 4 522 20⋅ ⋅

≠

�

�

The proportion 11 54 2= is false.



301

32. 5 73 5

5 5 3 725 21⋅ ⋅

≠

�

�

The proportion 5 73 5= is false.

33. 0.15 0.35

3 70.15 7 3 0.35

1.05 1.05⋅ ⋅

=

�

�

The proportion 0.15 0.35

3 7= is true.

34. 1.8 4.52 5

1.8 5 2 4.59 9⋅ ⋅

=

�

�

The proportion 1.8 4.52 5

= is true.

35. 2 23 51 15 9

2 1 1 23 9 5 5

2 227 25

⋅ ⋅

≠

�

�

The proportion 2 23 51 15 9

= is false.

36. 10 111 43 1

2410 1 3 111 2 4 4

5 311 16

⋅ ⋅

≠

�

�

The proportion 10 111 43 1

24

= is false.

37. 6

5 1010 5 6

10 3010 3010 10

3

x

xxx

x

=

⋅ = ⋅=

=

=

38. 12

3 99 3 12

9 369 369 9

4

x

xxx

x

=

⋅ = ⋅=

=

=

39. 18 3

5418 54 318 16218 16218 18

9

nnnn

n

−=

− ⋅ = ⋅− =−

=− −

= −

40. 25 7

10025 100 ( 7)25 70025 70025 25

28

nnnn

n

−=

⋅ = ⋅ −= −−

=

= −

41. 30 1510

30 10 1530 15030 15030 30

5

yyyy

y

=

⋅ = ⋅=

=

=

42. 1620 35

16 35 20560 20560 2020 2028

z

zzz

z

=

⋅ = ⋅=

=

=

43. 8

15 68 6 1548 1548 1515 153.2

z

zzz

z

=

⋅ = ⋅=

=

=



302

44. 1210 16

12 16 10192 10192 1010 10

19.2

z

zzz

z

=

⋅ = ⋅=

=

=

45. 24 60

9624 96 602304 602304 6060 6038.4

xxxx

x

=

⋅ = ⋅=

=

=

46. 26 28

4926 49 281274 281274 28

28 2845.5

xxxx

x

=

⋅ = ⋅=

=

=

47. 3.5 7

12.53.5 12.5 ( 7)3.5 87.53.5 87.53.5 3.5

25

nnnn

n

− −=

− ⋅ = ⋅ −− = −− −

=− −

=

48. 0.2 8

0.70.2 0.7 ( 8)0.2 5.60.2 5.60.2 0.2

28

nnnn

n

− −=

− ⋅ = ⋅ −− = −− −

=− −

=

49. 0.05

0.6 1212 0.6 0.05

12 0.03012 0.0312 12

0.0025

n

nnn

n

=

⋅ = ⋅=

=

=

50. 7.813 2.6

7.8 2.6 1320.28 1320.28 13

13 131.56

n

nnn

n

=

⋅ = ⋅=

=

=

51. 13

8 24

18 243

8 88 88 8

1

n

n

nn

n

=

⋅ = ⋅

=

=

=

52. 34

12 48

312 484

12 3612 3612 12

3

n

n

nn

n

=

⋅ = ⋅

=

=

=

53. 1 23 538

1 3 23 8 5

33 20

33 33 20

920

n

n

n

n

n

=

⋅ = ⋅

=

⋅ = ⋅

=

54. 7 19 4827

7 8 19 27 47 29 27

9 7 9 27 9 7 27

221

n

n

n

n

n

=

⋅ = ⋅

=

⋅ = ⋅

=



303

55. 2369

12

6 2129 3

283

3 8 2 32 1 3 2

12

n

n

n

n

n

=

⋅ = ⋅

= ⋅

⋅ = ⋅ ⋅

=

56. 8

1559

24

5 8249 15

40 83 15

15 40 8 158 3 15 8

25

n

n

n

n

n

=

⋅ = ⋅

= ⋅

⋅ = ⋅ ⋅

=

57.

16

1 25 3

4

1 62 1 16 1 43 5 6

20 6 253 5 6

20 53

3 20 3 520 3 20

34

n

n

n

n

n

n

=

⋅ = ⋅

⋅ = ⋅

=

⋅ = ⋅

=

58. 35

1 38 8

7

3 23 1 32 3 78 8 5

19 25 388 8 5

19 958 4

8 19 8 9519 8 19 4

10

n

n

n

n

n

n

=

⋅ = ⋅

⋅ = ⋅

⋅ =

⋅ ⋅ = ⋅

=

59. 730

25 3

725 33035 36

1 35 133 6 3

3518

n

n

n

n

n

=

⋅ = ⋅

=

⋅ = ⋅

=

60. 1115

9 5

119 51533 55

1 33 155 5 5

3325

n

n

n

n

n

=

⋅ = ⋅

=

⋅ = ⋅

=

61. 3.20.3 1.4

3.2 1.4 0.34.48 0.34.48 0.30.3 0.314.9

x

xxx

x

=

⋅ = ⋅=

=

≈

62. 1.8 2.5

8.41.8 8.4 2.5

15.12 2.515.12 2.5

2.5 2.56.0

zzzz

z

=

⋅ = ⋅=

=

≈

63. 0.08

5.2 66 5.2 0.08

6 0.4166 0.4166 6

0.07

z

zzz

z

=

⋅ = ⋅=

=

≈



304

64. 4.25 56.03

4.25 6.03 54.25 30.154.25 30.154.25 4.25

7.09

yyyy

y

=

⋅ = ⋅=

=

≈

65. 7

18 57 5 1835 1835 1818 181.9

x

xxx

x

=

⋅ = ⋅=

=

≈

66. 17 9

417 4 9

68 968 99 9

7.556

xxxx

x

=

⋅ = ⋅=

=

≈

67. 43 817

43 17 843 13643 13643 43

3.163

zzzz

z

=

⋅ = ⋅=

=

≈

68. 18

12 77 12 18

7 2167 2167 7

30.86

x

xxx

x

=

⋅ = ⋅=

=

≈

69. 8.01 8.1 ↑ ↑ 0 < 1 8.01 < 8.1

70. 7.26 7.026 ↑ ↑ 2 > 0 7.26 > 7.026

71. 1 12 3>

so 1 12 22 3>

72. 1 15 4<

so 1 19 95 4<

73. 75 3 25 3

125 5 25 5⋅

= =⋅

74. 11 1 11 199 9 11 9y yy y

⋅ ⋅= =

⋅ ⋅

75. 12 2 6 242 7 6 7x x x⋅ ⋅= =

⋅

76. 2

328 2 14 2

3 14 342y y y

y y y yy⋅ ⋅ ⋅

= =⋅ ⋅ ⋅ ⋅

77. 9 3

15 5=

9 153 5=

5 315 9

=

15 59 3=

78. 1 54 201 45 2020 54 14 201 5

=

=

=

=

79. 6 1

18 36 181 33 1

18 618 36 1

=

=

=

=



305

80. 2 47 1414 47 22 74 147 142 4

=

=

=

=

81. a cb d=

d cb a=

a bc d=

b da c=




85. 0

7 88 7 0

8 08 08 8

0

x

xxx

x

=

⋅ = ⋅=

=

=

86. 02 3.5

0 3.5 20 20 22 20

y

yyy

y

=

⋅ = ⋅=

=

=

87. 588

1150 483483 1150 588483 676,200483 676,200483 483

1400

z

zzz

z

=

⋅ = ⋅=

=

=

88. 585 117

474585 474 117277,290 117277,290 117

117 1172370

xxxx

x

=

⋅ = ⋅=

=

=

89. 222 37

1515222 1515 37222 56,055222 56,055222 222

252.5

yyyy

y

=

⋅ = ⋅=

=

=

90. 14251062 177

1425 177 1062252,225 1062252,225 1062

1062 1062237.5

z

zzz

z

=

⋅ = ⋅=

=

=

The Bigger Picture

1. 7 1 7 1 2 7 2 5 120 10 20 10 2 20 20 20 4

⋅− = − = − = =

⋅

2. 7 1 7 1 720 10 20 10 200

⋅⋅ = =

⋅

3. 7 1 7 10 7 10 7 1 or 320 10 20 1 2 10 1 2 2

⋅÷ = ⋅ = =

⋅ ⋅

4. 7 1 7 1 2 7 2 920 10 20 10 2 20 20 20

⋅+ = + = + =

⋅

5. 7.6 + 0.02 = 7.62

6. 7.60.02

0.152×

7. 4 50

1004 100 50

400 50400 5050 50

8

nnnn

n

=

⋅ = ⋅=

=

=



306

8. 60 1510

60 10 1560 10 1560 15060 15060 60

5 1 or 22 2

nnnnn

n

=

⋅ = ⋅= ⋅=

=

=

9. 0.06

0.8 1212 0.8 0.06

12 0.04812 0.04812 12

0.004

n

nnn

n

=

⋅ = ⋅=

=

=

10. 781 54 6

7 5 18 6 4

35 148 4

4 35 4 11 48 1 4

35 11 or 212 12

n

n

n

n

n n

=

⋅ = ⋅

=

⋅ = ⋅

= =

Integrated Review

1. The ratio of 27 to 30 is 27 9 3 9 .30 10 3 10

⋅= =

⋅

2. The ratio of 18 to 50 is 18 9 2 9 .50 25 2 25

⋅= =

⋅

3. The ratio of 9.4 to 10 is 9.4 9.4 10 94 47 2 47 .10 10 10 100 50 2 50

× ⋅= = = =

× ⋅

4. The ratio of 3.2 to 9.2 is 3.2 3.2 10 32 4 8 8 .9.2 9.2 10 92 4 23 23

× ⋅= = = =

× ⋅

5. The ratio of 8.65 to 6.95 is 8.65 8.65 100 865 173 5 173 .6.95 6.95 100 695 139 5 139

× ⋅= = = =

× ⋅

6. The ratio of 3.6 to 4.2 is 3.6 3.6 10 36 6 6 6 .4.2 4.2 10 42 7 6 7

× ⋅= = = =

× ⋅

7. The ratio of 72

to13 is

72 7 7 1 7 1 713 .

13 2 2 13 2 13 26⋅

= ÷ = ⋅ = =⋅

8. The ratio of 213

to 324

is

2334

1 2 3 5 11 5 4 5 4 201 2 .3 4 3 4 3 11 3 11 332

⋅= ÷ = ÷ = ⋅ = =

⋅

9. The ratio of 16 inches to 24 inches is 16 inches 2 8 2 .24 inches 3 8 3

⋅= =

⋅

10. The ratio of 5 hours to 40 hours is 5 hours 5 1 1 .40 hours 5 8 8

⋅= =

⋅

11. 12 inches 6 2 218 inches 6 3 3

⋅= =

⋅

12. a. 10 films were rated PG-13.

b. 10 films 10 1 120 films 10 2 2

⋅= =

⋅

13. 4 professors 1 professor

20 graduate assistants 5 graduate assistants=

14. 6 lights 3 lights20 feet 10 feet

=

15. 100 Senators 2 Senators

50 states 1 state=

16. 5 teachers 1 teacher

140 students 28 students=

17. 21 inches 3 inches7 seconds 1 second

=

18. $40 $8

5 hours 1 hour=



307

19. 76 households with computers

100 households19 households with computers

25 households=

20. 538 electoral votes 269 electoral votes

50 states 25 states=

21. 560 feet

4 seconds

1404 560

41616

00

−

−

The unit rate is 140 feet1 second

or 140 feet/second

22. 195 miles

3 hours

653 195

1815150

−

−

The unit rate is 65 miles1 hour

or 65 miles/hour.

23. 63 employees

3 fax lines

213 63

60330

−

−

The unit rate is 21 employees

1 fax line or

21 employees/fax line.

24. 85 phone calls

5 teenagers

175 85

535350

−

−

The unit rate is 17 phone calls

1 teenager or

17 phone calls/teenager.

25. 156 miles6 gallons

26

6 15612

36360

−

The unit rate is 26 miles1 gallon

or 26 miles/gallon.

26. 112 teachers7 computers

16

7 112742420

−

−

The unit rate is 16 teachers1 computer

or

16 teachers/computer.

27. 8125 books

1250 college students

6.51250 8125.0

7500625 0625 0

0

−

−

The unit rate is 6.5 books

1 college student or

6.5 books/student.

28. 2310 pounds

14 adults

16514 2310

149184

7070

0

−

−

−

The unit rate is 165 pounds

1 adult or

165 pounds/adult.

29. 0.27

8 2.161 6

56560

−

−

The 8-pound size costs $0.27 per pound.



308

0.2772 0.27718 4.9900

3 61 391 26

130126

40364

≈

−

−

−

−

The 18-pound size costs $0.277 per pound. The 8-pound size is the better buy.

30. 0.0198 0.020

100 1.98001 00

980900

800800

0

≈

−

−

−

The 100-plate size costs $0.20 per plate. 0.0179 0.018

500 8.99005 003 9903 500

49004500

400

≈

−

−

−

The 500-plate size costs $0.018 per plate. The 500 paper plates is the better buy.

31. 0.7966... 0.800

3 2.39002 1

2927

201820182

≈

−

−

−

−

The 3-pack size costs $0.800 per pack.

0.7487 0.7508 5.9900

5 63932

706460564

≈

−

−

−

−

The 8-pack size costs $0.750 per pack. The 8 pack is the better buy.

32. 0.9225 0.923

4 3.69003 6

09810

820200

≈

−

−

−

−

The 4 batteries cost $0.923 each. 0.989

10 9.8909 0

8980

90900

−

−

−

The 10 batteries cost $0.989 each. The 4 batteries are the better buy.

33. 7 54 3

7 3 4 521 20⋅ ⋅

≠

�

�


34. 8.2 16.42 4

8.2 4 2 16.432.8 32.8⋅ ⋅

=

�

�




309

35. 5 403

5 3 405 1205 1205 5

24

xxxx

x

=

⋅ = ⋅=

=

=

36. 13

10 44 10 13

4 1304 1304 4

32.5

y

yyy

y

=

⋅ = ⋅=

=

=

37. 6

11 56 5 1130 1130 1111 11

82.72 or 211

z

zzz

z z

=

⋅ = ⋅=

=

= =

38. 72213

721 32

7632

2 63 2 77 1 7 2

18

x

x

x

x

x

=

⋅ = ⋅

= ⋅

⋅ = ⋅ ⋅

=

Section 6.3

Practice Problems

1. Let x be the length of the wall.

14

4feet feet41inches inches

14 4 14

4 171 4

17

x

x

x

x

→ ←=→ ←

⋅ = ⋅

⋅ =

=

The wall is 17 feet long.

2. 5 8ounces ounces

16gallons gallons5 16 85 1285 1285 5

325.6 or 255

xxxx

x

→ ←=→ ←

⋅ = ⋅=

=

=

Therefore, 25.6 or 3255

gallons of gas can be

treated with 8 ounces of alcohol.

3. Let x be the number of gallons. 270 × 11 = 2970 square feet

1gallons gallons450 2970square feet square feet

1 2970 4502970 450450 450

6.67

x

xx

xx

→ ←=→ ←⋅ = ⋅

=

=≈

Therefore, 7 gallons are needed for the wall.

Exercise Set 6.3

1. Let x be the number of baskets made. 45baskets attempts

100 800attempts baskets45 800 10036,000 10036,000 100

100 100360

x

xxx

x

→ ←=→ ←⋅ = ⋅

=

=

=

He made 360 baskets.

2. Let x be the number of attempts. 45 225baskets baskets

100attempts attempts45 100 22545 22,50045 22,50045 45

500

xxxx

x

→ ←=→ ←

⋅ = ⋅=

=

=

He made 500 attempts.



310

3. Let x be the number of minutes. 30minutes minutes4 22pages pages

30 22 4660 4660 4

4 4165

x

xxx

x

→ ←=→ ←⋅ = ⋅

=

=

=

It will take her 165 minutes to word process and spell check 22 pages.

4. Let x be the number of pages. 30 4.5 60minutes minutes4pages pages

30 4 4.5 6030 108030 108030 30

36

xxxx

x

×→ ←=→ ←

⋅ = ⋅ ⋅=

=

=

She can word process and spell check 36 pages in 4.5 hours.

5. Let x be the number of applications received. 2 180accepted accepted7applied applied

2 7 1802 12602 12602 2

630

xxxx

x

→ ←=→ ←⋅ = ⋅

=

=

=

The school received 630 applications.

6. Let x be the number of applications received. 2 150accepted accepted7applied applied

2 7 1502 10502 10502 2

525

xxxx

x

→ ←=→ ←⋅ = ⋅

=

=

=

The school received 525 applications.

7. Let x be the length of the wall. 7821inches inches

8feet feet71 8 28

8 231 823

x

x

x

x

→ ←=→ ←

⋅ = ⋅

= ⋅

=


8. Let x be the length of the wall. 1451inches inches

8feet feet11 8 54

8 211 442

x

x

x

x

→ ←=→ ←

⋅ = ⋅

= ⋅

=


9. Let x be the number of square feet required. 9floor space floor space1 30students students

9 30 1270

x

xx

→ ←=→ ←⋅ = ⋅

=

30 students require 270 square feet of floor space.

10. Let x be the number of students. 9 21 15floor space floor space1students students

9 1 21 159 3159 3159 9

35

xxxx

x

×→ ←=→ ←⋅ = ⋅ ⋅

=

=

=

The room can accommodate 35 students.

11. Let x be the number of gallons. 627 1250miles miles12.3gallons gallons

627 12.3 1250627 15,375627 15,375627 627

25

xxxx

x

→ ←=→ ←

⋅ = ⋅=

=

≈

He can expect to burn 25 gallons.

12. Let x be the number of miles she can go. 627miles miles12.3 6.9gallons gallons

627 6.9 12.34326.3 12.34326.3 12.312.3 12.3

352

x

xxx

x

→ ←=→ ←

⋅ = ⋅=

=

≈

Yes, she should be able to go 352 miles (which is greater than 290 miles).



311

13. Let x be the distance between Milan and Rome. 30kilometers kilometers1 15cm on map cm on map

30 15 1450

x

xx

→ ←=→ ←⋅ = ⋅

=

Milan and Rome are 450 kilometers apart.

14. Let x be the distance between the small Italian village and the Mediterranean Sea.

30kilometers kilometers1 0.4cm on map cm on map

30 0.4 112

x

xx

→ ←=→ ←⋅ = ⋅

=

The actual distance is 12 kilometers.

15. Let x be the number of bags. 1bags bags

3000 260 180square feet square feet1 260 180 3000

46,800 300046,800 30003000 3000

15.616

x

xxx

xx

→ ←= ⋅→ ←⋅ ⋅ = ⋅

=

=

=≈

16 bags of fertilizer should be purchased.



25,600 300025,600 30003000 3000

8.59

x

xxx

xx

→ ←= ⋅→ ←⋅ ⋅ = ⋅

=

=

≈≈

9 bags of fertilizer should be purchased.

17. Let x be the number of hits the player is expected to get.

3hits hits8 40at bats at bats

3 40 8120 8120 8

8 815

x

xxx

x

→ ←=→ ←⋅ = ⋅

=

=

=

The player would be expected to get 15 hits.

18. Let x be the times at bat. 3 12hits hits8at bats at bats

3 8 123 963 963 3

32

xxxx

x

→ ←=→ ←⋅ = ⋅

=

=

=

He batted 32 times.

19. Let x be the number that prefer Coke. 2Coke Coke3 40Total Total

2 40 380 380 33 327

x

xxx

x

→ ←=→ ←⋅ = ⋅

=

=

≈

About 27 people are likely to prefer Coke.

20. Let x be the number that prefer Pepsi. 1Pepsi Pepsi3 36Total Total

1 36 336 33 312

x

xx

x

→ ←=→ ←⋅ = ⋅

=

=

12 students are likely to prefer Pepsi.

21. Let x be the number of applications she should expect.

4applications applications3 14ounces ounces

4 14 356 356 33 32183

x

xxx

x

→ ←=→ ←⋅ = ⋅

=

=

=

She should expect 18 applications from the 14-ounce bottle.

22. Let x be the number of ounces. 4 35applications applications3ounces ounces

4 3 354 1054 1054 4

27

xxxx

x

→ ←=→ ←⋅ = ⋅

=

=

≈

They should purchase 27 ounces.



312

23. Let x be the number of weeks. 5 8reams reams3weeks weeks

5 3 85 245 245 5

4.85

xxxx

xx

→ ←=→ ←⋅ = ⋅

=

=

=≈

The case is likely to last 5 weeks.

24. Let x be the number of reams. 5reams reams3 15weeks weeks

5 15 375 375 33 325

x

xxx

x

→ ←=→ ←⋅ = ⋅

=

=

=

They will need 25 reams or 25 3.125 4 cases.8= ≈

25. Let x be the number of servings he can make. 121 4milk milk

servings servings411 4 423 162

2 3 2 163 2 3

32 2103 3

x

x

x

x

x

→ ←=→ ←

⋅ = ⋅

=

⋅ = ⋅

= =

He can make 2103

servings.

26. Let x be the number of cups of flour. 2flour flour4 18servings servings

2 18 436 436 44 49

x

xxx

x

→ ←=→ ←⋅ = ⋅

=

=

=

It will take 9 cups of flour.

27. Let x be the number of calories. 16 24ounces ounces80calories calories

16 80 2416 192016 192016 16

120

xxxx

x

→ ←=→ ←

⋅ = ⋅=

=

=

There are 120 calories in 24 ounces.

28. Let x be the number of calories. 16 12ounces ounces220calories calories

16 220 1216 264016 264016 16

165

xxxx

x

→ ←=→ ←

⋅ = ⋅=

=

=

There are 165 calories in 12 ounces.

29. a. Let x be the number of teaspoons of granules needed.

25 450water water1granules granules

25 1 45025 45025 45025 25

18

xxxx

x

→ ←=→ ←

⋅ = ⋅=

=

=

18 teaspoons of granules are needed.

b. Let x be the number of tablespoons of granules needed.

3 18tsp tsp1tbsp tbsp

3 1 183 183 183 3

6

xxxx

x

→ ←=→ ←⋅ = ⋅

=

=

=

6 tablespoons of granules are needed.

30. a. Let x be the number of ounces. 3ounces ounces

150 5280feet feet3 5280 15015,840 15015,840 150

150 150105.6

x

xxx

x

→ ←=→ ←⋅ = ⋅

=

=

=

105.6 ounces of pesticide are needed.



313

b. Let x be the number of cups. 8 105.6ounces ounces1cups cups

8 1 105.68 105.68 8

13.213

xxx

xx

→ ←=→ ←⋅ = ⋅

=

=≈

This is about 13 cups of pesticide.

31. Let x be the number of people. 625 3750square feet square feet1people people

625 1 3750625 3750625 625

6

xxx

x

→ ←=→ ←

⋅ = ⋅

=

=

It will provide enough oxygen for 6 people.

32. Let x be the number of cars and light trucks with air bags.

71air bags air bags200 800total total

71 800 20056,800 20056,800 200

200 200284

x

xxx

x

→ ←=→ ←⋅ = ⋅

=

=

=

The expected number with air bags is 284 cars and light trucks.

33. Let x be the estimated head-to-toe height of the Statue of Liberty.

135height height

42arm length arm length212 42 53

162 423

2 2242 2242 2

112

x

x

x

xx

x

→ ←=→ ←

⋅ = ⋅

= ⋅

=

=

=

The estimated height is 112 feet. 1 11112 111

12 12− =

The difference is 1112

foot or 11 inches.

34. a. Let x be the length of your index finger. 8index finger index finger

111 5height height8 5 11140 11140 111

111 11140 0.36

111

x

xxx

x

→ ←=→ ←

⋅ = ⋅=

=

= ≈

The proposed length is 40

111 foot or about

0.36 foot.

b. answers may vary

35. Let x be the number of milligrams. 72milligrams milligrams3.5 5ounces ounces

72 5 3.5360 3.5360 3.53.5 3.5

102.9

x

xxx

x

→ ←=→ ←

⋅ = ⋅=

=

≈

There are about 102.9 milligrams of cholesterol in 5 ounces of lobster.

36. Let x be the number of milligrams. 76milligrams milligrams3 8ounces ounces

76 8 3608 3608 33 322023

x

xxx

x

→ ←=→ ←

⋅ = ⋅=

=

=

There are 22023

milligrams of cholesterol in

8 ounces of chicken.

37. Let x be the estimated height of the Empire State Building.

881height height102 72stories stories

72 102 88172 89,86272 89,86272 72

1248

x

xxx

x

→ ←=→ ←

⋅ = ⋅=

=

≈

The height of the Empire State Building is approximately 1248 feet.



314

38. Let x be the number of men that blame. 2blame blame5 40total total

2 40 580 580 55 516

x

xxx

x

→ ←=→ ←⋅ = ⋅

=

=

=

You would expect 16 men to blame their not eating well on fast food.

39. Let x be the number of visits needing a prescription for medication.

7medication medication10 620total total

7 620 104340 104340 1010 10434

x

xxx

x

→ ←=→ ←⋅ = ⋅

=

=

=

Expect 434 emergency room visits to need a prescription for medication.

40. Let x be the number of female residents. 145female female100 280male male

145 280 10040,600 10040,600 100

100 100406

x

xxx

x

→ ←=→ ←⋅ = ⋅

=

=

=

Expect 406 female residents.

41. Let x be the number expected to have worked in the restaurant industry.

1restaurant restaurant84 3workers workers

3 84 13 843 843 3

28

x

xxx

x

→ ←=→ ←

⋅ = ⋅=

=

=

You would expect 28 of the workers to have worked in the restaurant industry.

42. Let x be the number of pounds.

14

1pounds pounds2 6cups cups

11 6 24

964

4 6 4 99 1 9 4

83

223

x

x

x

x

x

x

→ ←=→ ←

⋅ = ⋅

= ⋅

⋅ = ⋅

=

=

The recipe requires 223

pounds of firmly

packed brown sugar.

43. Let x be the cups of salt. 5 12ice ice1salt salt

5 1 125 125 5

2.4

xxx

x

→ ←=→ ←⋅ = ⋅

=

=

Mix 2.4 cups of salt with the ice.

44. Let x be the cups of salt. 34185ice ice

1salt salt35 1 184

7554

1 1 7555 5 4

154334

x

x

x

x

x

x

→ ←=→ ←

⋅ = ⋅

=

⋅ = ⋅

=

=

Mix 334

cups of salt with the ice.



315

45. a. Let x be the number of gallons of oil needed.

1oil oil5 50gas gas

50 5 150 550 550 50

1 0.110

x

xxx

x

→ ←=→ ←⋅ = ⋅

=

=

= =

0.1 gallon of oil is needed.

b. Let x be the number of fluid ounces. 1 0.1gallons gallons

128fluid ounces fluid ounces1 128 0.1

12.8

xxx

→ ←=→ ←

⋅ = ⋅=

0.1 gallon is approximately 13 fluid ounces

46. a. Let x be the number of gallons of oil needed.

1oil oil10 20gas gas20 10 1

20 1020 20

0.5

x

xx

x

→ ←=→ ←⋅ = ⋅

=

=

0.5 gallon of oil is needed.

b. Let x be the number of quarts. 1 0.5gallons gallons4quarts quarts

1 4 0.52

xxx

→ ←=→ ←⋅ = ⋅

=

0.5 gallon is 2 quarts.

47. a. Let x be the milligrams of medicine. 150milligrams milligrams

275 20pounds pounds20 275 150

20 41,25020 41,25020 20

2062.5

x

xxx

x

→ ←=→ ←

⋅ = ⋅=

=

=

The daily dose is 2062.5 milligrams.

b. 24500 500 3 15008

× = × =

No, he is not receiving the proper dosage.

48. a. Let x be the milligrams of medicine. 80milligrams milligrams

190 25pounds pounds25 190 8025 15, 20025 15, 20025 25

608

x

xxx

x

→ ←=→ ←

⋅ = ⋅=

=

=

The daily dose is 608 milligrams.

b. 24608 608 4 1526

÷ = ÷ =

She should be given 152 milligrams every 6 hours.

49.

3 2

200 2 100

2 4 25

2 2 2 5 52 5

= ⋅↓ ↓

= ⋅ ⋅↓

= ⋅ ⋅ ⋅ ⋅

= ⋅

50.

2 2

300 2 150

2 2 75

2 2 3 25

2 2 3 5 52 3 5

= ⋅↓ ↓⋅ ⋅

↓ ↓ ↓⋅ ⋅ ⋅

↓ ↓ ↓ ↓⋅ ⋅ ⋅ ⋅

= ⋅ ⋅

51.

5

32 2 16

2 2 8

2 2 2 4

2 2 2 2 22

= ⋅↓ ↓⋅ ⋅

↓ ↓ ↓⋅ ⋅ ⋅

↓ ↓ ↓ ↓⋅ ⋅ ⋅ ⋅

=

52.

4

81 3 27

3 3 9

3 3 3 33

= ⋅↓ ↓⋅ ⋅

↓ ↓ ↓⋅ ⋅ ⋅

=



316

53. Let x be the number of ml. 15 12mg mg1ml ml

15 1 1215 1215 1215 15

4 0.85

xxxx

x

→ ←=→ ←⋅ = ⋅

=

=

= =

0.8 ml of the medicine should be administered.


15 1 3315 3315 3315 15

2.2

xxxx

x

→ ←=→ ←⋅ = ⋅

=

=

=



8 1 108 108 108 8

1.25

xxxx

x

→ ←=→ ←⋅ = ⋅

=

=

=



8 1 68 68 68 8

0.75

xxxx

x

→ ←=→ ←⋅ = ⋅

=

=

=


57. 11 muffins are approximately 1 dozen (12) muffins. 1.5 ⋅ 8 = 12 Approximately 12 cups of milk will be needed.

58. 2 dozen cookies = 24 cookies If we double 24, we have 48 ≈ 50, so double

122

cups of flour, or 5 cups of flour.

59. 7feet feet60 40pounds pounds

7 40 60280 60280 6060 60

243

n

nnn

n

→ ←=→ ←⋅ = ⋅

=

=

=

The distance is 243

feet.

60. 3feet feet

125 400pounds pounds3 400 1251200 1251200 125125 125

395

n

nnn

n

→ ←=→ ←⋅ = ⋅

=

=

=

The distance is 395

feet.


Section 6.4

Practice Problems

1. 100 10= because 210 100.=

2. 64 8= because 28 64.=

3. 121 11= because 211 121.=

4. 0 0= because 20 0.=

5. 1 14 2= because

21 1 .2 4

⎛ ⎞ =⎜ ⎟⎝ ⎠

6. 9 3

16 4= because

23 9 .4 16

⎛ ⎞ =⎜ ⎟⎝ ⎠

7. a. 10 3.162≈

b. 62 7.874≈



317

8. Let a = 12 and b = 16. 2 2 2

2 2 2

2

2

12 16144 256

400400

20

a b cccccc

+ =

+ =

+ =

===

The hypotenuse is 20 feet.

9. Let a = 9 and b = 7. 2 2 2

2 2 2

2

2

9 781 49

130130

11

a b cccccc

+ =

+ =

+ =

==≈

The hypotenuse is approximately 11 kilometers.

10. Let a = 7 and c = 13. 2 2 2

2 2 2

2

2

7 1349 169

120120

10.95

a b cbbbbb

+ =

+ =

+ =

==≈

The length of the hypotenuse is exactly

120 feet or approximately 10.95 feet.

11.

100 yd

53 yd

Let a = 53 and b = 100.

2 2 2

2 2 2

2

2

53 1002809 10,000

12,80912,809

113

a b cccccc

+ =

+ =

+ =

==≈

The diagonal is approximately 113 yards.

Calculator Explorations

1. 1024 32=

2. 676 26=

3. 15 3.873≈

4. 19 4.359≈

5. 97 9.849≈

6. 56 7.483≈


1. A square root of 100 is 10 and −10 because 10 ⋅ 10 = 100 and (−10)(−10) = 100.

2. 100 10= only because 10 ⋅ 10 = 100 and 10 is positive.

3. The radical sign is used to denote the positive square root of a nonnegative number.

4. The reverse process of squaring a number is finding a square root of a number.

5. The numbers 9, 1, and 125

are called perfect

squares.

6. Label the parts of the right triangle.

hypotenuseleg

leg

7. In the given triangle, 2 2 2.a c b+ =

Exercise Set 6.4

1. 4 2= because 22 4.=

2. 9 3= because 23 9.=

3. 121 11= because 211 121.=

4. 144 12= because 212 144.=

5. 1 181 9

= because 21 1 1 1 .

9 9 9 81⎛ ⎞ = ⋅ =⎜ ⎟⎝ ⎠

6. 1 164 8

= because 21 1 1 1 .

8 8 8 64⎛ ⎞ = ⋅ =⎜ ⎟⎝ ⎠



318

7. 16 4 164 8 2

= = because 24 4 4 16 .

8 8 8 64⎛ ⎞ = ⋅ =⎜ ⎟⎝ ⎠

8. 36 6 281 9 3

= = because 26 6 6 36 .

9 9 9 81⎛ ⎞ = ⋅ =⎜ ⎟⎝ ⎠

9. 3 1.732≈

10. 5 2.236≈

11. 15 3.873≈

12. 17 4.123≈

13. 31 5.568≈

14. 85 9.220≈

15. 26 5.099≈

16. 35 5.916≈

17. 256 16= because 216 256.=

18. 625 25= because 225 625.=

19. 92 9.592≈

20. 18 4.243≈

21. 49 7

144 12= because

27 7 7 49 .12 12 12 144⎛ ⎞ = ⋅ =⎜ ⎟⎝ ⎠

22. 121 11169 13

= because 211 11 11 121.

13 13 13 169⎛ ⎞ = ⋅ =⎜ ⎟⎝ ⎠

23. 71 8.426≈

24. 62 7.874≈

25. Let a = 5 and b = 12. 2 2 2

2 2 2

2

2

5 1225 144

169169

13

a b cccccc

+ =

+ =

+ =

===

The missing length is 13 inches.

26. Let a = 36 and b = 15. 2 2 2

2 2 2

2

2

36 151296 225

15211521

39

a b cccccc

+ =

+ =

+ =

===

The missing length is 39 feet.

27. Let a = 10 and c = 12. 2 2 2

2 2 2

2

2

10 12100 144

4444

6.633

a b cbbbbb

+ =

+ =

+ =

==≈

The missing length is approximately 6.633 centimeters.

28. Let a = 3 and c = 9. 2 2 2

2 2 2

2

2

3 99 81

7272

8.485

a b cbbbbb

+ =

+ =

+ =

==≈

The missing length is approximately 8.485 yards.

29. Let a = 22 and b = 48. 2 2 2

2 2 2

2

2

22 48484 23042788

278852.802

c a bccccc

= +

= +

= +

==≈

The missing length is approximately 52.802 meters.

30. Let a = 34 and b = 70. 2 2 2

2 2 2

2

2

34 701156 4900

60566056

77.820

a b cccccc

+ =

+ =

+ =

==≈

The missing length is approximately 77.820 miles.



319

31. Let a = 108 and b = 45. 2 2 2

2 2 2

2

2

108 4511,664 2025

13,68913,689

117

a b cccccc

+ =

+ =

+ =

===

The missing length is 117 millimeters.

32. Let a = 36 and b = 27. 2 2 2

2 2 2

2

2

36 271296 729

20252025

45

a b cccccc

+ =

+ =

+ =

===

The missing length is 45 kilometers.

33.

4

3 2 2

2 2

hypotenuse (leg) (other leg)

(3) (4)9 1625

5

= +

= += +==

The hypotenuse has length 5 units.

34.

12

9 2 2

2 2


9 1281 144225

15

= +

= += +==

The hypotenuse has length 15 units.

35.

513

2 2

2 2

leg (hypotenuse) (other leg)

13 5169 25144

12

= −

= −= −==

The leg has length 12 units.

36.

610

2 2

2 2


10 6100 3664

8

= −

= −= −==


37.

14

10 2 2

2 2


(10) (14)100 196296

17.205

= +

= += +=≈

The hypotenuse has length of about 17.205 units.

38.

16

2



320

2 2

2 2


2 164 256260

16.125

= +

= += +=≈


39.

28

35

2 2

2 2


35 281225 7842009

44.822

= +

= += +=≈


40.

15

30

2 2

2 2


30 15900 2251125

33.541

= +

= += +=≈


41.

30

30 2 2

2 2


(30) (30)900 9001800

42.426

= +

= += +=≈


42.

21

21

2 2

2 2


21 21441 441882

29.698

= +

= += +=≈


43.

12

2 2

2 2


2 14 13

1.732

= −

= −= −=≈

The leg has length of about 1.732 units.

44.

8

9

2 2

2 2


9 881 6417

4.123

= −

= −= −=≈


45.

7.5

4



321

2 2

2 2


(7.5) (4)56.25 1672.25

8.5

= +

= += +==

The hypotenuse has length 8.5 units.

46.

22.5

12

2 2

2 2


12 (22.5)144 506.25650.25

25.5

= +

= += +==

The hypotenuse has length 25.5 units.

47. 2 2

2 2


100 10010,000 10,00020,000

141.42

= +

= += +=≈

The length of the diagonal is about 141.42 yards.

48. 2 2

2 2


1 11 12

1.414

= +

= += +=≈

The length of the diagonal is about 1.414 miles.

49. 2 2

2 2


(32) (20)1024 400624

25.0

= −

= −= −=≈

The tree is about 25 feet tall.

50. 2 2

2 2

leg (hypotenuse) (leg)

168 6028, 224 360024,624

156.9

= −

= −= −=≈

The height of the antenna is about 156.9 feet.

51. 2 2

2 2


160 30025,600 90,000115,600

340

= +

= += +==

The length of the run was 340 feet.

52. 2 2

2 2


70 1104900 12,10017,000

130.4

= +

= += +=≈

The length of the diagonal is about 130.4 yards.

53. 10 5 2 512 6 2 6

⋅= =

⋅

54. 10 2 5 215 3 5 3

⋅= =

⋅

55. 2 260 2 30 30x x x⋅= =

⋅

56. 35 5 7 7

75 5 15 15y y y⋅

= =⋅ ⋅

57. 3 5 3 59 9 9x x −− =

58. 9 12 9 12 3

13 13 13 13y y y y−

− = = −

59. 9 9 98 8 8 8 64x x x⋅⋅ = =

⋅

60. 7 8 7 11 7 11 711 11 11 8 11 8 8x x x x

x x⋅ ⋅

÷ = ⋅ = =⋅ ⋅



322

61. Since 38 is between 36 = 6 ⋅ 6 and 49 = 7 ⋅ 7,

38 is between 6 and 7; 38 6.16.≈


27 is between 5 and 6; 27 5.20.≈

63. Since 101 is between 100 = 10 ⋅ 10 and

121 = 11 ⋅ 11, 101 is between 10 and 11;

101 10.05.≈


85 is between 9 and 10; 85 9.22.≈


66. 2 2 2

2 2 225 60 65625 3600 4225

4225 4225

a b c+ =

++

=

�

�

Yes, the set forms the lengths of the sides of a right triangle.

67. 2 2 2

2 2 220 45 50400 2025 2500

2425 2500

a b c+ =

++

≠

�

�

No; the set does not form the lengths of the sides of a right triangle.

68. Find the missing length in the large right triangle by letting a = 8 and c = 12.

2 2 2

2 2 2

2

2

8 1264 144

8080

a b cbbbb

+ =

+ =

+ =

==

Find the unlabeled length by letting a = 8 and c = 10.

2 2 2

2 2 2

2

2

2

8 1064 100

3636

6

a b cbbbbb

+ =

+ =

+ =

=

==

The unlabeled length is 6 inches. Thus

6 80x+ = or 80 6 2.944 inches.x = − ≈

The Bigger Picture

1. 81 9= because 29 81.=

2. 4 2

49 7= because

22 2 2 4 .7 7 7 49

⎛ ⎞ = ⋅ =⎜ ⎟⎝ ⎠

3. 9 25 3 5 15⋅ = ⋅ =

4. 16 4 4 2 8⋅ = ⋅ =

5. 6 9 3 4 6 3 3 2 18 6 24⋅ + ⋅ = ⋅ + ⋅ = + =

6. 3 25 2 81 3 5 2 9 15 18 33⋅ + ⋅ = ⋅ + ⋅ = + =

7. 4 49 0 100 4 7 0 1028 0 1028 028

⋅ − ÷ = ⋅ − ÷= − ÷= −=

8. 7 36 0 64 7 6 0 842 0 842 042

⋅ − ÷ = ⋅ − ÷= − ÷= −=

9.

4 25 2 53 6 3 6

2 2 53 2 64 56 6963 33 23 1or 12 2

+ = +

⋅= +

⋅

= +

=

⋅=

⋅

=

10. 9 49 3 7

16 8 16 83 7 2

16 8 23 14

16 1617 1or 116 16

+ = +

⋅= +

⋅

= +

=



323

Section 6.5

Practice Problems

1. a. The triangles are congruent by Side-Angle-Side.

b. The triangles are not congruent.

2. 9 mters 9

13 meters 13=

The ratio of corresponding sides is 9 .

13

3. 6

5 99 5 6

9 309 309 9

10 1 or 33 3

x

xxx

x

=

⋅ = ⋅=

=

=

4. 5 8

605 60 8300 8300 88 8

37.5

nnnn

n

=

⋅ = ⋅=

=

=

The height of the building is approximately 37.5 feet.


1. Two triangles that have the same shape, but not necessarily the same size are congruent. false

2. Two triangles are congruent if they have the same shape and size. true

3. Congruent triangles are also similar. true

4. Similar triangles are also congruent. false

5. For the two similar triangles, the ratio of

corresponding sides is 5 .6

false

Exercise Set 6.5

1. The triangles are congruent by Side-Side-Side.

2. The triangles are congruent by Side-Side-Side.

3. The triangles are not congruent.


5. The triangles are congruent by Angle-Side-Angle.


7. The triangles are congruent by Side-Angle-Side.

8. The triangles are congruent by Side-Angle-Side.

9. 22 14 12 211 7 6 1

= = =

The ratio of corresponding sides is 2 .1

10. 8 4 7 132 16 28 4

= = =


11. 10.5 9 12 3

7 6 8 2= = =

12. 12

6 8 46 34

= =


13. 9

3 66 3 96 276 276 6

4.5

x

xxx

x

=

⋅ = ⋅=

=

=

14. 60

3 55 3 60

5 1805 1805 5

36

x

xxx

x

=

⋅ = ⋅=

=

=



324

15. 4

18 1212 18 4

12 7212 7212 12

6

n

nnn

n

=

⋅ = ⋅=

=

=

16. 14

4 77 4 14

7 567 567 7

8

y

yyy

y

=

⋅ = ⋅=

=

=

17. 12

3.75 99 12 3.759 459 459 9

5

y

yyy

y

=

⋅ = ⋅=

=

=

18. 22.5

9 1515 9 22.5

15 202.515 202.515 15

13.5

z

zzz

z

=

⋅ = ⋅=

=

=

19. 30

18 4040 18 30

40 54040 54040 40

13.5

z

zzz

z

=

⋅ = ⋅=

=

=

20. 14

9 88 9 14

8 1268 1268 8

15.75

x

xxx

x

=

⋅ = ⋅=

=

=

21. 17.5

3.25 3.253.25 17.5 3.253.25 56.8753.25 56.8753.25 3.25

17.5

x

xxx

x

=

⋅ = ⋅=

=

=

22. 33.2

9.6 8.38.3 9.6 33.2

8.3 318.728.3 318.728.3 8.3

38.4

y

yyy

y

=

⋅ = ⋅=

=

=

23. 13

23

18

2 32 13 2 183 3

11 5523 3

11 1103 3

3 11 3 11011 3 11 3

10

y

y

y

y

y

y

=

⋅ = ⋅

⋅ = ⋅

⋅ =

⋅ = ⋅

=

24. 1213

6 919 6 132

279 62

9 819 819 9

9

z

z

z

zz

z

=

⋅ = ⋅

= ⋅

=

=

=

25. 15

60 3232 60 1532 90032 90032 32

28.125

z

zzz

z

=

⋅ = ⋅=

=

=



325

26. 13

13 2626 13 13

26 16926 16926 26

6.5

x

xxx

x

=

⋅ = ⋅=

=

=

27. 12

157 10110 7 152

10.5 10510.5 10510.5 10.5

10

x

x

xx

x

=

⋅ = ⋅

=

=

=

28. 8

37.5 2020 37.5 8

20 30020 30020 20

15

n

nnn

n

=

⋅ = ⋅=

=

=

29. 40

25 22 40 252 10002 10002 2

500

x

xxx

x

=

⋅ = ⋅=

=

=

The building is 500 feet tall.

30. 73

10 22 10 73

2 7302 7302 2

365

x

xxx

x

=

⋅ = ⋅=

=

=

The tree is 365 feet tall.

31. 100

28 55 28 100

5 28005 28005 5

560

x

xxx

x

=

⋅ = ⋅=

=

=

The fountain is approximately 560 feet high.

32. 75

6 99 6 75

9 4509 4509 9

50

x

xxx

x

=

⋅ = ⋅=

=

=

The building is approximately 50 feet tall.

33. 24

18 3030 24 1830 43230 43230 30

14.4

x

xxx

x

=

⋅ = ⋅=

=

=

The shadow of the tree is 14.4 feet long.

34. 44

32 2424 32 44

24 140824 140824 24

58.7

x

xxx

x

=

⋅ = ⋅=

=

≈

The shadow is approximately 58.7 feet.

35. 19

55 2020 55 19

20 104520 104520 20

52.2552

x

xxx

xx

=

⋅ = ⋅=

=

=≈

Pete can place 52 neon tetras in the tank.

36. 35 8

3 8 524 524 55 5

4.8

x

xxx

x

=

⋅ = ⋅=

=

=

45 − 4.8 = 40.2 His speed is 40.2 miles per hour.



326

37. Let a = 200 and c = 430. 2 2 2

2 2 2

2

2

200 43040,000 184,900

144,900144,900

381

a b cbbbbb

+ =

+ =

+ =

==≈

The gantry is approximately 381 feet tall.

38. 2 2

2 2


94 508836 250011,336

106

= +

= += +=≈

The length of the break is approximately 106 feet.

39. 3.600.414.01

+

40. Subtract the absolute values. 3.600.413.19

−

Attach the sign of the larger absolute value. 0.41 − 3.6 = −3.19

41. (0.41)(−3) = −1.23

42. 0.16

3 0.48318180

−

−

−0.48 ÷ 3 = −0.16

43. Let x be the new width. 5

7 99 7 5

9 359 359 9

839

x

xxx

x

=

⋅ = ⋅=

=

=

The new width is 839

inches. No, it will not fit

on a 3-by-5-inch card because 83 3.9>

44. 12.6

5.2 7.87.8 5.2 12.6

7.8 65.527.8 65.527.8 7.8

8.4

n

nnn

n

=

⋅ = ⋅=

=

=

45. 11.6

58.7 20.820.8 11.6 58.720.8 680.9220.8 680.9220.8 20.8

32.7

n

nnn

n

=

⋅ = ⋅=

=

≈


47. 14

1051 5 104

200

x

x

x

=

⋅ = ⋅

=

1 12 4

107

1 17 104 2

300

y

y

y

=

⋅ = ⋅

=

5 148

1010

1 510 104 8

425

z

z

z

=

⋅ = ⋅

=

The actual proposed dimensions are 200 feet by 300 feet by 425 feet.

48. 14

131 3 14

12

x

x

x

=

⋅ = ⋅

=



327

1 12 4

14

1 14 14 2

18

y

y

y

=

⋅ = ⋅

=

14

161 6 14

24

z

z

z

=

⋅ = ⋅

=

The lengths of the sides of the deck are 12 feet, 18 feet, and 24 feet.

Chapter 6 Vocabulary Check

1. A ratio is the quotient of two numbers. It can be written as a fraction, using a colon, or using the word to.

2. 7

2 16x= is an example of a proportion.

3. A unit rate is a rate with a denominator of 1.

4. A unit price is a “money per item” unit rate.

5. A rate is used to compare different kinds of quantities.

6. In the proportion 7 ,

2 16x= x ⋅ 16 and 2 ⋅ 7 are

called cross products.

7. If cross products are equal the proportion is true.

8. If cross products are not equal the proportion is false.

9. Congruent triangles have the same shape and the same size.

10. Similar triangles have exactly the same shape but not necessarily the same size.

11−13. Label the sides of the right triangle.

hypotenuseleg

leg

11.

12.

13.

14. A triangle with one right angle is called a right triangle.

15. In the right triangle,

ac

b 2 2 2a b c+ = is called the Pythagorean theorem.

Chapter 6 Review

1. The ratio of 23 to 37 is 23 .37

2. The ratio of $121 to $143 is $121 11 11 11.$143 11 13 13

⋅= =

⋅

3. The ratio of 4.25 yards to 8.75 yards is 4.25 yards 4.25 100 425 25 17 17 .8.75 yards 8.75 100 875 25 35 35

× ⋅= = = =

× ⋅

4. The ratio of 124

to 348

is

1438

2 1 3 9 35 9 8 182 4 .4 8 4 8 4 35 354

= ÷ = ÷ = ⋅ =

5. length 4.5 meters 4.5 10 45 5 9 9width 2 meters 2 10 20 5 4 4

× ⋅= = = = =

× ⋅

6. width 2 meter 2

perimeter (4.5 2 4.5 2) meters 13s

= =+ + +

7. 6000 people 1200 5 people 5 people

2400 pets 1200 2 pets 2 pets⋅

= =⋅

8. 15 pages 3 5 pages 5 pages6 minutes 3 2 minutes 2 minutes

⋅= =

⋅

9. 468 miles9 hours

52

9 46845

18180

−

−


or 52 miles/hour.



328

10. 180 feet

12 seconds

1512 180

126060

0

−

−

The unit rate is 15 feet

1 second or 15 feet/second.

11. $6.96

4 diskettes

1.744 6.96

42928

16160

−

−

−

The unit rate is $1.74

1 diskette or $1.74/diskette.

12. 104 bushels

8 trees

138 104

824240

−

−

The unit rate is 13 bushels

1 tree or 13 bushels/tree.

13. 0.1237 0.124

8 0.990081916

3024

60564

≈

−

−

−

−


0.1408 0.14112 1.6900

1 24948100

964

≈

−

−

−


14. 0.0827 0.083

18 1.49001 44

5036140136

4

≈

−

−

−


28 2.39002 24

150140

1008416

≈

−

−

−


15. 24 uniforms 3 uniforms

8 players 1 player=

16. 12 tires 4 tires3 cars 1 car

=

17. 19 148 6

19 6 8 14114 112⋅ ⋅

≠

�

�


18. 3.75 7.5

3 63.75 6 3 7.5

22.5 22.5⋅ ⋅

=

�

�




329

19. 30

3 1818 3 30

18 9018 9018 18

5

x

xxx

x

=

⋅ = ⋅=

=

=

20. 7

9 33 9 7

3 633 633 3

21

x

xxx

x

=

⋅ = ⋅=

=

=

21. 8 9

58 5 98 458 458 8

5.625

xxxx

x

−=

− ⋅ = ⋅− =−

=− −

= −

22. 94

275

927 ( 5)491354

4 135 4 99 1 9 4

60

x

x

x

x

x

−=−

− ⋅ − = ⋅

= ⋅

⋅ = ⋅ ⋅

=

23. 0.4 2

4.70.4 4.7 2

1.88 21.88 2

2 20.94

xxxx

x

=

⋅ = ⋅=

=

=

24.

110

1 22 5

2

4 82 1 18 4 25 2 10

42 9 215 2 10

42 1895 20

5 42 5 18942 5 42 20

9 1or 18 8

x

x

x

x

x

x

=

⋅ = ⋅

⋅ = ⋅

⋅ =

⋅ ⋅ = ⋅

=

25. 4.7

0.4 33 0.4 4.7

3 1.883 1.883 3

0.63

x

xxx

x

=

⋅ = ⋅=

=

≈

26. 0.07 7.20.3

0.07 0.3 7.20.07 2.160.07 2.160.07 0.07

30.9

nnnn

n

=

⋅ = ⋅=

=

≈

27. Let x be the number of completed passes. 3completed completed7 32attempted attempted

3 32 796 796 77 714

x

xxx

x

→ ←=→ ←⋅ = ⋅

=

=

≈

He completed 14 passes.

28. Let x be the number of attempted passes. 3 15completed completed7attempted attempted

3 7 153 1053 1053 3

35

xxxx

x

→ ←=→ ←⋅ = ⋅

=

=

=

He attempted 35 passes.



330



31,500 400031,500 40004000 40007.875

x

xxx

x

→ ←= ⋅→ ←⋅ ⋅ = ⋅

=

=

=

Purchase 8 bags of pesticide.



62,500 400062,500 40004000 400015.625

x

xxx

x

→ ←= ⋅→ ←⋅ ⋅ = ⋅

=

=

=

Purchase 16 bags of pesticide.

31. Let x be the miles. 80miles miles

2 0.75inches inches0.75 2 80

0.75 1600.75 1600.75 0.75

640 1 or 2133 3

x

xxx

x

→ ←=→ ←⋅ = ⋅

=

=

=

The distance is 12133

miles.

32. Let x be the inches. 1025 80miles miles

0.75inches inches1025 0.75 80

768.75 80768.75 80

80 809.6

xxxx

x

→ ←=→ ←⋅ = ⋅

=

=

≈

The distance is about 9.6 inches.

33. 64 8= because 28 64.=

34. 144 12= because 212 144.=

35. 12 3.464≈

36. 15 3.873≈

37. 0 0= because 20 0.=

38. 1 1= because 21 1.=

39. 50 7.071≈

40. 65 8.062≈

41. 4 225 5

= because 22 2 2 4 .

5 5 5 25⎛ ⎞ = ⋅ =⎜ ⎟⎝ ⎠

42. 1 1

100 10= because

21 1 1 1 .10 10 10 100⎛ ⎞ = ⋅ =⎜ ⎟⎝ ⎠

43. 2 2

2 2


12 5144 25169

13

= +

= += +==


44. 2 2

2 2


20 21400 441841

29

= +

= += +==


45. 2 2

2 2


14 9196 81115

10.7

= −

= −= −=≈


46. 2 2

2 2


66 564356 31367492

86.6

= +

= += +=≈




331

47. 2 2

2 2


20 20400 400800

28.28

= +

= += +=≈

The diagonal is about 28.28 centimeters.

48. 2 2

2 2


126 9015,876 81007776

88.2

= −

= −= −=≈

The height is about 88.2 feet.



51. 20

20 3030 20 20

30 40030 40030 30

40 1 or 133 3

x

xxx

x

=

⋅ = ⋅=

=

=

52. 24

5.8 88 5.8 24

8 139.28 139.28 8

17.4

x

xxx

x

=

⋅ = ⋅=

=

=

53. 42

5.5 77 5.5 42

7 2317 2317 7

33

x

xxx

x

=

⋅ = ⋅=

=

=

The height of the building is approximately 33 feet.

54. 2

10 2424 10 2

24 2024 2024 24

56

x

xxx

x

=

⋅ = ⋅=

=

=

2

26 2424 26 2

24 5224 5224 24

13 1or 26 6

y

yyy

y

=

⋅ = ⋅=

=

=

The unknown lengths are 56

x = inch and

126

y = inches.

55. The ratio of 15 to 25 is 15 3 .25 5

=

56. The ratio of 3 pints to 81 pints is 3 pints 1 .

81 pints 27=

57. 2 teachers 2 1 teachers 1 teacher

18 students 2 9 students 9 students⋅

= =⋅

58. 6 nurses 1 6 nurses 1 nurse

24 patients 4 6 patients 4 patients⋅

= =⋅

59. 136 miles 4 34 miles 34 miles

4 hours 4 1 hour 1 hour⋅

= =⋅


or 34 miles/hour.

60. 12 gallons 6 2 gallons 2 gallons

6 cows 6 1 cow 1 cow⋅

= =⋅

The unit rate is 2 gallons

1 cow or 2 gallons/cow.

61. 29 medals 29Germany256 medals 256Total

→ =→



332

62. 1576 steps9.5 minutes

9.5 1576 becomes 165.8

95 15760.095626570

56047585076090

−

−

−

−

He ran 166 steps/minute.

63. 4 ounces: 1.235

4 4.94040 9

814122020

0

−

−

−

−

The 4-ounce size costs $1.235 per ounce. 8 ounces:

1.2475 1.2488 9.9800

81 91 6

38326056

4040

0

≈

−

−

−

−

−


64. 12 ounces: 0.0541 0.054

12 0.650060

50482012

8

≈

−

−

−

The 12-ounce size costs $0.054 per ounce. 64 ounces:

0.0465 0.04764 2.9800

2 56420384360320

40

≈

−

−

−


65. 2 cups cookie dough 4 cups cookie dough

30 cookies 60 cookies=

66. 5 nickels 20 nickels3 dollars 12 dollars

=

67. 3 15

83 8 1524 1524 1515 151.6

xxxx

x

=

⋅ = ⋅=

=

=

68. 54 20

5 20 4100 4100 4

4 425

x

xxx

x

=

⋅ = ⋅=

=

=



333

69. 7.5

3 66 3 7.5

6 22.56 22.56 6

3.75

x

xxx

x

=

⋅ = ⋅=

=

=

70. 13

25 301 30 253

10 2510 2525 2525

x

x

xx

x

=

⋅ = ⋅

=

=

=

71. 36 6= because 26 36.=

72. 16 481 9

= because 24 4 4 16 .

9 9 9 81⎛ ⎞ = ⋅ =⎜ ⎟⎝ ⎠

73. 105 10.247≈

74. 32 5.657≈

75. 2 2

2 2


66 564356 31367492

86.6

= +

= += +=≈

76. 2 2

2 2


24 12576 144432

20.8

= −

= −= −=≈

77. 10

6 55 6 10

5 605 605 5

12

n

nnn

n

=

⋅ = ⋅=

=

=

78.

38

2 13 2

9

8 121 2 312 8 92 3 8

25 26 752 3 8

25 3252 4

2 25 2 32525 2 25 4

13 1or 62 2

n

n

n

n

n

n

=

⋅ = ⋅

⋅ = ⋅

=

⋅ = ⋅

=

Chapter 6 Test

1. The ratio of 4500 trees to 6500 trees is 4500 trees 9 500 9 .6500 trees 13 500 13

⋅= =

⋅

2. The ratio of 9 inches of rain in 30 days is 9 inches 3 3 inches 3 inches .30 days 3 10 days 10 days

⋅= =

⋅

3. The ratio of 8.6 to 10 is 8.6 8.6 10 86 43.10 10 10 100 50

×= = =

×

4. The ratio of 758

to 394

is

7834

5 7 3 47 39 47 4 475 9 .8 4 8 4 8 39 789

= ÷ = ÷ = ⋅ =

5. The ratio 414 feet to 231 feet is 414 feet 3 138 138 .231 feet 3 77 77

⋅= =

⋅



334

6. 650 kilometers

8 hours

81.258 650.00

6410

82 01 6

40400

−

−

−

−

The unit rate is 81.25 kilometers

1 hour or

81.25 kilometers/hour.

7. 140 students

5 teachers

285 140

1040400

−

−

The unit rate is 28 students1 teacher

or

28 students/teacher.

8. 108 inches12 seconds

9

12 108108

0−

The unit rate is 9 inches1 second

or 9 inches/second.

9. 0.1487 0.149

8 1.1900839327064

60564

≈

−

−

−

−

The 8-ounce size costs $0.149/ounce.

0.1575 0.15812 1.8900

1 26960

908460600

≈

−

−

−

−

The 12-ounce size costs $0.158/ounce. The 8-ounce size is the better buy.

10. 0.09312 0.0931

16 1.490001 44

5048

2016

4032

8

≈

−

−

−

−


24 2.390002 16

230216

140120200192

8

≈

−

−

−

−


11. 28 1416 8

28 8 16 14224 224⋅ ⋅

=

�

�


12. 3.6 1.92.2 1.2

3.6 1.2 2.2 1.94.32 4.18⋅ ⋅

≠

�

�




335

13. 15

3 99 3 15

9 459 459 9

5

n

nnn

n

=

⋅ = ⋅=

=

=

14. 8 11

68 6 1148 1148 1111 1144

11

xxxx

x

=

⋅ = ⋅=

=

=

15. 3 147

4

1 344 7

317

7 7 313 3 7

73

y

y

y

y

y

=

⋅ = ⋅

=

⋅ = ⋅

=

16. 1.5 2.45

1.5 5 2.41.5 121.5 121.5 1.5

8

nnnn

n

=

⋅ = ⋅=

=

=

17. Let x be the length of the home in feet. 9feet feet

11 2inches inches2 11 9

2 992 992 2

1492

x

xxx

x

→ ←=→ ←⋅ = ⋅

=

=

=

The home is 1492

feet long.

18. Let x be the number of hours. 80 100miles miles3hours hours

80 3 10080 30080 30080 80

334

xxxx

x

→ ←=→ ←

⋅ = ⋅=

=

=

It will take 334

hours to travel 100 miles.

19. Let x be the number of grams. 10grams grams15 80pounds pounds

10 80 15800 15800 1515 15

1533

x

xxx

x

→ ←=→ ←⋅ = ⋅

=

=

=

The standard dose for an 80-pound dog is 1533

grams.

20. 49 7= because 27 49.=

21. 157 12.530≈

22. 64 8 4

100 10 5= = because

28 8 8 64 .10 10 10 100⎛ ⎞ = ⋅ =⎜ ⎟⎝ ⎠

23. 2 2

2 2


4 416 1632

5.66

= +

= += +=≈

The hypotenuse is 5.66 centimeters.

24. 5

12 88 12 5

8 608 608 8

7.5

n

nnn

n

=

⋅ = ⋅=

=

=



336

25. Let x be the height of the tower.

34

484534 5 484

234 484

4 2764 2764 4

69

x

x

x

xx

x

=

⋅ = ⋅

= ⋅

=

=

=

The tower is approximately 69 feet tall.

Cumulative Review Chapters 1−6

1. a. 12 − 9 = 3 Check: 3 + 9 = 12

b. 22 − 7 = 15 Check: 15 + 7 = 22

c. 35 − 35 = 0 Check: 0 + 35 = 35

d. 70 − 0 = 70 Check: 70 + 0 = 70

2. a. 20 ⋅ 0 = 0

b. 20 ⋅ 1 = 20

c. 0 ⋅ 20 = 0

d. 1 ⋅ 20 = 20

3. To round 248,982 to the nearest hundred, observe that the digit in the tens place is 8. Since this digit is at least 5, we add 1 to the digit in the hundreds place. The number 248,982 rounded to the nearest hundred is 249,000.

4. To round 248,982 to the nearest thousand, observe that the digit in the hundreds place is 9. Since this digit is at least 5, we add 1 to the digit in the thousands place. The number 248,982 rounded to the nearest thousand is 249,000.

5. a.

4258

200×

b.

2 3246

51230×

6. 373 R 24

28 10,4688 42 061 96

1088424

−

−

−

7. 1 ( 10) ( 8) 9 9 ( 8) 917 98

+ − + − + = − + − += − += −

8. −12(7) = −84

9.

4

80 2 40

2 2 20

2 2 2 10

2 2 2 2 52 5

= ⋅↓ ↓

= ⋅ ⋅↓ ↓ ↓

= ⋅ ⋅ ⋅↓ ↓ ↓ ↓

= ⋅ ⋅ ⋅ ⋅

= ⋅

10. 2 23 1 9 1 8− = − =

11. 12 4 3 320 4 5 5

⋅= =

⋅

12. 29 3 81 3 243⋅ = ⋅ =

13. 6 26 6 26 6 13 2 2

13 30 13 30 13 6 5 5⋅ ⋅ ⋅

− ⋅− = = =⋅ ⋅ ⋅

14. 3 5 27 41 9 3 41 123 33 4 or 158 9 8 9 8 9 8 8

⋅ ⋅⋅ = ⋅ = =

⋅

15. 7 6 3 7 6 3 16 28 8 8 8 8

+ ++ + = = =

16. 7 3 4 7 3 4 8 4

10 10 10 10 10 5− +

− + = = =

17. 7 = 7 14 = 2 ⋅ 7 LCD = 2 ⋅ 7 = 14



337

18. 17 3 17 2 3 525 10 25 2 10 5

17 2 3 525 2 10 534 1550 5034 15

504950

+ = ⋅ + ⋅

⋅ ⋅= +

⋅ ⋅

= +

+=

=

19. 4 ⋅ 5 = 20, so 3 3 5 154 4 5 20

⋅= =

⋅

20. 10 5 2 255 5 11 116 3 2 233 3 11 11

⋅= =

⋅⋅

= =⋅

Yes, they are equivalent.

21. 2 10 2 11 10 33 11 3 11 11 3

22 3033 3322 30

33833

− = ⋅ − ⋅

= −

−=

= −

22. 51724599

−

15177240972

−

8716724097247772

−

23. 5 1 5 3 5 3 2 1

12 4 12 12 12 12 6−

− = − = = =

There is 16

hour remaining.

24. 80 ÷ 8 ⋅ 2 + 7 = 10 ⋅ 2 + 7 = 20 + 7 = 27

25. 123358

+

8224952417724

+

26. 3 4 115 9 15 27 20 33 3

3 45 45 4580 145 316 19 31627

+ + ⎛ ⎞= + + ÷⎜ ⎟⎝ ⎠

= ⋅

= ⋅

=

27. 3 3 11 334 4 11 44

⋅= =

⋅

9 9 4 3611 11 4 44

⋅= =

⋅

Since 33 < 36, 33 3644 44

< and 3 9 .4 11<

28. 5 8 243 243 243 3

8

y yyy

y

− =− =−

=− −

= −

29. 5 2 65 8

5 5 8 53

yy

yy

− = − −− = −

− + = − += −

30. 3 6 7 63 6 6 7 6 6

3 73 7 7 7

4 04 04 4

0

y yy y

y yy y y y

yy

y

− = −− + = − +

=− = −− =−

=− −

=

31. 3 6 43 6 4

2 6 42 6 6 4 6

2 102 102 2

5

a aa a a aa

aaa

a

− = +− − = + −

− =− + = +

=

=

=



338

32. 4( 1) 3 214 4 3 21

4 1 214 1 1 21 1

4 204 204 4

5

yyy

yyy

y

+ − =+ − =

+ =+ − = −

=

=

=

33. 3(2 6) 6 06 18 6 0

6 12 06 12 12 0 12

6 126 126 6

2

xxx

xxx

x

− + =− + =

− =− + = +

=

=

=

34. Seventy-five thousandths written in standard form is 0.075.

35. To round 736.2359 to the nearest tenth, observe that the digit in the hundredths place is 3. Since this digit is less than 5, we do not add 1 to the digit in the tenths place. The number 736.2359 rounded to the nearest tenth is 736.2.

36. To round 736.2359 to the nearest thousandth, observe that the digit in the ten-thousandths place is 9. Since this digit is at least 5, we add 1 to the digit in the thousandths place. The number 736.2359 rounded to the nearest thousandth is 736.236.

37. 23.8501.604

25.454+

38. 700.0018.76

681.24−

39. 0.053116

31865310

0.8496

×

40. 0.375

8 3.0002 4

6056

4040

0

−

−

−

3 0.3758=

41. 0.052

115 5.9805 75

230230

0

−

−

−5.98 ÷ 115 = −0.052

42. 9 7 10 9 797.9 7

10 10 10⋅ +

= = =

43. −0.5(8.6 − 1.2) = −0.5(7.4) = −3.7

44. 12

4 1616 4 12

16 4816 4816 16

3

n

nnn

n

=

⋅ = ⋅=

=

=

45. 9 0.45020

=

4 0.4 0.444...9= =

0.456 = 0.456 4 9, ,9 20

0.456

46. 700 meters 5 140 meters 140 meters5 seconds 5 1 second 1 second

⋅= =

⋅

The unit rate is 140 meters/second.

47. The ratio of $15 to $10 is $15 15 5 3 3 .$10 10 5 2 2

⋅= = =

⋅

48. The ratio of 7 to 21 is 7 1 7 1 .21 3 7 3

⋅= =

⋅



339

49. The ratio of 2.5 to 3.15 is 2.5 2.5 100 250 50 5 50 .

3.15 3.15 100 315 63 5 63⋅ ⋅

= = = =⋅ ⋅

50. The ratio of 900 to 9000 is 900 900 1 1 .9000 900 10 10

⋅= =

⋅


chapter 6 · chapter 6: ratio, proportion, and ... the ratio of medals won by canada to the total...

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