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CHAPTER ONE CRITICAL THINKING SKILLS Exercise Set 1.1 1. Counting 2. Divisible 3. Hypothesis 4. Counterexample 5. Inductive 6. Deductive 7. Deductive 8. Inductive 9. Inductive reasoning, because a general conclusion was made from observation of specific cases. 10. Inductive reasoning, because a general conclusion was made from observation of specific cases. 11.

5×5 = 25

12.

12×14 = 168

13.

1 5 (= 1 + 4) 10 (= 4 + 6) 10 (= 6 + 4) 5(= 4 + 1) 1

14.

100,000 = 105

15.

16.

17.

18.

19. 21.

23. 25.

10, 12, 14 (Add 2 to previous number.) 3, −3, 3 (Alternate 3 and −3.)

20.

19, 23, 27 (Add 4 to the previous number.)

22.

−3, −5, −7 (Subtract 2 from previous number.) 2500, −12,500, 62,500 (Multiply previous number by –5.)

24.

1 1 1 , , (Increase the denominator value by 1.) 5 6 7 36, 49, 64 (The numbers in the sequence are the squares of the counting numbers.)

26.

21, 28, 36 (15 + 6 = 21, 21 + 7

= 28, 28 + 8 = 36)

CHAPTER 1 Critical Thinking Skills

27.

34, 55, 89 (Each number in the sequence is the sum of the previous two numbers.)

29. There are three letters in the pattern. 39× 3 = 117 , so the 117th entry is the second R in the pattern. Therefore, the 118th entry is Y.

31.

a) 36, 49, 64 b) Square the numbers 6, 7, 8, 9 and 10. c) 8×8 = 64 9 × 9 = 81 72 is not a square number since it falls between the two square numbers 64 and 81.

33. Blue: 1, 5, 7, 10, 12

Purple: 2, 4, 6, 9, 11

28.

243 729 2187 ,− , 256 1024 4096

(Multiply previous number 3 by − .) 4

30.

a) Answers will vary. b) The sum of the digits is 9. c) When a one- or two-digit number is multiplied by 9, repeated summing of the digits in the product yields the number 9. 32. a) 28 and 36 b) To find the 7th triangular number, add 7 to the 6th triangular number. To find the 8th triangular number, add 8 to the 7th triangular number. To find the 9th triangular number, add 9 to the 8th triangular number. To find the 10th triangular number, add 10 to the 9th triangular number. To find the 11th triangular number, add 11 to the 10th triangular number. c) 36 + 9 = 45; 45 + 10 = 55; 55 + 11 = 66; 66 + 12 = 78 72 is not a triangular number since it falls between the consecutive triangular numbers 66 and 78. Yellow: 3, 8

34. a) 19 (Each new row has two additional triangles.) b) 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 = 100 35. a) ≈ $200, 000 b) We are using observation of specific cases to make a prediction. 36. a) ≈ $3.7 trillion

b) We are using observation of specific cases to make a prediction. 38.

37.

39. a) You should obtain the original number. b) You should obtain the original number. c) Conjecture: The result is always the original number. 4n + 12 4n 12 = + = n + 3, n + 3 − 3 = n 4 4 4 40. a) You should obtain twice the original number. b) You should obtain twice the original number. c) Conjecture: The result is always twice the original number. d) n, 4n, 4n + 12,

d) n, 4n, 4n + 6,

4n + 6 4n 6 = + = 2n + 3, 2n + 3 − 3 = 2n 2 2 2

SECTION 1.2

41. a) You should obtain the number 5. b) You should obtain the number 5. c) Conjecture: No matter what number is chosen, the result is always the number 5. 2n + 10 2n 10 = + = n + 5, n + 5 − n = 5 d) n, n + 1, n + ( n + 1) = 2n + 1, 2n + 1 + 9 = 2n + 10, 2 2 2 42. a) You should obtain the number 0. b) You should obtain the number 0. c) Conjecture: No matter what number is chosen, the result is always the number 0. n + 10 ⎛⎜ n + 10 ⎞⎟ , 5⎜ = n + 10, n + 10 − 10 = n, n − n = 0 d) n, n + 10, ⎜⎝ 5 ⎠⎟⎟ 5

43. 7 − 5 = 2 is one counterexample. 44. 5 ÷ 2 = 2 12 , which is not a counting number.

5 , which is not an even number. 2 46. 900 is a three-digit number. The product of 900 and 900 is 810,000, which is not a five-digit number. 47. One and two are counting numbers. The difference of 1 and 2 is 1− 2 = −1 , which is not a counting number. 48. The sum of the odd numbers 1 and 5 is 6, which is not divisible by 4. 49. a) The sum of the measures of the interior angles should be 180° . b) Yes, the sum of the measures of the interior angles should be 180° . c) Conjecture: The sum of the measures of the interior angles of a triangle is 180° . 50. a) The sum of the measures of the interior angles should be 360° . b) Yes, the sum of the measures of the interior angles should be 360° . c) Conjecture: The sum of the measures of the interior angles of a quadrilateral is 360° . a b 51. 129, the numbers in positions are found as follows: c a +b+ c

45. Two is a counting number. The sum of 2 and 3 is 5. Five divided by two is

52. 1881, 8008, 8118 (They look the same when looked at in a mirror.) 53. c

Exercise Set 1.2 (Note: Answers in this section will vary depending on how you round your numbers. The answers may differ from the answers in the back of the textbook. However, your answers should be something near the answers given. All answers are approximate.)

1. 2.

Estimation Equal

261 + 127.4 + 273.9 + 16.2 + 81.5 ≈ 260 + 127 + 274 + 16 + 82 = 759

4. 2.57 + 212.6 +176.2 + 83

198, 600×3.072 ≈ 200, 000×3.000 = 600, 000

1854 ×0.0096 ≈ 1900×0.01 = 19

405 400 ≈ = 8000 0.049 0.05

0.63×1523 ≈ 0.6×1500 = 900

≈ 0 + 210 +180 + 80 = 470

CHAPTER 1 Critical Thinking Skills

10. 51, 608× 6981 ≈ 50, 000× 7000 = 350, 000,000

11% of 8221 ≈ 10% of 8000 = 0.10×8000 = 800

11.

18% ×1576 ≈ 20%×1600 = 0.20×1600 = 320

12.

296.3 ÷ 0.0096 ≈ 300 ÷ 0.01 = 30, 000

13.

$10.49 $10 ≈ = $2 5 5

14.

$37.80 $40 ≈ = $2 20 20

15.

12 months ×$120.80 ≈12×$120 = $1440

16.

8% of $11, 250 ≈ 0.08 × $11, 000 = $880

17.

One third of an annual profit of $8795 1 ≈ × $9, 000 = $3000 3

18.

$1.29 + $6.86 + $12.43 + $25.62 + $8.99 ≈ $1+ $7 + $12 + $26 + $9 = $55

19.

95lb +127 lb + 210 lb ≈100 +100 + 200 = 400 lb

21.

20. 22.

15% of $26.32 ≈ 15% of $26 = 0.15×$26 = $3.9

3.25 lb 3.00 lb ≈ = 0.5 lb 6 6 $400 $400 ≈ = 16 $23 $25

24.

23. ($65.99 + $49.99 + $49.95) − $114.99 ≈ ($66 + $50 + $50) − $115 = $166 − $115 = $5

Team A: 189 + 172 + 191 ≈ 190 + 170 + 190 = 550 Team B: 183 + 229 + 167 ≈ 180 + 230 + 170 = 580 580 − 550 = 30 lb

25.

11 × 8 × $1.50 ≈ 10 × 8 × $1.50 = 10 × $12 = $120

26.

6 min, 25 sec×26.2 mi ≈ 6.5 min × 26 mi =169 min 169 min ≈ 3hours 60 min

27. 100 Mexican pesos = 100× 0.083 U.S. dollars

28. $973 + 6 ($61) + 6 ($97) + 6 ($200)

≈ 100× 0.08 U.S. dollars = 8 U.S. dollars $50 − $8 = $42

29. ≈ 60 miles

≈ $970 + 6 ($60) + 6 ($100) + 6 ($200) = $970 + $360 + $600 + $1200 = $3130

30.

≈ 55 miles

SECTION 1.3

32.

a) 100 b) 50 c) 125

33. a) 5 million b) 98 million c) 98 million − 33 million = 65 million d) 19 million + 79 million + 84 million + 65 million + 33 million = 280 million

34.

a) 19% b) 25% c) 20% of 179 lb ≈ 20% of 180 = 0.2 ×180 = 36 lb

35. a) 85% b) 68% − 53% = 15% c) 85% of 70 million acres = 59,500, 000 acres

36.

a) 2 ( 410) + 4 (545)

31. a) 33% of 700 ≈ 30% of 700 = 0.30×700 = 210 b) 9% of 700 ≈10% of 700 = 0.10×700 = 70 c) 24% of 700 ≈ 25% of 700 = 0.25×700 =175

≈ 2 (400) + 4 (550) = 800 + 2200 = 3000 calories b) Running: 4 (920) ≈ 4 (925) = 3700 calories

d) No, since we are not given the area of each state.

Casual bike riding: 4 (300) = 1200 calories , 3700 −1200 = 2500 calories c) 3(545) + 3(545) ≈ 3(550) + 3(550) = 1650 + 1650 = 3300 calories per week , 3300 calories per week (52 weeks) ≈ 3000×50 = 150,000 calories 38.

37.

39. 41. 43. 45. 47.

≈ 120 bananas 150° 10% 9 square units 150 feet

40. 42. 44. 46.

≈ 160 berries 315° 25% 12 square units

48.

4 (60) = 240 in. or

49.-57. Answers will vary. 59. a) Answers will vary. b) Answers will vary.

58.

There are 118 ridges around the edge.

1 in. 20.1 in. = 2.5 yd x yd

Exercise Set 1.3 1 in. 5.75 in. 1. = 12 mi x mi 1x = 12 (5.75)

1x = 2.5 (20.1)

x = 69 mi

x = 50.25 yd

240 ≈ 20 ft 12

CHAPTER 1 Critical Thinking Skills

3 ft x ft = 1.2 ft 19.36 ft 3(19.36) = 1.2 x

x bags 1 bag = 2 4000 ft 35, 000 ft 2 4000 x = 1(35, 000)

58.08 1.2 x = 1.2 1.2 58.08 = 48.4 ft x= 1.2

4000 x 35, 000 = 4000 4000 35, 000 = 8.75 bags x= 4000

5. 6.4% of $7605 = 0.064 × $7605 = $486.72 $7605 + $486.72 = $8091.72 ≈ $8092

30% of $117 = 0.30 × $117 = $35.10 $117 − $35.10 = $81.90

7. a) Ent./Misc.: 19.4% of $1750 = 0.194 × $1750 8. a) 11 − 20 years: 25% of 6.2 million = 0.25 × 6.2 = 1.55 million = $339.50 0 − 3 years: 16% of 48 million = 0.16 × 6.2 Food: 12.7% of $1750 = 0.127 × $1750 = $222.25 = 0.992 million $339.50 − $222.25 = $117.25 1.55 − 0.992 = 0.558 million = 558,000 b) Housing: 33.9% of $1750 = 0.339 × $1750 b) 4 −10 years: 33% of 6.2 million = 0.33 × 6.2 = $593.25 = 2.046 million Transportation: 17% of $1750 = 0.17 × $1750 11− 20 years: 25% of 6.2 million = 0.25 × 6.2 = $297.50 = 1.55 million $593.25 − $297.50 = $295.75 2.046 − 1.55 = 0.496 million = 496,000

54.46% of $200, 000 = 0.5446 × 200, 000 = $108,920 $200, 000 + $108,920 = $308,920

b) 2.9% of $180, 000 = 0.029 ×180,000 = $5220 $180, 000 + $5220 = $185, 220

10. 40 rides × $2 per ride = $80. In order for the cost of rides with the $81 MetroCard to be less than the cost of the rides without the MetroCard, Chandler would have to take 41 rides per month. $81 ≈ $1.98per ride 41rides

c) Flagstaff, AZ: − 1.85% of $200, 000 = −0.0185 × $200, 000 = − $3700 $200, 000 − $3700 = $196,300 Bellingham, WA: − 1.04% of $200,000 = −0.0104×$200, 000 =− $2080 $200, 000 − $2080 = $197,920 $197,920 − $196,300 = $1620 11.

$120 + $80 (15) = $120 + $1200 = $1320 Savings: $1320 − $1250 = $70

12.

2005: $20 × 2 million = $40 million 2006: $20 + 0.25 × $20 = $25 $25 × 2 million = $50 million 50 − 40 = $10 million or $10,000,000

SECTION 1.3

13.

14. 15 year mortgage: $777.83(12)(15) = $140, 009.4

Points needed for 80 average: 80 (5) = 400 points Wallace’s points so far: 79 + 93 + 91 + 68 = 331 points Grade needed on fifth exam: 400 − 331 = 69

30 year mortgage: $521.65(12)(30) = $187, 794.0 Savings: $187, 794.00 − $140, 009.40 = $47, 784.6

15.

a) 10×10×10×10 = 10, 000

16.

b) 1 in 10,000

17.

38,687.0 mi − 38, 451.4 mi = 235.6 mi

18. a) 40 ×$8.50×52 = $17, 680 b) Each week he makes 40×$8.50 = $340. $1275 = 3.75 weeks $340

235.6 mi ≈ 18.698 ≈ 18.7 mpg 12.6 gal

19.

460 = 9.2 min 50 1550 b) = 62 min 25 1400 c) = 40 min 35 1550 2200 3750 d) + = ≈ 47 min 80 80 80

By mail: ($52.80 + $5.60 + $8.56)× 4 = $66.96× 4 = $267.84 Tire store: $324 + 0.08×$324

$885 − $25 (15) = $885 − $375 = $510 20. $510 =17 hours $30

= $324 + $25.92 = $349.92 Savings: $349.92 − $267.84 = $82.08

21.

15,000 ft − 3000 ft = 12,000 ft decrease in elevation. Temperature increases 2.4° F for every 1000 ft decrease in elevation. 2.4° F×12 = 28.8° F −6° F + 28.8° F = 22.8° F The precipitation at the airport will be snow.

22. a) $620 (0.12) = $74.40 b) $1200 (0.22) = $264 c) The store lost $1200 − $1000 = $200 on the purchase. Store's profit: $264 − $200 = $64

CHAPTER 1 Critical Thinking Skills

24. a) 0.1 cm 3 × 60 sec × 60 min × 24 hr × 365 days

23. Steve and Maureen paid more than $9362.50 but less than $26,687.50, so they paid $9362.50 plus 25% of the amount over $68,000.

= 3,153,600 cm 3 b) 30 cm × 20 cm × 20 cm=12,000 cm 3

$13,365 − $9362.50 = $4002.50

0.1 cm3 × 60 sec × 60 min × 24 hr = 8640 cm3

$4002.5 = $16, 01 The amount over $68,000 was 0.25 $68, 000 + $16, 010 = $84, 010

25. a) 1 oz × 60 min × 24 hr × 365 days = 525,600 oz 525,600 = 4106.25 gal 128 b) 4106.25 ×$11.20 = 4.10625× $11.20 = $45.99 1000

27. a)

20,000 20,000 − ≈ 961.538 − 925.926 20.8 21.6 = 35.612 ≈ 35.61 gal

b) 35.61×$3.00 = $106.83 c) 140,000,000 ×35.61 = 4,985, 400, 000 gal

29. Cost after 1 year: $799 + 0.06($799) = $799 + $47.94 = $846.94 Cost after 2 years: $846.94 + 0.06 ($846.94) = $846.94 + $50.82 = $897.76

12, 000 = 1.38 ≈ 1.4 days 8640

26.

a) Short: $33 × 5 = $165 Long: $18 × 5 = $90 $165 − $90 = $75; Jeff saves $75. b) $6 for first hour, plus 6 × $3 for remaining 3 hours, for a total of $24. c) Short: $6 + 8 × $3 = $30 Long: $18 Long term is cheaper by $12.

28. a) Yes, divide the total amount spent by the amount spent per capita.

$45,592.59 ≈ 301.14 million $151.40

$11, 237.53 ≈ 60.78 million $184.90

30. Value after first year: $1000 + 0.10($1000) = $1000 + $100 = $1100 Value after second year: $1100 − 0.10($1100) = $1100 − $110 = $990

31. After paying the $100 deductible, Yungchen must pay 20% of the cost of x-rays. First x-ray: $100 + 0.20 ($540) = $100 + $108 = $208

$990 is less than the intial investment of $1000. 32. $3000 is the difference between one-fourth of the cost and one-fifth of the cost. 1 1 1 − = ; 20 × $3000 = $60,000. 4 5 20

Second x-ray: 0.20 ($920) = $184 Total: $208 + $184 = $392 ⎛1⎞ 3 salt: 3⎜⎜ ⎟⎟⎟ = tsp ⎜⎝ 8 ⎠ 8

33. a) water/milk: 3(1) = 3 cups Cream of wheat: 3(3) = 9 tbsp =

9 cup (because 16 tbsp = 1 cup) 16

SECTION 1.3

2 + 3.75 5.75 7 = = 2.875 cups = 2 cups 2 2 8 0.25 + 0.5 0.75 3 = = 0.375 tsp = tsp salt: 2 2 8 0.5 + 0.75 1.25 5 5 = = 0.625 cups = cup = (16 tbsp) = 10 tbsp cream of wheat: 2 2 8 8 3 15 4 11 3 c) water/milk: 3 −1 = − = = 2 cups 4 4 4 4 4 1 1 4 1 3 3 3 12 3 9 cream of wheat: − = − = cup = 9 tbsp salt: − = − = tsp 2 8 8 8 8 4 16 16 16 16 d) Differences exist in water/milk because the amount for 4 servings is not twice that for 2 servings. 1 Differences also exist in Cream of Wheat because cup is not twice 3 tbsp. 2 1 b) rice: 1(2) = 2 cups 34. a) rice: ( 4) = 2 cups 2 1 4 16 1 1 9 18 2 1 water: 2 (2) = ( 2) = = 4 = 4 cups water: 1 (4) = ( 4) = = 5 cups 3 3 3 3 4 4 4 4 2 1 1 salt: ( 2) = 1 tsp salt: ( 4) = 1 tsp 4 2 b) water/milk:

butter/margarine: 1(4) = 4 tsp

butter/margarine: 2 ( 2) = 4 tsp

1 1 1 3 4 + 1 = + = = 2 cups 2 2 2 2 2 1 1 4 10 14 2 water: 1 + 3 = + = = 4 cups 3 3 3 3 3 3 1 3 4 salt: + = = 1 tsp 4 4 4 butter/margarine: 1 tsp + 1 tbsp = 1 tsp + 3 tsp = 4 tsp d) rice: 3 −1 = 2 cups 1 24 9 15 3 water: 6 − 2 = − = = 3 cups 4 4 4 4 4 1 1 salt: 1 − = 1 tsp 2 2 butter/margarine: 2 tbsp = 2 (3tsp) = 6 tsp

c) rice:

6 tsp − 2 tsp = 4 tsp e)

Differences exist in water because the amount for 4 servings is not twice that for 2 servings.

$425 − $240 (one box of 20 DVDs) = $185 $185 − $180 (one box of 12 DVDs) = $5 One box of 20 DVDs and one box of 12 DVDs are the maximum number of DVDs that can be purchased. b) $240 + $180 = $420

35. a)

36. Mark will win. 38. 1 ft 3 = 12 in.×12 in.×12 in. = 1728 in.3

37. 1 ft 2 would be 12 in. by 12 in. Thus, 1 ft 2 = 12 in.×12 in. = 144 in.2

CHAPTER 1 Critical Thinking Skills

39. Area of original rectangle = lw

20 ft ×20 ft = 400 ft 2

Area of new rectangle = ( 2l )( 2 w) = 4lw Thus, if the length and width of a rectangle are doubled, the area is 4 times as large. 41. Volume of original cube = lwh Volume of new cube = (2l )(2 w)( 2h ) = 8lwh Thus, if the length, width, and height of a cube are doubled, the volume is 8 times as large or increases eight- fold.

43.

5 ft ×5 ft = 25 ft 2

40.

400 ft 2 =16 squares 25ft 2 42. 11 ft is one-sixth of the pole, so the length is 6 × 11 ft = 66 ft.

10 pieces 1000 pieces = $x $10 1000 x = 10 (10) 1000 x 100 = 1000 1000 100 x= = $0.10 = 10¢ 1000

44. Left side: 1(−6) = −6

Right side: 1(2) = 2

2 (−2) = −4

1(3) = 3

− 6 +−4 = −10

1(6) = 6

45. 3

2 + 3 + 6 = 11 Place it at −1 so the left side would total −10 +−1 = −11 46. 10;2002 , 2112 , 2222 , 2332 , 2442 , 2552 ,

47. a) ( 4× 4) + (3×3) + (2 × 2) + (1×1)

2662 , 2772 , 2882 , 2992

= 16 + 9 + 4 + 1 = 30 b) (7×7) + (6×6) + (5×5) + 30 = 49 + 36 + 25 + 30 = 140 49.

48. a) Place the object, 1 g, and 3 g on one side and 9 g on the other side. b) Place the object, 9 g, and 3 g on one side and 27 g and 1 g on the other side. 50. Eight pieces

51.

52.

53. 8 + 6 + 2 + 4 = 20;3 + 7 + 5 + 1 = 16; 10 +14 +12 + 8 = 44

The sum of the four corner entries is 4 times the number in the center of the middle row.

SECTION 1.3

54. 15,12,33 Multiply the number in the center of the middle row by 3. 56. 35 −15 = 20 cubes

45,36,99 Multiply the number in the center of the middle row by 9. 57. 3× 2×1 = 6 ways

58. Each shakes with four people.

59.

55.

Other answers are possible, but 1 and 8 must appear in the center. 61.

60.

(The diagram shows the number of times each part is used.) 62.

With umbrella policy: Mustang reduced premium: $1648 − $90 = $1558 Focus reduced premium: $1530 − 0.12 ($1530) = $1530 − $183.60 = $1346.40 Total for umbrella policy: $1558 + $1346.40 + $450 = $3354.40 Without umbrella policy: $1648 + $1530 = $3178 Net amount for umbrella policy: $3354.40 − $3178 = $176.40 64. 16 + 16 + 4 + 4 + 4 = 44

Other answers are possible. 63.

Mary is the skier.

65.

Areas of the colored regions are: 1×1, 1×1, 2× 2, 3× 3, 5×5, 8×8, 13×13, 21× 21 ; 1 + 1 + 4 + 9 + 25 + 64 + 169 + 441 = 714 square units

66.

Let x be the amount Samantha had to start. 1 1 After first store: x − x − 20 = x − 20 2 2 ⎞⎟ 1 ⎛⎜ 1 1 After second store: ⎜ x − 20⎟⎟ − 20 = x − 30 ⎠ 2 ⎜⎝ 2 4 This is equal to $0, so the original amount was $120.

67.

Thomas would have opened the box labeled grapes and cherries. Because all the boxes are labeled incorrectly, whichever fruit he pulls from the box of grapes and cherries, will be the only fruit in that box. If he pulled a grape, he labeled the box grape. If he pulled a cherry, he labeled the box cherries. That left two boxes whose original labels were incorrect. Because all labels must be changed, there was only one way for Thomas to assign the two remaining labels.

CHAPTER 1 Critical Thinking Skills

Review Exercises 1. 27, 32, 37 (Add 5 to previous number.)

−48, 96, −192 (Multiply previous number by –2.)

10, 4, − 3 (subtract 1, then 2, then 3, ...)

26, 37, 50 (17 + 9 = 26, 26 + 11 = 37, 37 + 13 = 50) 25, 32, 40 (19 + 6 = 25, 25 + 7 = 32, 32 + 8 = 40) 3 3 3 1 , , (Multiply previous number by .) 8 16 32 2

6. 8.

9. c 10. a) The final number is twice the original number. b) The final number is twice the original number. c) Conjecture: The final number is twice the original number. 10n + 5 10n 5 d) n, 10n, 10n + 5, = + = 2n + 1, 2n + 1 − 1 = 2n 5 5 5 11. This process will always result in an answer of 3. n, n + 5,6 (n + 5) = 6n + 30, 6n + 30 −12 = 6n + 18,

6n + 18 6n 18 3n + 9 3n 9 = + = 3n + 9, = + = n + 3, n + 3− n = 3 2 2 2 3 3 3

12. 12 + 22 = 5,5 is an odd number. Other answers are possible. (Note: Answers for Ex. 13 - 25 will vary depending on how you round your numbers. The answers may differ from the answers in the back of the textbook. However, your answers should be something near the answers given. All answers are approximate.) 13. 14. 215.9 + 128.752 + 3.6 + 861 + 792 ≈ 200 + 100 + 0 + 900 + 800 = 2000 210,302 ×1992 ≈ 210,000× 2000 = 420,000,000

15.

19% of 1025 ≈ 20% of 1000 = 0.20×1000 = 200

16.

17.

52 shovels ×$99.97 ≈ 50×100 = $5000

18.

Answers will vary.

7% of $1999 ≈ 7% of 2000 = 0.07 × 2000 = $140 19.

1.1 mi 1 mi 3 mi ≈ = = 3 mph 22 min 20 min 60 min

21. 5 in. =

20.

$2.49 + $0.79 + $1.89 + $0.10 + $2.19 + $6.75

22.

≈ $2 + $1 + $2 + $0 + $2 + $7 = $14.00 2.35 million − 1.95 million = 0.4 million

24.

13 square units

26.

$50 + $40 (12)= $530 Savings: $530 − $500 = $30

⎛1⎞ 20 in. = 20 ⎜⎜ ⎟⎟⎟ in. = 20 (0.1) mi = 2 mi ⎜⎝ 4 ⎠ 4

23.

2.8 million − 1.8 million = 1.0 million

25.

Length = 1.75 in., 1.75(12.5) = 21.875 ≈ 22 ft Height = 0.625 in., 0.625(12.5) = 7.8125 ≈ 8 ft

REVIEW EXERCISES

27.

4 ($2.69) = $10.76 for four six-packs Savings: $10.76 − $9.60 = $1.16

$445 = $89 5 $510 = $85 Cost per person with 6 people: 6 $89 − $85 = $4 savings

29. Cost per person with 5 people:

31.

10% of $1030 = 0.10 × $1030 = $103

30 x 30 × 24,000 = ; x= = 288 lb 2500 24,000 2500 150 b) = 5 bags, and 5 × 2500 = 12,500 ft 2 30

30. a)

32.

Savings: $721 − $60 = $661

1.5 mg x mg = 10 lb 47 lb 10 x = 47 (1.5) 10 x 70.5 = 10 10 x = 7.05 mg

$5000 − 0.30 ($5000) = $5000 − $1500

34.

9 A.M. Eastern is 6 A.M. Pacific, from 6 A.M. Pacific to 1:35 P.M. Pacific is 7 hr 35 min , 7 hr 35 min − 50 min stop = 6 hr 45 min

36.

a) 5280 ft 1hr 1min 5280 ft × × = ≈1.47 ft/sec 1 60 min 60 sec 3600sec

= $3500 take-home 28% of $3500 = 0.28×$3500 = $980 35.

Taylor: $45 for 8 hours Admar: $8 × 6 hours = $48 $48 − $45 = $3 Taylor Rental is cheaper by $3.

28.

$103× 7 = $721

33.

3 P.M. − 4 hr = 11 A.M. July 26, 11:00 A.M.

b) 55 mi 5280 ft 1hr 1min 290, 400 ft × × × = 1hr 1mi 60 min 60sec 3600sec ≈ 80.67 ft/sec 37. Each figure has an additional two dots. To get the hundredth figure, 97 more figures must be drawn, 97 ( 2) = 194 dots added to the third

38.

figure. Thus, 194 + 7 = 201.

39.

40.

59 min 59 sec Since it doubles every second, the jar was half full 1 second earlier than 1 hour.

41. 6 42. Nothing. Each friend paid $9 for a total of $27; $25 to the hotel, $2 to the clerk. $25 for the room + $3 for each friend + $2 for the clerk = $30 43. Let x = the total weight of the four women

CHAPTER 1 Critical Thinking Skills

x 520 + 180 700 = 130, x = 520, = = 140 lb 4 5 5 44. Yes; 3 quarters and 4 dimes, or 1 half dollar, 1 quarter and 4 dimes, or 1 quarter and 9 dimes. 45. 6 cm × 6 cm × 6 cm = 216 cm 3 46. Place six coins in each pan with one coin off to the side. If it balances, the heavier coin is the one on the side. If the pan does not balance, take the six coins on the heavier side and split them into two groups of three. Select the three heavier coins and weigh two coins. If the pan balances, it is the third coin. If the pan does not balance, you can identify the heavier coin.

n (n + 1)

500 (501)

250,500 = 125, 250 2 2 2 48. 16 blue: 4 green → 8 blue, 2 yellow → 5 blue, 2 white → 3 blue 49. 90: 101, 111, 121, 131, 141, 151, 161, 171, 181, 191, … 50. The fifth figure will be an octagon with sides of equal length. Inside the octagon will be a seven sided figure with each side of equal length. The figure will have one antenna. 51. 61: The sixth figure will have 6 rows of 6 tiles and 5 rows of 5 tiles (6× 6 + 5× 5 = 36 + 25 = 61). 52. Some possible answers are given below. There are other possibilities.

47.

53. a) 2 b) There are 3 choices for the first spot. Once that person is standing, there are 2 choices for the second spot and 1 for the third. Thus, 3× 2×1 = 6 . c) 4× 3× 2 ×1 = 24 d) 5× 4×3× 2 ×1 = 120 e) n (n −1)( n − 2)"1, (or n !), where n = the number of people in line

Chapter Test 1. 27, 33, 39 (Add 6 to previous number.)

1 1 1 1 , , (Multiply previous number by .) 16 32 64 2

3. a) The result is the original number plus 1. b) The result is the original number plus 1. c) Conjecture: The result will always be the original number plus 1. 5n + 10 5n 10 = + = n + 2, n + 2 −1 = n + 1 d) n,5n,5n + 10, 5 5 5 (Note: Answers for #4 - #6 will vary depending on how you round your numbers. The answers may differ from the answers in the back of the textbook. However, your answers should be something near the answers given. All answers are approximate.)

CHAPTER TEST

0.21×82, 000 ≈ 0.2×80, 000 = 16, 000

9 square units

a) 3.5 million b) 0.3 million

10.

$15 ≈ 5.79 $2.59 The maximum number of 6 packs is 5.

175, 000 170, 000 ≈ ≈ 1, 700, 000 0.09 0.1 130 lb ≈ 2.0635 a) 63 in. 2.0635 = 0.032754 63 in. 0.032754×703 ≈ 23.03 b) He is in the at risk range.

$85.99 − $59.99 = $26 9. $26 = 65 additional minutes $0.40 11. 1 cut yields 2 equal pieces. Cut each of these 2 equal pieces to get 4 equal pieces. 3 cuts → 3(2.5 min) = 7.5 min

$15.00 − (5×$2.59) = $15.00 − $12.95 = $2.05 $2.05 = 2.5625 $0.80 Thus, two individual cans can be purchased. Number of cans 6 packs Indiv. cans 5 2 32 4 5 29 3 9 27 2 12 24 1 15 21 0 18 18 The maximum number of cans is 32. 12. 2.5 in. by 1.875 in. ≈ 2.5×15.8 by 1.875×15.8 = 39.5 in. by 29.625 in. ≈ 39.5 in. by 29.6 in. (The actual dimensions are 100.5 cm by 76.5 cm.)

13. $12.75 × 40 = $510 $12.75 × 1.5 × 10 = $191.25 $510 + $191.25 = $701.25 $701.25 − $652.25 = $49.00

14.

15.

Mary drove the first 15 miles at 60 mph which took

15 1 = hr, and the second 15 miles at 30 mph which 60 4

15 1 3 = hr for a total time of hr. If she drove the entire 30 miles at 45 mph, the trip would take 30 2 4 30 2 3 = hr (40 min) which is less than hr (45 min). 45 3 4

took

CHAPTER 1 Critical Thinking Skills

16.

6 lb 1 3 1 = 3; 3× tsp = tsp or 1 tsp 2 lb 2 2 2 1

17.

1 1 tsp = tbsp 2 2

Area of lawn including walkway: (10 + 2)×(12 + 2)=12×14 =168 m 2 Area of lawn only: 10×12 = 120 m 2 Area of walkway: 168 −120 = 48 m 2

18. 243 jelly beans; 260 −17 = 243, 234 + 9 = 243, 274 − 31 = 243 19. a) 3 × $3.99 = $11.97 b) 9 ($1.75×0.75) = 11.8125 ≈ $11.81 c) $11.97 − $11.81 = $0.16 Using the coupon is least expensive by $0.16. 20. 24 (The first position can hold any of four letters, the second any of the three remaining letters, and so on. 4 × 3 × 2 × 1 = 24

Group Projects

$325 ≈ $108.33 3 b) Let x = the amount before tax x + 0.07 x = 325

1. a)

1.07 x 325 = 1.07 1.07 x = 303.7383178 ≈ $303.74 $303.74 = 101.246 ≈ $101.25 3 c) Inductive reasoning - arriving at a general conclusion from specific cases d) Combination set: $62.00 − ($62.00×0.10) = $62.00 − $6.20 = $55.80 Individual sets: 2×$36.00 = $72.00,$72.00 −($72.00× 0.20) = $72.00 − $14.40 = $57.60 Therefore, the combination set is cheaper. e) Combination with tax: $55.80 × 1.07 ≈ $59.71 Individual set with tax: $57.60 × 1.07 ≈ $61.63 $61.63 − $59.71 = $1.92 2. a) – d) Answers will vary. 400 mi ÷ 50 mi hr = 8 hrs, 9 A.M. + 8 hrs = 5 P.M. e) f) – h) Answers will vary. 3. Order 1 2 3 4

Name Ernie Zeke Jed Tex

Apparel holster vest chaps Stetson

CHAPTER TWO SETS Exercise Set 2.1 1. Set 2. Ellipsis 3. Description, Roster form, Set-builder notation 4. Finite 5. Infinite 6. Equal 7. Equivalent 8. Cardinal 9. Empty or null 10. { } ,∅ 11. Universal 12. One-to-one 13. Not well defined, “best” is interpreted differently by different people. 14. Not well defined, “most interesting” is interpreted differently by different people. 15. Well defined, the contents can be clearly determined. 16. Well defined, the contents can be clearly determined. 17. Well defined, the contents can be clearly determined. 18. Not well defined, “most interesting” is interpreted differently by different people. 19. Infinite, the number of elements in the set is not a natural number. 20. Finite, the number of elements in the set is a natural number. 21. Infinite, the number of elements in the set is not a natural number. 22. Infinite, the number of elements in the set is not a natural number. 23. Infinite, the number of elements in the set is not a natural number. 24. Finite, the number of elements in the set is a natural number. ⎧San Marino, Scotland, Serbia, Slovakia,⎪ ⎫ 25. { Maine, Maryland, Massachusetts, Michigan, ⎪ ⎪ 26. ⎪⎨ ⎬ ⎪ ⎪ Minnesota, Misssissippi, Missouri, Montana } Slovenia, Spain, Sweden, Switzerland ⎪ ⎪ ⎩ ⎭

27. 29.

{ 11,12,13,14, … ,177 } B = { 2, 4, 6, 8, …}

28.

C = {4}

30.

{ } or ∅

CHAPTER 2 Sets

31. 33.

{ } or ∅ E = { 14, 15, 16, 17, …, 84}

32. 34.

{Idaho, Oregon } { Alaska, Hawaii }

35. {Metropolitan Museum of Art, Tate Modern, National Gallery of Art, British Museum, Louvre Museum} 36. { Musee d'Art Moderne Prado, Museum of Modern Art } 37. { Museum of Modern Art, Musee d'Art Moderne Prado, Musee d'Orsay } 38.

{ National Gallery, Vatican Museums, Metropolitan Museum of Art, Tate Modern, National Gallery of Art } 39.

{ 2007, 2008 }

40.

{ 2003 }

41.

{ 2004, 2005, 2006, 2007 }

42.

{ } or ∅

43.

B = { x x ∈ N and 6 < x < 15} or

44.

A = { x x ∈ N and x < 10} or

B = { x x ∈ N and 7 ≤ x ≤ 14}

A = { x x ∈ N and x ≤ 9}

45.

C = { x x ∈ N and x is a multiple of 3}

46.

D = { x x ∈ N and x is a multiple of 5}

47.

E = { x x ∈ N and x is odd}

48.

A = { x x is Independence Day}

49.

C = { x x is February}

50.

F = { x x ∈ N and 14 < x < 101} or F = { x x ∈ N and 15 ≤ x ≤ 100}

51. 52. 53. 54. 55. 56. 57. 58.

Set A is the set of natural numbers less than or equal to 7. Set D is the set of natural numbers that are multiples of 3. Set V is the set of vowels in the English alphabet. Set S is the set of the seven dwarfs in Snow White and the Seven Dwarfs. Set T is the set of species of trees. Set E is the set of natural numbers greater than or equal to 4 and less than 11. Set S is the set of seasons. Set B is the set of members of the Beatles.

59.

{ China, India, United States }

60.

{ Pakistan, United Kingdom }

61.

{ Russia, Brazil, Indonesia, Japan, Germany }

62.

{ India, United States }

63.

{ 2008, 2009, 2010, 2011 }

64.

{ 1996, 1997, 1998, 1999 }

65. {2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007}

66.

{2008, 2010}

67.

False; {e} is a set, and not an element of the set.

68.

True; b is an element of the set.

69. 71. 73.

False; h is not an element of the set. False; 3 is an element of the set. True; Titanic is an element of the set.

70. 72. 74.

True; Mickey Mouse is an element of the set. False; the Amazon is a river in South America. False; 2 is an even natural number.

75.

n ( A) = 4

76.

n ( B) = 6

77.

n (C ) = 0

78.

n (D) = 5

79. Both; A and B contain exactly the same elements. 80. Equivalent; both sets contain the same number of elements, 3. 81. Neither; the sets have a different number of elements.

154

CHAPTER 5 Number Theory and the Real Number System

3. Co-pay for prescriptions = 50% Co-pay for office visits = $10 01/10: $10 + .50 ($44) = $32.00 02/27: $10 + .20 (188) = $47.60 04/19: $10 + .20 (348) + .50 (76) = $117.60 a) Total = $197.20 b) .50 (44) + .80 (188) + .80 (348) + .50 (76) +3(40) = $608.80 c) $500.00 – 197.20 = $302.80 4. a) 1 branch b) 5 branches c) 233 branches

Co-pay for medical tests = 20%

d) Produces the same sequence.

CHAPTER SIX ALGEBRA, GRAPHS, AND FUNCTIONS Exercise Set 6.1 1. Variable 2. Constant 3. Expression 4. 4 5. 3 6. Solution 7. Evaluate 8. Division 9. x = −6, x 2 = (−6) = 36

10. x = 5, x 2 = 52 = 25

11. x = −4, − x 2 = −(−4) = −16

12. x = −2, − x 2 = −(−2) = −4

13. x = −7, − 2 x 3 = −2 (−7) = −2 (−343) = 686

14. x = −4, − x 3 = −(−4) = −(−64) = 64

15. x = 4, x − 7 = 4 − 7 = −3

⎛5⎞ 5 16. x = ,8 x − 3 = 8⎜⎜ ⎟⎟⎟− 3 = 20 − 3 = 17 ⎜ ⎝2⎠ 2

17. x = −2, −3x + 7 = −3(−2) + 7 = 6 + 7 = 13

18. x = −1, 4 x − 3 = 4 (−1) − 3 = −4 − 3 = −7

19. x = 3, x 2 − 5 x + 12 = (3) − 5(3) + 12

20. x = −2, −x 2 + 3x −10 = −(−2) + 3(−2)−10

= −4 − 6 −10 = −20

= 9 −15 + 12 = 6

1 2 2 ⎛1⎞ 1 22. x = , x 2 + x −1 = ⎜⎜ ⎟⎟⎟ + −1 2 3 3 ⎜⎝ 2 ⎠ 2 2

2 ⎛2⎞ 2 1 1 ⎛2⎞ 21. x = , x 2 − 5 x + 2 = ⎜⎜ ⎟⎟⎟ − 5⎜⎜ ⎟⎟⎟ + 2 ⎜ ⎝⎜ 3 ⎠ 3 2 2 ⎝ 3⎠

2 ⎛1⎞ 1 = ⎜⎜ ⎟⎟⎟ + −1 3 ⎜⎝ 4 ⎠ 2

1 ⎛ 4 ⎞ 10 = ⎜⎜ ⎟⎟⎟ − + 2 2 ⎜⎝ 9 ⎠ 3

4 60 36 20 10 − + =− =− 18 18 18 18 9

155

2 1 + −1 12 2 2 6 12 = + − 12 12 12 4 1 =− =− 12 3 =

157

CHAPTER 6 Algebra, Graphs, and Functions

⎛ 1 ⎞3 ⎛ 1 ⎞2 1 23. x = ,8 x 3 − 4 x 2 + 7 = 8⎜⎜ ⎟⎟⎟ − 4 ⎜⎜ ⎟⎟⎟ + 7 ⎜⎝ 2 ⎠ ⎝⎜ 2 ⎠ 2

24. x = 2, y = −3, − x 2 + 5 xy = −(2) + 5( 2)(−3) 2

= −4 − 30 = −34

⎛1⎞ ⎛ 1 ⎞ = 8⎜⎜ ⎟⎟⎟ − 4 ⎜⎜ ⎟⎟⎟ + 7 ⎝⎜ 8 ⎠ ⎝⎜ 4 ⎠ = 1−1 + 7 = 7

25. x = −2, y = 1, 3 x 2 − xy + 2 y 2 = 3(−2) − (−2)(1) + 2 (1) = 12 + 2 + 2 = 16 2

27. x = 2, y = −1, 4 x 2 −10 xy + 3 y 2 = 4 ( 2) −10 ( 2)(−1) + 3(−1) = 16 + 20 + 3 = 39 2

29. 5 x + 2 = 17, x = 3

2 1 26. x = 2, y = 5, 3x 2 + xy − y 2 5 5 2 1 2 2 = 3(2) + (2)(5) − (5) 5 5 = 12 + 4 − 5 = 11 2 2 28. x = 4, y = −2, (2 x − y ) = ⎡⎣ 2 ( 4) − (−2)⎤⎦

= 102 = 100 30. 6 x − 3 = −27, x = −4

5(3) + 2 = 15 + 2 = 17

6 (−4) − 3 = −24 − 3 = −27

17 = 17, x = 3 is a solution.

−27 = −27, x = −4 is a solution. 32. 4 x + 2 y = −2, x = −2, y = 3

31. x − 3 y = 0, x = 6, y = 3 6 − 3(3) = 6 − 9 = −3

4 (−2) + 2 (3) = −8 + 6 = −2

−3 ≠ 0, x = 6, y = 3 is not a solution.

−2 = −2, x = −2, y = 3 is a solution.

33. −x 2 + 3x + 6 = 5, x = 2

34. −2 x 2 + x + 5 = 0, x = 3

−( 2) + 3 ( 2 ) + 6 = − 4 + 6 + 6 = 8

−2 (3) + 3 + 5 = −2 (9) + 3 + 5 = −10

8 ≠ 5, x = 2 is not a solution.

−10 ≠ 20, x = 3 is not a solution.

35. 3x 2 + 2 x = 40, x = −4

36. y = x 2 + 3 x − 6, x = −1, y = −8

3(−4) + 2 (−4) = 3(16) − 8 = 48 − 8 = 40

(−1) + 3(−1)− 6 = 1− 3 − 6 = −8

40 = 40, x = −4 is a solution.

−8 = −8, x = −1, y = −8 is a solution.

37. y = −x 2 + 3x −1, x = 3, y = −1

38. y = x 3 − 3x 2 + 1, x = 2, y = −3

−(3) + 3(3) −1 = −9 + 9 −1 = −1

(2) − 3(2) + 1 = 8 −12 + 1 = −3

−1 = −1, x = 3, y = −1 is a solution.

−3 = −3, x = 2, y = −3 is a solution.

39. x = 120 mi, 0.55 x = 0.55(120 mi ) = $66

40. n = 8,000, 000,000,000 0.000002n = 0.000002 (8,000,000,000, 000) = 16,000,000 sec

41. 2012 is 8 years since 2004. 6.8(8) + 61 = 115.4; 115.4 million taxpayers

42. x = 75, 220 + 2.75 x = 220 + 2.75(75)

43. 2(0.60) + 80(0.60) + 40 = 0.72 + 48 + 40 = 88.72; 88.72 min 2

= 220 + 206.25 = $426.25

158

CHAPTER 6 Algebra, Graphs, and Functions

44. s = 72 0.08s 2 + 0.24 s − 7.10 = 0.08(72)2 + 0.014(72) − 7.10 = 0.08(5184) + 17.28 − 7.10 = 414.72 + 17.28 − 7.10 = 424.9 ft 45. R = 2, T = 70,0.2 R 2 + 0.003RT + 0.0001T 2 = 0.2 (2) + 0.003(2)(70) + 0.0001(70) 2

= 0.8 + 0.42 + 0.49 = 1.71 in. 46. (−1) = 1 for any even number, n , since there will be an even number of factors of (−1) , and when n

these are multiplied, the product will always be 1. 47.

( x + y)

x2 + y 2

52 = 25

4 + 9 = 13

−2

−3

(−5) = 25

4 + 9 = 13

−2

12 = 1

4 + 9 = 13

−3

(−1) = 1

4 + 9 = 13

The two expressions are not equal. 48. 1 = 1 for all natural numbers since 1 multiplied by itself any number of times will always be 1. 49. Answers will vary. n

Exercise Set 6.2 1. Terms 2. Like 3. Coefficient 4. Linear 5. Simplify 6. Algorithm 7. Ratio 8. Proportion 9. Identity 10. Contradiction 11. Isolate 12. Equivalent 13. 5 x + 3 x = 8 x 14. 7 x − 2 x = 5 x 15. −8 x − 3 x + 12 = −11x + 12 17. 7 x + 3 y − 4 x + 8 y = 3 x + 11 y

16. 18.

−6 x + 2 x + 21 = −4 x + 21 x − 4 x + 3 = −3x + 3

19. 21. 23.

−3 x + 2 − 5 x = − 8 x + 2 2 − 3 x − 2 x + 1 = −5 x + 3 6.2 x − 8.3 + 7.1x = 13.3x − 8.3

20. 22. 24.

−3 x + 4 x − 2 + 5 = x + 3 −0.2 x + 1.7 x − 4 = 1.5 x − 4 4 s + 3t + 8 − 2t = 4s + t + 8

25.

2 1 4 1 5 x + x −5 = x + x −5 = x −5 3 6 6 6 6

26.

1 1 3 5 2 x− x−4 = x− x−4 = − x −4 5 3 15 15 15

SECTION 6.2

27.

7x − 4 y −8 y + 6x + 2 = 7x + 6x − 4 y −8 y + 2

28.

= 13 x −12 y + 2

30.

0.2( x + 4) + 1.2( x − 3) = 0.2 x + 0.8 + 1.2 x − 3.6 = 1.4 x − 2.8 3 3 3 3 3 3 3 ( x + 1) − x = x − x + = x + 5 10 5 10 5 10 5

35.

1 4 2 3 8 4 5 4 x+ − x = x− x+ =− x+ 4 5 3 12 12 5 12 5

37.

38.

39.

40.

6 (r − 3) − 2 (r + 5) + 10 = 6r −18 − 2r −10 + 10 = 4r −18

2 ( s + 3) + 6 ( s − 4) + 1 = 2 s + 6 + 6s − 24 + 1 = 8s −17

33.

2 p − 4q − 3 p + 4q −15 = 2 p − 3 p − 4q + 4q −15 = − p −15

29.

31.

32.

0.9 ( 2.3x − 2) + 1.7 (3.2 x − 5) = 2.07 x −1.8 + 5.44 x − 8.5 = 7.51x −10.3

2 1 1 5 (3x + 9)− (2 x + 5) = 2 x + 6 − x − 3 4 2 4 4 1 24 5 3 19 = x− x+ − = x+ 2 2 4 4 2 4 3 5 1 9 5 9 2 36. n − + n − = n + n − − 4 9 6 9 9 12 12 14 11 = n− 9 12 34.

y − 4 = 13 y − 4 + 4 = 13 + 4 y = 17

Add 4 to both sides of the equation.

y + 6 = 12 y + 6 − 6 = 12 − 6 y=6

Subtract 6 from both sides of the equation.

21 = −2t + 9 21− 9 = −2t + 9 − 9 12 = −2t 12 −2t = −2 − 2 −6 = t 12 = 3 x + 6 12 − 6 = 3 x + 6 − 6 6 = 3x 6 3x = 3 3 2= x

159

Subtract 9 from both sides of the equation.

Divide both sides of the equation by -2.

Subtract 6 from both sides of the equation.

Divide both sides of the equation by 3.

160

41.

CHAPTER 6 Algebra, Graphs, and Functions

3 7 = x 8 3(8) = 7 x

Cross multiplication

24 = 7 x 24 7 x = 7 7 24 =x 7 42.

Divide both sides of the equation by 7.

x −1 x + 5 = 5 15 15( x −1) = 5 ( x + 5) 15 x −15 = 5 x + 25 15 x − 5 x −15 = 5 x − 5 x + 25 10 x −15 = 25 10 x −15 + 15 = 25 + 15 10 x = 40 10 x 40 = 10 10 x=4

43.

1 1 2 x+ = 2 3 3 ⎛1 ⎛ 2⎞ 1⎞ 6 ⎜⎜ x + ⎟⎟⎟ = 6 ⎜⎜ ⎟⎟⎟ ⎝⎜ 2 ⎝⎜ 3 ⎠ 3⎠

Add 15 to both sides of the equation.

Divide both sides of the equation by 10.

Distributive Property Subtract 2 from both sides of the equation.

Divide both sides of the equation by 3.

1 1 1 y+ = 2 3 4 ⎛1 ⎞ ⎛1⎞ 1 12 ⎜⎜ y + ⎟⎟⎟ = 12 ⎜⎜ ⎟⎟⎟ ⎜⎝ 2 ⎝⎜ 4 ⎠ 3⎠ 6y + 4 = 3 6 y + 4 − 4 = 3− 4 6 y = −1 6 y −1 = 6 6 1 y =− 6

Distributive Property Subtract 5 x from both sides of the equation.

Multiply both sides of the equation by the LCD.

3x + 2 = 4 3x + 2 − 2 = 4 − 2 3x = 2 3x 2 = 3 3 2 x= 3

44.

Cross multiplication

Multiply both sides of the equation by the LCD. Distributive Property Subtract 4 from both sides of the equation.

Divide both sides of the equation by 6.

SECTION 6.2

45.

46.

47.

48.

49.

0.9 x −1.2 = 2.4 0.9 x + 1.2 −1.2 = 2.4 + 1.2 0.9 x = 3.6 0.9 x 3.6 = 0.9 0.9 x=4 5 x + 0.050 = −0.732 5 x + 0.050 − 0.050 = −0.732 − 0.050 5 x = −0.782 5 x −0.782 = 5 5 x = −0.1564 6t − 8 = 4t − 2 6t − 4t − 8 = 4t − 4t − 2 2t − 8 = −2 2t − 8 + 8 = −2 + 8 2t = 6 2t 6 = 2 2 t=3 6t − 7 = 8t + 9 6t − 8t − 7 = 8t − 8t + 9 −2t − 7 = 9 −2t − 7 + 7 = 9 + 7 −2t = 16 −2t 16 = −2 − 2 t = −8

1 x + 2x = 4 3 ⎛x ⎞⎟ ⎛1⎞ 12 ⎜⎜ + 2 x⎟⎟ = 12 ⎜⎜ ⎟⎟⎟ ⎝⎜ 4 ⎠ ⎝⎜ 3⎠ 3 x + 24 x = 4 27 x = 4 27 x 4 = 27 27 4 x= 27

Add 1.2 to both sides of the equation.

Divide both sides of the equation by 0.9.

Subtract 0.050 from both sides of the equation.

Divide both sides of the equation by 5.

Subtract 4t from both sides of the equation. Add 8 to both sides of the equation.

Divide both sides of the equation by 2.

Subtract 8t from both sides of the equation. Add 7 to both sides of the equation.

Divide both sides of the equation by -2.

Mulitply both sides of the equation by the LCD. Distributive Property

Divide both sides of the equation by 27.

161

162

50.

CHAPTER 6 Algebra, Graphs, and Functions

r + 2r = 7 3 ⎛r ⎞ 3⎜⎜ + 2r ⎟⎟⎟ = 3(7) ⎜⎝ 3 ⎠

Mulitply both sides of the equation by the LCD.

r + 6r = 21

Distributive Property

7r = 21 7r 21 = 7 7 x=3

51.

Divide both sides of the equation by 7.

x−3 x + 4 = 2 3 3( x − 3) = 2 ( x + 4)

Cross multiplication

3x − 9 = 2 x + 8

Distributive Property

3x − 2 x − 9 = 2 x − 2 x + 8

Subtract 2 x from both sides of the equation.

x−9 = 8 x−9+ 9 = 8+9

Add 9 to both sides of the equation.

x = 17

52.

x −5 x −9 = 4 3 3( x − 5) = 4 ( x − 9)

Cross multiplication

3x −15 = 4 x − 36

Distributive Property

3x − 3x −15 = 4 x − 3x − 36

Subtract 3x from both sides of the equation.

−15 = x − 36 −15 + 36 = x − 36 + 36

Add 36 to both sides of the equation.

21 = x

53.

2 ( x + 3) − 4 = 2 ( x − 4) 2x + 6 − 4 = 2x −8 2x + 2 = 2x −8 2x − 2x + 2 = 2x − 2x −8 2 = −8 No solution

54.

Distributive Property Subtract 2 x from both sides of the equation. False

3( x + 2) + 2 ( x −1) = 5 x − 7 3x + 6 + 2 x − 2 = 5 x − 7 5 x + 4 = 5x − 7 5 x − 5 x + 4 = 5x − 5x − 7 4 = −7 No solution

Distributive Property Subtract 5 x from both sides of the equation. False

SECTION 6.2

55.

4 ( x − 4) + 12 = 4 ( x −1) 4 x −16 + 12 = 4 x − 4 4x − 4 = 4x − 4

Distributive Property

This equation is an identity. Therefore, the solution is all real numbers.

56.

6(t + 2) −14 = 6t − 2 6t + 12 −14 = 6t − 2 6t − 2 = 6t − 2

Distributive Property Combine like terms.

This equation is an identity. Therefore, the solution is all real numbers

57.

1 2 ( x + 3) = ( x + 2) 3 5 ⎛ 1 ⎞⎟ ⎛ 2⎞ 15⎜⎜ ⎟⎟( x + 3) = 15⎜⎜ ⎟⎟⎟( x + 2) ⎝⎜ 3 ⎠ ⎝⎜ 5 ⎠

Multiply both sides of the equation by the LCD.

5 ( x + 3) = 6 ( x + 2) 5 x + 15 = 6 x + 12 5 x − 5 x + 15 = 6 x − 5 x + 12 15 = x + 12 15 −12 = x + 12 −12 3= x

58.

2 1 ( x − 4) = ( x + 1) 3 4 ⎛ 2 ⎞⎟ ⎛1⎞ 12 ⎜⎜ ⎟⎟( x − 4) = 12 ⎜⎜ ⎟⎟⎟( x + 1) ⎜⎝ 3 ⎠ ⎝⎜ 4 ⎠

Distributive Property Subtract 5x from both sides of the equation. Subtract 12 from both sides of the equation.

Multiply both sides of the equation by the LCD.

8( x − 4) = 3( x + 1) 8 x − 32 = 3x + 3 8 x − 3x − 32 = 3x − 3 x + 3 5 x − 32 = 3 5 x − 32 + 32 = 3 + 32 5 x = 35 5 x 35 = 5 5 x=7

Distributive Property Subtract 3 x from both sides of the equation. Add 32 to both sides of the equation.

Divide both sides of the equation by 5.

163

164

59.

60.

61.

CHAPTER 6 Algebra, Graphs, and Functions

3x + 2 − 6 x = −x −15 + 8 − 5 x −3 x + 2 = − 6 x − 7 −3 x + 6 x + 2 = − 6 x + 6 x − 7 3 x + 2 = −7 3 x + 2 − 2 = −7 − 2 3 x = −9 3 x −9 = 3 3 x = −3

6 x + 8 − 22 x = 28 + 14 x −10 + 12 x −16 x + 8 = 26 x + 18 −16 x − 26 x + 8 = 26 x − 26 x + 18 −42 x + 8 = 18 −42 x + 8 − 8 = 18 − 8 −42 x = 10 −42 x 10 = −42 −42 10 5 x=− =− 42 21

Subtract 2 from both sides of the equation.

Divide both sides of the equation by 3.

Subtract 26 x from both sides of the equation. Subtract 8 from both sides of the equation. Divide both sides of the equation by − 42.

5(3n + 1) = 2 (5n − 4) + 6n 15n + 5 = 10n − 8 + 6n 15n + 5 = 16n − 8 15n −15n + 5 = 16n −15n − 8 5 = n −8 5 + 8 = n −8 + 8 13 = n

62.

Add 6 x to both sides of the equation.

Distributive Property Subtract 15n from both sides of the equation. Add 8 to both sides of the equation.

4 (t − 3) + 8 = 4 (2t − 6) 4t −12 + 8 = 8t − 24 4t − 4 = 8t − 24 4t − 4t − 4 = 8t − 4t − 24 −4 = 4t − 24 −4 + 24 = 4t − 24 + 24 20 = 4t 20 4t = 4 4 5=t

Distributive Property Subtract 4t from both sides of the equation. Add 24 to both sides of the equation.

Divide both sides of the equation by 4.

SECTION 6.2

1.98 x = 750 27, 000

63.

1.98 200 = 750 x 1.98 x = (200)(750) 1.98 x 150, 000 = 1.98 1.98 x ≈ 75, 757 gal

64.

1.98( 27, 000) = 750 x 1.98(27, 000) 750 x = 750 750 x = $71.28

825 6600 = 1 x 825 x = (6600)(1) 825 x 6600 = 825 825 x = 8 gal

65.

67. a)

6 16 = 9 x 6 x = 9 (16)

1 21 = 1,102, 000 x

66.

x = 1,102, 000 (21) x = 23,142, 000 households

x 40 = 12 480 40(480) = 12 x 40(480) 12 x = 12 12 x = 1600 lb 480 = 40 bags 12

68. a)

6 x = 144 6 x 144 = 6 6 x = 24 oz x 6 = 32 16 16 x = 32 (6)

16 x = 192 16 x 192 = 16 16 x = 12 servings 69. a)

50 1 = 80 x 50 x = 80 50 x 80 = 50 50 x = 1.6 kph x 50 = 80 90 80 x = 50 (90)

70.

40 250 = 0.6 x 40 x = 0.6 (250) 40 x = 150 40 x 150 = 40 40 x = 3.75 mA

80 x = 4500 80 x 4500 = 80 80 x = 56.25 mph

165

166

CHAPTER 6 Algebra, Graphs, and Functions

40 15 = 1 x 40 x = 15 40 x 15 = 40 40 x = 0.375 cc 73. a) Answers will vary. 71.

72.

40 35 = 1 x 40 x = 35 40 x 35 = 40 40 x = 0.875 cc

2 ( x + 3) = 4 x + 3 − 5 x

2 x + 6 = −x + 3 Distributive Property 2 x + x + 6 = −x + x + 3 Add x to both sides of the equation. 3x + 6 = 3 3x + 6 − 6 = 3 − 6 Subtract 6 from both sides of the equation. 3 x = −3 3x -3 = Divide both sides of the equation by 3. 3 3 x = −1 74. a) P = 14.70 + 0.43 x 148 = 14.70 + 0.43 x Given P = 148, find x. 148 −14.70 = 14.70 −14.70 + 0.43 x Subtract 14.70 from both sides of the equation. 133.3 = 0.43x 133.3 0.43x = Divide both sides of the equation by 0.43. 0.43 0.43 x = 310 ft P = 14.70 + 0.43 x b) 128.65 = 14.70 + 0.43 x Given P = 128.65, find x. 128.65 −14.70 = 14.70 −14.70 + 0.43x Subtract 14.70 from both sides of the equation. 113.95 = 0.43x 113.95 0.43x = Divide both sides of the equation by 0.43. 0.43 0.43 x = 265 ft below sea level

75. a) 2:5; There are 2x males and a total of 2x + 3x = 5x students. b) m : ( m + n) 76. a) Yes (multiply the first equation on both sides by18, and you get the second equation). b) Yes (multiply the second equation by 4 on both sides and you get the first equation).

166

CHAPTER 6 Algebra, Graphs, and Functions

Exercise Set 6.3 1. Formula 2. Subscripts

3. A = bh = 12 ( 4) = 48

4. P = a + b + c = 25 + 53 + 32 = 110

5. P = 2l + 2w

F = ma 40 = m (5)

P = 2 (12) + 2 (16) = 24 + 32 = 56

40 5m = 5 5 8= m 7.

1 2 mv 2 1 4500 = m(30)2 2 4500 = 450m K=

62,500 = (5) r 2

62,500 = 25r 62,500 25r = 25 25 2500 = r

4500 450m = 450 450 10 = m 9.

S = π r ( r + h) S = 3.14(8)(8 + 2)

p = i2r

10.

S = 3.14(8)(10)

B= B=

S = 3.14(80) S = 251.2

11.

x−µ σ 2.5 42.1− µ = 1 2 2.5( 2) = 42.1− µ z=

5 = 42.1− µ 5 − 42.1 = 42.1− 42.1− µ −37.1 = −µ

12.

703w h2 703(130)

(67)

91,390 = 20.35865449 ≈ 20.36 4489 S = 4lw + 2wh

336 = 4l (8) + 2(8)(9) 336 = 32l + 144 336 −144 = 32l + 144 −144 192 = 32l 192 32l = 32 32 6=l

−37.1 −µ = −1 −1 37.1 = µ

SECTION 6.3

13.

PV k P (20)

a+b+c 3 a + 77 + 92 85 = 3 85 a + 169 = 1 3 85(3) = a + 160 m=

14.

80 = 1 0.5 80 (0.5) = 20 P 40 = 20 P 40 20 P = 20 20 2= P

225 = a + 169 225 −169 = a + 169 −169 86 = a

A = P (1 + rt )

15.

3600 = P (1 + 0.04 (5))

16.

3600 = P (1 + 0.2) 3600 = 1.2 P 3600 1.2 P = 1.2 1.2 3000 = P

4608 = 144a 4608 144a = 144 144 −32 = a

V = πr 2 h

17.

942 = 3.14 (5) h 2

18.

942 = 3.14(25)h 942 = 78.5h 942 78.5h = 78.5 78.5 12 = h 19.

21.

5 C = ( F − 32) 9 5 C = (77 − 32) 9 5 C = (45) = 25 9 m= m= m=

1 V = − at 2 2 1 2 2304 = a (12) 2 2304 144a = 1 2 2304 ( 2) = 144a

y2 − y1 x2 − x1 8 − (−4) −3 − (−5) 8+ 4 12 = =6 −3 + 5 2

9 F = C + 32 5 9 F = (7) + 32 5 63 F = + 32 = 12.6 + 32 = 44.6 5

F − 32 + 273.1 1.8 100 − 32 K= + 273.1 1.8 68 K= + 273.1 1.8 K = 37.7 + 273.1 = 310.87 ≈ 310.88

20.

22.

x −µ σ n 66 − 60 z= 15 25 6 6 = =2 z= 15 3 5

167

168

23.

25.

CHAPTER 6 Algebra, Graphs, and Functions

S = R − rR 186 = 1R − 0.07 R 186 = 0.93R 186 0.93R = 0.93 0.93 200 = R

24.

S = C + rC 115 = 1C + 0.15C 115 = 1.15C 115 1.15C = 1.15 1.15 100 = C

E = a1 p1 + a2 p2 + a3 p3

26.

E = 5(0.2) + 7 (0.6) + 10 (0.2)

−b + b 2 − 4ac 2a −(−5) + (−5) − 4 (2)(−12) 2

E = 1 + 4.2 + 2 = 7.2

2 ( 2)

5 + 25 + 96 4 5 + 121 5 + 11 16 x= = = =4 4 4 4 R = O + (V − D ) r x=

27.

s = −16t 2 + v0 t + s0

28.

670 = O + (100 −10)(4)

s = −16 (4) + 30 (4) + 150 2

670 = O + 360 670 − 360 = O + 360 − 360 310 = O

s = −16 (16) + 120 + 150 s = −256 + 120 + 150 = 14 29.

f 1+ i f 3000 = 1 + 0.08 3000 f = 1 1.08 3000 (1.08) = f P=

30.

c = a 2 + b2 c = 52 + 122 c = 25 + 144 c = 169 c = 13

3240 = f 31.

RT =

R1 R2 R1 + R2

(100)(200) 100 + 200 20, 000 RT = 300 RT = 66.67 RT =

32.

P= 12 =

nRT V (10)(60)(8)

V 12 4800 = 1 V 12V = 4800 12V 4800 = 12 12 V = 400

SECTION 6.3

33.

sn =

a1 (1− r n )

⎛ r⎞ A = P ⎜⎜1 + ⎟⎟⎟ ⎜⎝ n ⎠

1− r ⎛ ⎛ 1 ⎞4 ⎞⎟ 10 ⎜⎜⎜1− ⎜⎜ ⎟⎟⎟ ⎟⎟ ⎝⎜ ⎝⎜ 2 ⎠ ⎠⎟⎟ sn = 1 1− 2 ⎛ 1⎞ 10 ⎜⎜1− ⎟⎟⎟ ⎜⎝ 16 ⎠ sn = 1 2 ⎛15 ⎞ 10 ⎜⎜ ⎟⎟⎟ ⎛15 ⎞ ⎝⎜16 ⎠ 3 sn = = 20 ⎜⎜ ⎟⎟⎟ = 18 ⎜⎝16 ⎠ 1 4 2

34.

⎛ 0.06 ⎞⎟ ( ) A = 100 ⎜⎜1 + ⎟ ⎜⎝ 1 ⎠⎟ 13

A = 100 (1 + 0.06)

A = 100 (1.06)

A = 100 (1.191016) A = 119.1016 ≈ 119.10

5 x − 4 y = 14 5 x − 5 x − 4 y = −5 x + 14 Subtract 5x from both sides of the equation. −4 y = −5 x + 14 −4 y −5 x + 14 Divide both sides of the equation by −4. = −4 −4 −5 x + 14 −(−5 x + 14) y= = 4 −4 5 x −14 5 7 = = x− 4 4 2 36. 7 x − 3 y = 23 7 x − 7 x − 3 y = −7 x + 23 Subtract 7x from both sides of the equation. −3 y = −7 x + 23 −3 y −7 x + 23 Divide both sides of the equation by −3. = −3 −3 23 −7 x + 23 7 y= = x− 3 3 −3 37. 8x + 7 y = 21 Subtract 8x from both sides of the equation. −8 x + 8 x + 7 y = −8 x + 21 7 y = −8 x + 21 7 y −8 x + 21 Divide both sides of the equation by 7. = 7 7 8 −8 x + 21 −8 x 21 y= = + = − x +3 7 7 7 7

35.

169

170

CHAPTER 6 Algebra, Graphs, and Functions

38.

39.

40.

41.

42.

− 9 x + 4 y = 11 Add 9x to both sides of the equation. −9 x + 9 x + 4 y = 9 x + 11 4 y = 9 x + 11 4 y 9 x + 11 Divide both sides of the equation by 4. = 4 4 9 x + 11 9 x 11 9 11 y= = + = x+ 4 4 4 4 4 2x − 3 y + 6 = 0 2x − 3 y + 6 − 6 = 0 − 6 Subtract 6 from both sides of the equation. 2 x − 3 y = −6 −2 x + 2 x − 3 y = − 2 x − 6 Subtract 2 x from both sides of the equation. −3 y = − 2 x − 6 − 3 y −2 x − 6 = Divide both sides of the equation by -3. −3 −3 −2 x − 6 −(−2 x − 6) 2 x + 6 2 x 6 2 y= = = = + = x+2 −3 3 3 3 3 3 10 x − 3 y = 0 10 x −10 x − 3 y = −10 x + 0 Subtract 10x from boths sides of the equation. −3 y = −10 x −3 y −10 x = Divide both sides of the equation by -3. −3 −3 10 y= x 3 −2 x + 3 y + z = 15 Add 2 x to both sides of the equation. −2 x + 2 x + 3 y + z = 2 x + 15 3 y + z = 2 x + 15 3 y + z − z = 2 x − z + 15 Subtract z from both sides of the equation. 3 y = 2 x − z + 15 3 y 2 x − z + 15 Divide both sides of the equation by 3. = 3 3 2 x − z + 15 2 1 = x− z +5 y= 3 3 3 5 x + 3 y − 2 z = 22 5 x − 5 x + 3 y − 2 z = −5 x + 22 Subtract 5 x from both sides of the equation. 3 y − 2 z = −5 x + 22 3 y − 2 z + 2 z = −5 x + 2 z + 22 Add 2 z to both sides of the equation. 3 y = −5 x + 2 z + 22 3 y −5 x + 2 z + 22 Divide both sides of the equation by 3. = 3 3 −5 x + 2 z + 22 5 2 22 y= =− x+ z+ 3 3 3 3

SECTION 6.3

43.

44.

45.

46.

47.

48.

9x + 4z = 7 + 8 y 9x + 4z − 7 = 7 − 7 + 8 y 9x + 4z − 7 = 8 y 9x + 4z − 7 8 y = 8 8 9x + 4z − 7 9 1 7 y= = x+ z− 8 8 2 8 2 x − 3 y + 5z = 0 2 x − 3 y + 3 y + 5z = 0 + 3 y 2 x + 5z = 3 y 2x + 5z 3 y = 3 3 2x + 5z 2 5 y= = x+ z 3 3 3 A = lw A lw = l l A w= l V = lwh V lwh = lh lh V w= lh p = a+b+c p −b = a + b−b + c p −b = a + c p −b−c = a + c −c a = p −b −c

Subtract 7 from both sides of the equation.

Divide both sides of the equation by 8.

Add 3 y to both sides of the equation.

Divide both sides of the equation by 3.

Divide both sides of the equation by l.

Divide both sides of the equation by lh.

Subtract b from both sides of the equation. Subtract c from both sides of the equation.

p = a + b + s1 + s2 p − a = a − a + b + s1 + s2

Subtract a from both sides of the equation.

p − a = b + s1 + s2 p − a − b = b − b + s1 + s2

Subtract b from both sides of the equation.

p − a − b = s1 + s2 p − a − b − s2 = s1 + s2 − s2

Subtract s2 from both sides of the equation.

s1 = p − a − b − s2

171

172

CHAPTER 6 Algebra, Graphs, and Functions

49.

50.

51.

52.

53.

1 V = Bh 3 ⎛1 ⎞ 3V = 3⎜⎜ Bh⎟⎟⎟ ⎜⎝ 3 ⎠ 3V = Bh 3V Bh = h h 3V B= h

Multiply both sides of the equation by 3.

Divide both sides of the equation by h.

V = πr 2 h V πr 2 h = π π V = r2h π V r 2h = πr 2 r2 V h= 2 πr C = 2π r C 2π r = 2 2 C = πr 2 C πr = π 2π C r= 2π y = mx + b y − b = mx + b − b y − b = mx y − b mx = x x y −b m= x y = mx + b y − mx = mx − mx + b b = y − mx

Divide both sides of the equation by π.

Divide both sides of the equation by r 2 .

Divide both sides of the equation by 2.

Divide both sides of the equation by π.

Subtract b from both sides of the equation.

Divide both sides of the equation by x.

Subtract mx from both sides of the equation.

SECTION 6.3

54.

55.

1 V = πr 2 h 3 1 2 πr h V =3 1 2 1 2 πr πr 3 3 3V h= 2 πr P = 2l + 2w

Divide both sides of the equation by

P − 2l = 2l − 2l + 2w

1 2 πr . 3

Subtract 2l from both sides of the equation.

P − 2l = 2w P − 2l 2w = 2 2 P − 2l w= 2 56.

Divide both sides of the equation by 2.

1 A = bh 2 ⎛1 ⎞ 2 A = 2 ⎜⎜ bh⎟⎟⎟ ⎜⎝ 2 ⎠ 2 A = bh

Multiply both sides of the equation by 2.

2 A bh = h h 2A b= h 57.

Divide both sides of the equation by h.

c + 2d 4 ⎛ c + 2d ⎞⎟ 4 L = 4 ⎜⎜ ⎜⎝ 4 ⎠⎟⎟ 4 L = c + 2d L=

4 L − c = c − c + 2d

Multiply both sides of the equation by 4.

Subtract c from both sides of the equation.

4 L − c = 2d 4 L − c 2d = 2 2 4L − c d= 2

Divide both sides of the equation by 2.

173

174

CHAPTER 6 Algebra, Graphs, and Functions

58.

w1 f −f = 2 w2 f − f1 w1 =

59.

w2 ( f 2 − f ) f − f1

Multiply both sides of the equation by w2 .

2 gm c2 ⎛ 2 gm ⎞ c 2 r = ⎜⎜ 2 ⎟⎟⎟ c 2 ⎜⎝ c ⎠ r=

Multiply both sides of the equation by c 2 .

c 2 r = 2 gm c 2 r 2 gm = 2g 2g m= 60.

Divide both sides of the equation by 2 g.

c2r 2g

PV PV 1 1 = 2 2 T1 T2 PV 1 1T2 = P2V2T1

Cross multiplication

PV PV T 1 1T2 = 2 2 1 T1 T1

Divide both sides of the equation by T1 .

PV 1 1T2 = P2V2 T1 PV PV 1 1T2 = 2 2 T1 P2 P2 V2 =

61.

Divide both sides of the equation by P2 .

PV 1 1T2 T1 P2

9 F = C + 32 5 9 F − 32 = C + 32 − 32 5 9 F − 32 = C 5 5 5 ⎛9 ⎞ ( F − 32) = ⎜⎜⎜ C ⎟⎟⎟ 9 9 ⎝5 ⎠

Subtract 32 from both sides of the equation.

Multiply both sides of the equation by

5 C = ( F − 32) 9

5 . 9

SECTION 6.3

5 C = ( F − 32) 9 9 9 ⎛ 5⎞ C = ⎜⎜ ⎟⎟⎟( F − 32) 5 5 ⎜⎝ 9 ⎠

62.

Multiply both sides of the equation by

9 . 5

9 C = F − 32 5 9 C + 32 = F − 32 + 32 5 9 F = C + 32 5

Add 32 to both sides of the equation.

S = π r 2 + π rs

63.

S − π r 2 = π r 2 − π r 2 + π rs

Subtract π r 2 from both sides of the equation.

S − π r 2 = π rs

64.

S − π r 2 π rs = πr πr 2 S − πr =s πr 1 A = h (b1 + b2 ) 2 ⎛1 ⎞ 2 A = 2 ⎜⎜ h (b1 + b2 )⎟⎟⎟ ⎜⎝ 2 ⎠

Divide both sides of the equation by π r.

Multiply both sides of the equation by 2.

2 A = h (b1 + b2 ) 2 A h (b1 + b2 ) = h h 2A = b1 + b2 h

65.

Divide both sides of the equation by h.

2A − b1 = b1 − b1 + b2 h 2A b2 = − b1 h d = rt

Subtract b1 from both sides of the equation.

549 = 61t 549 =t 61 t = 9 hours

66. a) i = prt i = 4500 (0.025)(1) = $112.50 b) $4500 + $112.50 = $4612.50

175

GROUP PROJECTS

Group Projects 1. Answers will vary. 2. Let x = # of Model 01 bookcases y = # of Model 02 bookcases a) 5x + 4y ≤ 1000 2x + 3y ≤ 600 y ≥ 50 x ≥ 40

2. c)

200

b) P = 75x + 125y d) Determine the maximum profit.

283

(40,520/3) (600/7,1000/7)

At (40,173) P = 75(40) + 125(173) = $24,625

100

At (40,50) P = 75(40) + 125(50) = $9,250 At (160,50) P = 75(160) + 125(50) = $18,250 At (86,143) P 75(86) + 125(143) = $24,325 Maximum profit occurs at 40 Model 01 bookcases and 173 Model 02 bookcases. e) Maximum profit = $24,625 3. Answers will vary.

(40,50) (160,50) 80

160

240

CHAPTER EIGHT THE METRIC SYSTEM Exercise Set 8.1 1. Metric 3. a) Meter 4. a) Measurement 5. a) Deka b) Deci 6. a) Hecto

b) Kilogram b) Quantity

c) Liter c) 10

b) Centi

7. a) Kilo

2. Customary d) Celsius

b) Milli

8. a) Mega b) Micro 9. 100 10. 10,000 11. a) 0o C b) 100o C c) 37o C 15. 22o C 13. a) Yard b) Quart 14. 30o C 12. 2.2 16. 2 m 17. kilo d 18. milli b 19. hecto c 20. deka e 21. deci f 22. centi a 23. a) 0.001 gram b) 100 grams c) 1000 grams d) 0.01 gram e) 10 grams f) 0.1 gram 24. a) 1/100 liter b) 10 liters c) 1/1000 liter d) 1/10 liter e) 1000 liters f) 100 liters 25. mg 1/1000 gm 26. cg 1/100 gm 27. dg 1/10 gm 28. dag 10 gm 29. hg 100 gm 30. kg 1000 gm 31. Max. mass 3000 kg = (3000 x 1000) g = 3 000 000 g

32. Max. mass 3000 kg = (3000 x 1,000,000) mg = 3 000 000 000 mg

33. 7 m = (7 x 1000) mm = 7000 mm

34. 42.9 hg = (42.9 x 100) g = 4290 g

35. 0.085 hl = (0.085)(0.1) kl = 0.0085 kl 37. 186.2 cm = (186.2)(0.0001) hm = 0.01862 hm

36. 8 dam = (8 x 10) m = 80 m 38. 9.32 mA = (9.32)(0.001) A = 0.00932 A

39. 9.2 m = (9.2)(1000) mm = 9200 mm 41. 895 l = 895,000 ml 43. 40,302 ml = 4.0302 dal 45. 500 m = (500)(0.001) km = 0.5 km 47. 500 m = (500) (0.01) hm = 5 hm 49. 80 km/hr = (80)(1 000 000) mm/km 80 000 000 mm/hr 51. 620 cm, 4.4 dam, 0.52 km 53. 1.4 kg, 1600 g, 16,300 dg 55. 105,000 mm, 2.6 km, 52.6 hm

40. 69.3 kg = (69.3)(10) hg= 693 hg 42. 24 dm = 0.0024 km 44. 0.034 ml = 0.000034 l 46. 500 m = (500)(100) cm = 50 000 cm 48. 80km = (80)(1000) m = 80 000 m/hr 50. 80 km/hr = (80)(10) hm/km = 800 hm/hr 52. 680 m, 514 hm, 62 km 54. 420 cl, 4.3 l, 0.045 kl 56. 0.032 kl, 460 dl, 48,000 cl 285

286

CHAPTER 8

The Metric System

57. Jim, since a meter is longer than a yard.

58. 1 hectometer in 10 min. 1 hm > 1 dm

59. The pump that removes 1 daA of water per min. 1 dekaliter > 1 deciliter

60. The side with the 15 lb. weight would go down. 5 kg = 5(2.2 lbs.) = 11 lbs.

61. a) Perimeter= 2l + 2w= 2(74) + 2(99)= 346 cm b) 346 cm = (346 x 10) mm = 3,460 mm

62. a) (2)(250)(7) = 3,500 mg / week b) 3,500 mg / week = 3.5 g / week

63. a) 200 km / 12.21 l ≈ 16.4 km/l b) 12.21 km/l(1000) m/km= 12 210 m/l

64. 8 (400) m = 3,200 m;

65. a) 6(360) ml = 2,160 ml b) 2160(1000) = 2.16 l c) 2.45 / 2.16 = $1.13 per liter

66. a) (4)(27 m) = 108 m b) 108 m = 0.108 km c) 108 m = 108 000 mm

3,200 m = 3.2 km

67. a) 1944 – 1558 = 386 m b) 386 m = 0.386 km 68. a) (16950 – 5830) km = 11,120 km b) 11,120,000 m c) 386 m = 38 600 cm 69. 1 gigameter = 1000 megameters 70. 1 nanogram = .001 microgram 71. 1 teraliter = 1 x 1024 picoliters 72. 1 megagram = 1 x 1015 73. 9000 cm = 9 dam nanogms 74. 2000 mm = 2 m 75. 0.00006 hg = 6 mg 76. 3000 dm = 3 hm 77. 0.02 kl = 2 dal 78. 500 cm = 5 m 79. magr gram 80. migradec decigram 83. terem meter 86. timenceret centimeter

Exercise Set 8.2 1. Volume 5. Length 9. Volume 13. Centimeters 17. Centimeters 21. Kilometers 25. c 130 km 29. c 828 m

33. cm or m AWV 37. sq. m 41. sq. mm or sq. cm 45. a 50 cm2 49. a 3 cm2 53. AWV 57. AWV 61. Kiloliters

81. rteli liter 84. leritililm milliliter 87. greseed sulesic degrees celsius

2. Length 6. Volume 10. Length 14. Kilometers 18. Meters 22. cm or m 26. a 800 cm 30. b 1000 m 34. mm or cm AWV 38. sq. cm 42. sq. cm 46. a 800 m2 50. c 1200 cm2 54. AWV 58. AWV 62. Cubic meters or cubic centimeters

82. raktileed dekaliter 85. reketolim kilometer 88. togmeharc hectogram

3. Area 7. Volume 11. Volume 15. cm or mm 19. Kilometers 23. c 110 m 27. b 4 cm 31. mm Answers will vary (AWV). 35. cm AWV 39. sq. cm or sq. m 43. sq. km or hectares 47. b 1/8 ha 51. c 4900 km2 55. AWV 59. Milliliters 63. Cubic meters

4. Length 8. Area 12. Volume 16. Centimeters 20. cm or mm 24. a 2 cm x 3 cm 28. b 8 cm 32. cm AWV 36. mm AWV 40. sq. m or hectares 44. sq. m 48. c 930 cm2 52. b 2.2 m2 56. AWV 60. Liters 64. Liters or milliliters

SECTION 8.2

65. Cubic meters 69. c 55 kA 73. b 5000 cm3

66. Cubic centimeters 70. b 355 mA 74. b 2600 m3

67. c 7780 cm3 71. c 0.04 m3 75. a) AWV

68. a 3 mA 72. b 120 mA 75. b) 10,580 cm3

76. a) AWV b) (2)(1.5)(.25) = 0.75 m3 78. a) AWV b) v = π r2h ≈ (3.14)(0.20)2(2) = 0.25m3 80. A kiloliter

77. a) AWV b) v ≈ (3.14)(0.25)2 (1) = 0.20 m3 79. 1 cubic decimeter

82. 2.5 acres

83. Longer side = 4cm; Shorter side = 2.2 cm A = lw = (4)(2.2) = 8.8 cm2

84. r ≈ 1.2 cm, A = π r 2 ≈ 3.14 (1.22 ) ≈ 4.52 cm 2

85. A = π r 2 ≈ 3.14 (152 ) ≈ 706.86 m 2

86. (37)(28) = 1036 cm2 2540 – 1036 = 1504 cm2 88. a) (3.75)(1.4) = 5.25 km2 b) (5.25)(100 ha) = 525 ha 90. a) (18)(10)(2.5) = 450 m3 b) 450 m3 = 450 kl

87. a) (73)(53) = 3869 m2 b) 3869 – (70)(50) = 3869 – 3500 = 369 m2 89. a) (42.4)(32.5) = 1378 m2 b) (1378)(0.0001) ha = 0.1378 ha

81. A cubic centimeter

91. Total Surface Area of 4 walls = 2lh + 2wh = 2(20)(6) + 2(12)(6) = 384 m2 ⎛ 1 A ⎞⎟ ⎛ 1 A ⎞⎟ = 38.4 l Liters for second coat = (384 m2) ⎜⎜ = 25.6 l Liters for first coat = (384 m2) ⎜⎜ ⎜⎝10 m 2 ⎠⎟⎟ ⎜⎝15 m 2 ⎠⎟⎟

Total liters = 38.4 + 25.6 = 64 l

Total cost = (64)($4.75) = $304

92. V = π r2h ≈ (3.14)(9.0)2(9) = 2290 cm3 93. a) V = lwh = (70)(40)(20) = 56,000 cm3

b) 56,000 cm3 = 56,000 ml

⎛ 56000 ⎞⎟ l = 56 l c) 56 000 ml = ⎜⎜ ⎜⎝ 1000 ⎠⎟⎟

0.56 m = 0.28 m; V = πr 2 h ≈ 3.14 (0.282 ) (1.17) ≈ 0.29 m3 2 b) (0.29)(1000) l = 290 l 95. 102 = 100 times larger 96. 1002 = 10,000 times larger 94. a) radius =

97. 103 = 1000 times larger 100. 10,000 dam2 103. 1,000,000 cm3 106. 435 cm3 = 0.435 l 109. 60 m3 = 60 kl

98. 103 = 1000 times larger 101. 0.0001 m2 104. 1 hm3 = 0.001 km3 107. 76 kl = 76 m3 110. (600,000)(100) = 60,000,000 ml = 60,000,000 cm3

111. 6.7 kl = 6.7 m3 = (6.7 x 103) dm3 = 6,700 dm3

99. 100 mm2 102. 0.000001 dm3 105. 419 cm3 = 419 mA 108. 4.2 l = 4,200 cm3

112. 1.4 ha = 14,000 m2 = (14000 x 1002) cm2 = 140,000,000 cm2

287

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ft 113. a) 1 sq mi = (1 mi2)(5280)2 2 = 27,878,400 ft2 mi in 2 = 4,014,489,600 in2 ft 2 b) It is easier to convert in the metric system because it is a base 10 system. 27,878,400 ft2 x (12)2

114. a) AWV; the average use is 590 liters / day b) AWV; the average use is 75 liters / day

Exercise Set 8.3 1. Kilogram 5. Celsius

9. Kilograms 13. Grams 17. Kilograms or metric tonnes 21. b 1.4 kg 25. AWV 29. c 0o C 33. c 1200o C 37. c 40o C

2. Grams 6. 0o C

3. Mass 7. 100 o C

4. Kilogram 8. 37 o C

10. Grams or milligrams 14. Kilograms or metric tonnes 18. Grams

11. Grams 15. Kilograms

12. Grams 16. Milligrams

19. b

20. c

22. c 0.45 kg 26. AWV 30. b 21o C

23. b 2800 kg 27. AWV 31. b Dress warmly and walk. 35. c 40o C

34. b

5o C

38. b

−5o C

9 (20) + 32 = 36 + 32 = 68o F 5 5 5 41. C = (92 − 32) = (60) = 33.3o C 9 9 5 5 43. C = (0 − 32) = (−32) ≈ −17.8o C 9 9 9 45. F = (37) + 32 = 66.6 + 32 = 98.6o F 5 5 5 47. C = (17 − 32) = (−15) ≈ –8.3o C 9 9 9 49. F = (0) + 32 = 0 + 32 = 32o F 5 5 5 51. C = (−20 − 32) = (−52) = −28.9o C 9 9 9 53. F = (22) + 32 = 39.6 + 32 = 71.6o F 5 9 55. F = (33.6) + 32 = 60.48 + 32 = 92.48o F 5 39. F =

2.27 kg

24. c 1.6 t 28. AWV 32. b 96o C 36. c

177o C

9 (−5) + 32 = – 9 + 32 = 23o F 5 5 5 42. C = (−10 − 32) = (−42) = –23.3o C 9 9 5 5 44. C = (100 − 32) = (68) = 37.8o C 9 9 9 46. F = (−4) + 32 = – 7.2 + 32 = 24.8o F 5 5 5 48. C = (75 − 32) = (43) = 23.9o C 9 9 9 50. F = (50) + 32 = 90 + 32 = 122o F 5 5 5 52. C = (425 − 32) = (393) = 218.3o C 9 9 9 54. F = (35.1) + 32 = 63.2 + 32 = 95.2o F 5 9 56. F = (32.3) + 32 = 58.1 + 32 = 90.1o F 5 40. F =

SECTION 8.3

9 (17.8) + 32 = 32 + 32 = 64.04o F 5 9 high: F = (23.5) + 32 = 42.3 + 32 = 74.3o F 5 Range = 74.30 – 64.04 = 10.26o F

57. low: F =

59. a) cost = (9)(2.50) = 22.5 euros

9 (22) + 32 = 39.6 + 32 = 71.6o F 5 9 high: F = (34) + 32 = 61.2 + 32 = 93.2o F 5 Range = 93.2 – 71.6 = 21.6o F

58. low: F =

60. (180)(.001)t + 2.92t = 3.1t

b) cost = (9)($1.46) = $13.14

3.5t – 3.1t = 0.4t

c) $5.97/lb 61. total mass = 45 g + 29 g + 370 ml = 45 g + 29 g + 370 g = 444 g

62. fuel used = (4320)(17) = 73,440 kg ⎛ 1 t ⎞⎟ ⎟ = 73.44 t 73, 440 kg ⎜⎜ ⎜⎝1000 kg ⎠⎟⎟

63. a) V= lwh,

l = 16 m, w = 12 m, h = 12 m V = (16)(12)(12) = 2304 m3

b) 2304 m3= 2304 kl c) 2304 kl = 2304 t

64. a) V = π r2h

r = 50 cm = 0.50 m h = 150 cm = 1.5 m V = (3.14)(0.50)2(1.50) = 1.1775 m3

b) 1.1775 m3 = 1.1775 kl = 1177.5 l c) 1177.5 l = 1177.5 kg

65. a) Yes; mass is a measure of the amount of matter in an object. b) No; weight is a measure of gravitational force. ⎛ 1 t ⎞⎟ ⎟ = 0.0062 t 67. 6.2 kg = (6.2 kg) ⎜⎜ ⎜⎝1000 kg ⎠⎟⎟ ⎛1000 kg ⎞⎟ = 17,400 kg = 69. 17.4 t = (17.4 t) ⎜⎜ ⎜⎝ 1 t ⎠⎟⎟

66. Yes, 78o F =

⎛1000 kg ⎞⎟ = 9520 kg 68. 9.52 t = (9.52 t) ⎜⎜ ⎜⎝ 1 t ⎠⎟⎟ ⎛ 1 t ⎞⎟ ⎟= 70. 1,460,000 mg = 1.46 kg = (1.46 kg) ⎜⎜ ⎜⎝1000 kg ⎠⎟⎟

17,400,000 g

0.00146 t

71. 1.2 l = 1200 ml

a) 1200 g

72. a) V = lwh

b) 1200 cm3

l = 1 yd = 3 ft w = 15 in = 1.25 ft h = 1.5 ft V = (3)(1.25)(1.5) = 5.625 cubic feet ⎛ ⎝

b) (5.625 ft3) ⎜⎜⎜62.5

lbs ⎞⎟ ⎟ = 351.6 lb ft 3 ⎠⎟

⎛ 1 gal ⎞⎟ c) (351.6 lb) ⎜⎜ = 42.4 gal ⎜⎝ 8.3 lb ⎠⎟⎟ 74. a) −62.11o C b) 2.5o C c) 56.7o C

5 (78 – 32) ≈ 25.6o C, not 20o C 9

73. 3 kg

(3)(2) = 6 = 4x x = 6/4 = 3/2 = 1.5 1.5 kg = 1500 g

9 (−62.11) + 32 = −111.798 + 32 = −79.8o F 5

9 (2.5) + 32 = 4.5 + 32 = 36.5o F 5 9 F = (56.7) + 32 = 102.06 + 32 ≈ 134.1o C 5

1.5 kg

289

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Exercise Set 8.4 1. Dimensional 2. Unit 60 seconds 1 minute 3. or because 60 seconds = 1 minute 1 minute 60 seconds 12 in 1 ft , 4. 1 ft 12 in 5.

100 cm 1 m , 1 m 100 cm

1000 mA 1A , 1A 1000 mA

2.2 lb Since we need to eliminate kilograms, kg must appear in the denominator. Since we 1 kg need to convert to pounds, lb must appear in the numerator.

1 kg Since we need to eliminate pounds; lb must appear in the denominator. Since we 2.2 lb need to convert to kilograms, kg must appear in the numerator. 1 ft 8. a) Since we need to eliminate centimeters, cm must appear in the denominator. Since we 30 cm need to convert to feet, ft must appear in the numerator. b)

30 cm Since we need to eliminate feet, ft must appear in the denominator. Since we 1 ft need to convert to centimeters, cm must appear in the numerator. 3.8 A 9. a) Since we need to eliminate gallons, gal must appear in the denominator. Since we 1 gal b)

need to convert to liters, l must appear in the numerator. 1 gal Since we need to eliminate liters, l must appear in the denominator. Since we 3.8 A need to convert to gallons, gal must appear in the numerator.

10. a)

0.8 m 2 Since we need to eliminate square yards, yd 2 must appear in the denominator. Since we 1 yd 2

need to convert to square meters, m 2 must appear in the numerator. b)

1 yd 2 Since we need to eliminate square meters, m 2 must appear in the denominator. Since we 0.8 m 2

need to convert to square yards, yd 2 must appear in the numerator.

⎛ 2.54 cm ⎞⎟ = 208.28 cm 11. 82 in. = (82 in.)⎜⎜ ⎜⎝ 1 in. ⎠⎟⎟ ⎛ 0.45 kg ⎞⎟ 12. 9 lb = (9 lb)⎜⎜ = 4.05 kg ⎜⎝ 1 lb ⎠⎟⎟ ⎛ 30 cm ⎞⎛ ⎟⎟⎜⎜ 1 m ⎞⎟⎟ = 1.26 m 13. 4.2 ft = (4.2 ft )⎜⎜ ⎟⎜100 cm ⎠⎟ ⎝⎜ 1 ft ⎠⎝

SECTION 8.4

⎛ 1 oz ⎞⎟ ⎟ = 15.25 oz 14. 427 g = ( 427 g)⎜⎜ ⎜⎝ 28 g ⎠⎟⎟ ⎛ 1 lb ⎞⎟ ⎟ = 360 lb 15. 162 kg = (162 kg )⎜⎜ ⎜⎝ 0.45 kg ⎠⎟⎟ ⎛ 0.8 m 2 ⎞⎟ ⎟ = 16 m 2 16. 20 yd 2 = ( 20 yd 2 )⎜⎜ ⎜⎝ 1 yd 2 ⎠⎟⎟ ⎛1.6 km ⎞⎟ = 62.4 km 17. 39 mi = (39 mi)⎜⎜ ⎜⎝ 1 mi ⎠⎟⎟ ⎛ 1 cm ⎞⎛ ⎟⎟⎜⎜ 1 in. ⎞⎟⎟ = 30.11811024 ≈ 30.12 in. 18. 765 mm = (765 mm)⎜⎜ ⎟⎜ 2.54 cm ⎠⎟ ⎝⎜10 mm ⎠⎝ ⎛ 1 acre ⎞⎟ 19. 675 ha = (675 ha )⎜⎜ = 1687.5 acres ⎜⎝ 0.4 ha ⎠⎟⎟ ⎛ 28 g ⎞⎟ 20. 253 oz = (253 oz)⎜⎜ = 7084 g ⎜⎝ 1 oz ⎠⎟⎟ ⎛ 1 pt ⎞⎟ = 43.19148936 ≈ 43.19 pints 21. 20.3 A = (20.3 A)⎜⎜ ⎜⎝ 0.47 A ⎠⎟⎟ ⎛ 0.9 t ⎞⎟ = 3.6 t 22. 4 T = ( 4 T )⎜⎜ ⎜⎝ 1 T ⎠⎟⎟ ⎛ 1 mi 2 ⎞⎟ ⎟ = 1.4615... ≈ 1.46 mi 2 23. 3.8 km2 = (3.8 km 2 )⎜⎜ ⎜⎝ 2.6 km 2 ⎠⎟⎟ ⎛1 fl oz ⎞⎟ = 0.853 ≈ 0.85 fl oz 24. 25.6 mA = ( 25.6 mA)⎜⎜ ⎜⎝ 30 mA ⎠⎟⎟ ⎛ 0.45 kg ⎞⎟ = 54 kg 25. 120 lb = (120 lb)⎜⎜ ⎜⎝ 1 lb ⎠⎟⎟ ⎛ 0.4 ha ⎞⎟ 26. 6.2 acres = (6.2 acres)⎜⎜ = 2.48 ha ⎜⎝ 1 acre ⎠⎟⎟ ⎛ 1 yd ⎞⎟ 27. 505 m = (505 m)⎜⎜ = 561. 1 ≈ 561.11 yd ⎜⎝ 0.9 m ⎠⎟⎟ ⎛ 1 yd ⎞⎟ 28. 175 m = (175 m)⎜⎜ = 194.4 ≈ 194.44 yd ⎜⎝ 0.9 m ⎠⎟⎟ ⎛100 cm ⎞⎛ ⎟⎟⎜⎜ 1 ft ⎞⎟⎟ = 1146.6 ≈ 1146.67 ft 29. 344 m = (344 m)⎜⎜ ⎟⎜ 30 cm ⎠⎟ ⎝⎜ 1 m ⎠⎝ ⎛100 cm ⎞⎛ ⎟⎟⎜⎜ 1 ft ⎞⎟⎟ = 1010 ft 30. 303 m = (303 m)⎜⎜ ⎟⎜ 30 cm ⎠⎟ ⎝⎜ 1 m ⎠⎝ ⎛ 1 mi ⎞⎟ 31. 30 kph = (30 km)⎜⎜ = 18.75 mph ⎜⎝1.6 km ⎠⎟⎟ ⎛1.6 km ⎞⎟ = 168 km 32. 105 mi = (105 mi)⎜⎜ ⎜⎝ 1 mi ⎠⎟⎟

291

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33. (6 yd )(9 yd ) = 54 yd 2

⎛ 0.8 m 2 ⎞⎟ ⎟ = 43.2 m 2 54 yd 2 = (54 yd 2 )⎜⎜ ⎜⎝ 1 yd 2 ⎠⎟⎟ ⎛1.6 km ⎞⎟ = 1147.2 km 34. 717 mi = (717 mi)⎜⎜ ⎜⎝ 1 mi ⎠⎟⎟ ⎛ 1 mi ⎞⎟ 35. 80 km = (80 km)⎜⎜ = 50 mph ⎜⎝1.6 km ⎠⎟⎟ ⎛ 30 mA ⎞⎟ 36. 8 fl oz = (8 fl oz )⎜⎜ = 240 mA ⎜⎝1 fl oz ⎠⎟⎟ ⎛ 1 oz ⎞⎟ ⎟ ≈ 0.21 oz 37. 6 g = (6 g)⎜⎜ ⎜⎝ 28 g ⎠⎟⎟ ⎛ 3.8 A ⎞⎟⎛ 1 kA ⎞ ⎟⎟ = 47.5 kA ⎟⎜⎜ 38. 12,500 gal = (12,500 gal)⎜⎜ ⎝⎜ 1 gal ⎠⎟⎟⎝⎜1000 A ⎠⎟ ⎛ 0.4 ha ⎞⎟ 39. (91,696 acres)⎜⎜ ⎟ = 36 678.4 ha ⎝⎜ 1 acre ⎠⎟ 40. (50 ft )(30 ft )(8 ft ) = 12,000 ft 3 ⎛ 0.03 m3 ⎞⎟ ⎟⎟ = 360 m3 12,000 ft 3 = (12, 000 ft 3 )⎜⎜⎜ 3 ⎝ 1 ft ⎠⎟ ⎛ 1 lb ⎞⎟ ⎟ = 2.2 lb 41. 1 kg = (1 kg)⎜⎜ ⎜⎝ 0.45 kg ⎠⎟⎟ $1.10 = $0.495 per pound 2.2 ⎛ 1 T ⎞⎟ = 8.3 ≈ 8.33 T 42. a) 7.5 t = (7.5 t )⎜⎜ ⎜⎝ 0.9 t ⎠⎟⎟ ⎛ 2000 lb ⎞⎟ b) 8.3 T = (8.3 T)⎜⎜ = 16, 666.7 ≈ 16, 666.67 lb ⎜⎝ 1 T ⎠⎟⎟ ⎛1000 A ⎞⎛ ⎟⎟⎜⎜ 1 gal ⎞⎟⎟ = 9078.947368 ≈ 9078.95 gal 43. 34.5 kA = (34.5 kA)⎜⎜ ⎟⎜ 3.8 A ⎠⎟ ⎜⎝ 1 kA ⎠⎝ ⎛ 28 g ⎞⎟ 44. 0.25 oz = (0.25 oz )⎜⎜ =7 g ⎜⎝ 1 oz ⎠⎟⎟

$80 = 11.42857143 ≈ $11.43 per gram 7 ⎛ 1 qt ⎞⎟ = 6 qt 45. 5.7 A = (5.7 A)⎜⎜ ⎜⎝ 0.95 A ⎠⎟⎟ ⎛ 30 cm ⎞⎟ 46. a) −282 ft = (−282 ft )⎜⎜ = −8460 cm ⎜⎝ 1 ft ⎠⎟⎟ ⎛ 1 m ⎞⎟ b) −8460 cm = (−8460 cm)⎜⎜ = −84.6 m ⎜⎝100 cm ⎠⎟⎟

SECTION 8.4

⎛ 1 km ⎞⎟ 47. a) 550 m = (550 m)⎜⎜ = 0.55 km ⎜⎝1000 m ⎠⎟⎟ ⎛ 1 km ⎞⎛ ⎟⎟⎜⎜ 1 mi ⎞⎟⎟ ≈ 0.34 mi b) 550 m = (550 m)⎜⎜ ⎟⎜1.6 km ⎠⎟ ⎜⎝1000 m ⎠⎝

48.

⎛ 1 g ⎞⎟ 1 carat = (0.125 carat )⎜⎜ = 0.025 g ⎜⎝ 5 carat ⎠⎟⎟ 8

⎛(3.3)2 ft 2 ⎞⎟ ⎜ ⎟⎟ = 10.89 ft 2 49. a) 1 m 2 = (1 m 2 )⎜⎜ ⎜⎜⎝ 1 m 2 ⎠⎟⎟ ⎛ (3.3)3 ft 3 ⎞⎟ ⎜ 3 b) 1 m3 = (1 m3 )⎜⎜ ⎟⎟ = 35.937 ft 3 ⎜⎜⎝ 1 m ⎠⎟⎟ ⎛ 0.4 ha ⎞⎟ ⎟ = 6.12 ha 50. a) 15.3 acres = (15.3 acres)⎜⎜ ⎜⎝ 1 acre ⎠⎟⎟ ⎛10,000 m 2 ⎞⎟ ⎟ = 61, 200 m 2 b) 6.12 ha = (6.12 ha )⎜⎜ ⎜⎝ 1 ha ⎠⎟⎟ ⎛ 0.45 kg ⎞⎟⎛⎜1 mg ⎞⎟ ⎟ = 25.2 mg 51. 56 lb = (56 lb)⎜⎜ ⎟⎜ ⎝⎜ 1 lb ⎠⎟⎜⎝ 1 kg ⎠⎟⎟ ⎛ 0.45 kg ⎞⎟⎛⎜1.5 mg ⎞⎟ 52. 170 lb = (170 lb)⎜⎜ ⎟⎜⎝ 1 kg ⎠⎟⎟⎟ = 114.75 mg ⎜⎝ 1 lb ⎠⎟⎜ ⎛ 1 g ⎞⎟ ⎛ 0.45 kg ⎞⎟⎛⎜ 200 mg ⎞⎟ ⎜ 53. 76 lb = (76 lb)⎜⎜ ⎟⎜⎝ 1 kg ⎠⎟⎟⎟ = 6840 mg; 6840 mg = (6840 mg)⎝⎜⎜1000 mg ⎠⎟⎟⎟ = 6.84 g ⎜⎝ 1 lb ⎠⎟⎜ ⎛ 0.45 kg ⎞⎟⎛⎜ 5 mg ⎞⎟ ⎟ = 184.5 mg 54. 82 lb = (82 lb)⎜⎜ ⎟⎜ ⎝⎜ 1 lb ⎠⎟⎝⎜ 1 kg ⎠⎟⎟ ⎛ 12.5 mg ⎞⎟ ⎟ = 25 mg 55. a) 2 teaspoons = ( 2 teaspoons)⎜⎜ ⎜⎝1 teaspoon ⎠⎟⎟ ⎛ 30 mA ⎞⎛ ⎟⎟⎜⎜12.5 mg ⎞⎟⎟ = 900 mg b) 12 fl oz = (12 fl oz )⎜⎜ ⎟⎜ 5 mA ⎠⎟ ⎝⎜1 fl oz ⎠⎝ ⎛ 236 mg ⎞⎟ ⎟ = 472 mg 56. a) 2 tablespoons = (2 tablespoons)⎜⎜ ⎜⎝1 tablespoon ⎠⎟⎟ ⎛ ⎞ ⎛ 30 mA ⎞⎛ ⎟⎜1 tablespoon ⎞⎟⎜ 236 mg ⎟ b) 8 fl oz = (8 fl oz )⎜⎜ ⎟⎟⎜⎜ 15 mA ⎠⎟⎟⎜⎜⎝1 tablespoon ⎠⎟⎟⎟ = 3776 mg ⎜⎝ 1 fl oz ⎠⎝ ⎛ 1 mi ⎞⎟ 57. a) 1.2 km = (1.2 km)⎜⎜ ≈ 0.75 mi ⎜⎝1.6 km ⎠⎟⎟ ⎛ 1 m 2 ⎞⎟ ⎟ ≈ 2.63 m 2 b) 6.84 km 2 = (6.84 km 2 )⎜⎜⎜ 2⎟ ⎝ 2.6 km ⎠⎟ ⎛ 1 yd ⎞⎟⎛⎜ 3 ft ⎞⎟ c) 426 m = (426 m)⎜⎜ ⎟⎜⎝1 yd ⎠⎟⎟⎟ = 1420 ft ⎜⎝ 0.9 m ⎠⎟⎜

293

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⎛ 30 cm ⎞⎛ ⎟⎟⎜⎜ 1 m ⎞⎟⎟ = 289.2 m 58. a) 964 ft = (964 ft )⎜⎜ ⎟⎜100 cm ⎠⎟ ⎜⎝ 1 ft ⎠⎝ ⎛ 0.9 tonne ⎞⎟ b) 85,000 tons = (85,000 tons)⎜⎜ = 76 500 t ⎜⎝ 1 ton ⎠⎟⎟ ⎛1.6 km ⎞⎟ c) 28 mi = (28 mi)⎜⎜ = 44.8 kph ⎜⎝ 1 mi ⎠⎟⎟ ⎛ 1 km ⎞⎛ ⎟⎟⎜⎜ 1 mi ⎞⎟⎟ ≈ 6.08 mi 59. a) 9730 m = (9730 m)⎜⎜ ⎟⎜1.6 km ⎠⎟ ⎝⎜1000 m ⎠⎝ ⎛ 1 mi ⎞⎟ b) 709 kph = (709 kph )⎜⎜ ≈ 443.13 kph ⎜⎝1.6 km ⎠⎟⎟ ⎛ 1 mi ⎞⎟ c) 7181 kph = (181 kph )⎜⎜ ≈ 113.13 kph ⎜⎝1.6 km ⎠⎟⎟ 9 d) −45° C = (−45) + 32 = −49°F 5 ⎛ 2.00 €€ 2.00 €€ ⎞⎛ ⎟⎟⎜⎜ 1 kg ⎞⎟⎟ ≈ 0.9 €€ 60. a) = ⎜⎜ ⎟⎟⎜ 2.2 lb ⎟⎟ 1 lb ⎜⎝ 1 kg ⎠⎝ 1 kg ⎠ 61. a)

⎛ 7.00 €€ ⎞⎛ 7.00 €€ ⎟⎟⎜⎜1000 g ⎞⎟⎟ ≈ 70 €€ = ⎜⎜ ⎟ ⎟ ⎜ 100 g ⎝ 100 g ⎠⎝⎜ 1 kg ⎠⎟⎟ 1 kg

⎛ 31.5 €€ ⎞⎛ 31.5 €€ ⎟⎟⎜⎜ $1.40 ⎞⎟⎟ = $44.10 = ⎜⎜⎜ ⎟⎟⎜ 1 €€ ⎠⎟⎟ 1 lb 1 lb ⎝ 1 lb ⎠⎝

62. a)

⎛ 0.9 €€ ⎞⎛ 0.9 €€ ⎟⎟⎜⎜ $1.40 ⎞⎟⎟ = $1.26 = ⎜⎜ ⎟⎟⎜ 1 €€ ⎟⎟ ⎜⎝ 1 lb ⎠⎝ 1 lb 1 lb ⎠

⎛ 70 €€ ⎞⎛ 70 €€ ⎟⎟⎜⎜ 0.45 kg ⎞⎟⎟ ≈ 31.5 €€ = ⎜⎜ ⎟ ⎟ ⎜ 1 kg ⎝ 1 kg ⎠⎝⎜ 1 lb ⎠⎟⎟ 1 lb

12.85 pesos 17.35 pesos ≈ 0.74 euro 1 euro

0.74 euro 0.70 euro ≈ 1.06 Canadian dollars 1 Canadian dollar

12.85 pesos 12.12 pesos ≈ 1.06 Canadian dollars 1 Canadian dollar

63. a)

745 kmph 1.85 kmph ≈ 402 knots 1 knot

402 knots 0.87 knots ≈ 463 mph 1 mph

463 mph 0.62 mph ≈ 745 kph 1 kmph

⎛ 0.24 A ⎞⎟ 64. (0.5 c)⎜⎜ = 0.12 A graham cracker crumbs ⎜⎝ 1 c ⎠⎟⎟ ⎛ 28 g ⎞⎟ ⎟ = 336 g nuts ⎝ 1 oz ⎠⎟

(12 oz)⎜⎜⎜

⎛ 28 g ⎞⎟ ⎟ = 224 g chocolate pieces ⎝ 1 oz ⎠⎟

(8 oz )⎜⎜⎜

⎛ 4 ⎞⎛ ⎞ ⎜⎜ c⎟⎟⎜⎜ 0.24 A ⎟⎟ = 0.32 A flaked coconut ⎟ ⎝⎜ 3 ⎠⎝⎜ 1 c ⎠⎟

REVIEW EXERCISES

295

⎛ 4 ⎞⎛ ⎞ ⎜⎜ c⎟⎟⎜⎜ 0.24 A ⎟⎟ = 0.32 A condensed milk ⎟⎜ 1 c ⎠⎟ ⎝⎜ 3 ⎠⎝ ⎛ 2.54 cm ⎞⎟ ⎛ 2.54 cm ⎞⎟ ⎟×(13 in.)⎜⎜⎜ ⎟ = 22.86 cm × 33.02 cm baking pan ⎝ 1 in. ⎠⎟ ⎝ 1 in. ⎠⎟

(9 in.)⎜⎜⎜

5 350° F = (350 − 32) = 176.6 ≈ 176.7° C 9 ⎛ 2.54 cm ⎞⎟ ⎛ 2.54 cm ⎞⎟ (1.5 in.)⎜⎜⎜ ⎟⎟×(3 in.)⎜⎜⎜ ⎟ = 3.81 cm × 7.62 cm bars ⎝ 1 in. ⎠ ⎝ 1 in. ⎠⎟

⎛ ⎞⎟ ⎜ ⎛1 grain ⎞⎟⎜⎜ 1 mA ⎟⎟⎟ ⎟⎜ ⎟⎟ = 1.0 cc, or b) 65. (0.2 mg)⎜⎜ ⎜⎝ 60 mg ⎠⎟⎟⎜⎜ 1 ⎟⎟ ⎜ grain ⎜⎝ 300 ⎠⎟⎟ ⎛ 0.18 kg ⎞⎟⎜⎛ 1 lb ⎞⎟ ⎟ = 7.8 lb 66. 15(130 lb) = 1950 lb; (1950 lb)⎜⎜ ⎜⎝ 100 lb ⎠⎟⎟⎜⎜⎝ 0.45 kg ⎠⎟⎟ ⎛1000 mA ⎞⎟⎛⎜1 cm3 ⎞⎟ ⎟ = 3000 cc 67. a) (3.0 A )⎜⎜ ⎟⎜ ⎝⎜ 1 l ⎠⎟⎜⎜⎝ 1 mA ⎠⎟⎟ ⎛ 1 in.3 ⎞⎟ ⎜ ⎟⎟ = 3000 = 183.071... ≈ 183.07 in.3 b) (3000 cm3 )⎜⎜ ⎜⎜⎝(2.54)3 cm3 ⎠⎟⎟ 16.387064 68. 1 microscope

69. wonton

⎛ ⎞ ⎛ 28 g ⎞⎟ = 448 g⎟⎟⎟ 70. 1 pound cake ⎜⎜1 lb = 16 oz; 16 oz ⎜⎜ ⎜⎝ 1 oz ⎠⎟⎟ ⎜⎝ ⎠⎟

71. 1 kilohurtz

72. 2 megacycles 74. 1 decacards

73. 1 megaphone 75. 2 kilomockingbirds

Review Exercises

1. 1000× base unit 4. 100× base unit 7. 82 mg = 0.082 g 10. 1 000 000 mg = 1 kg 13. 2.67 kl = 2 670 000 ml 14 630 cl = 146 300 ml 3000 ml, 14 630 cl, 2.67 kl 16. Degrees Celsius 19. Millimeters 22. Centimeters 25. c 28. a

1 of base unit 100 5. 10 times base unit

8. 3.2 l = 320 cl 11. 4.62 kl = 4620 l 14. 0.047 km = 47 m 47 000 cm = 470 m 0.047 km, 47 000 cm, 4700 m 17. Square meters 20. Kilograms or tonnes 23. a) and b) Answers will vary. 26. b 29. a

1 of base unit 1000 1 6. of base unit 10 9. 0.0035 cm = 0.035 mm 12. 192.6 dag = 19 260 dg 15. Meters 3.

18. Milliliters or cubic cm 21. Kilometers 24. a) and b) AWV 27. c 30. c

296

CHAPTER 8

The Metric System

⎛ 1 t ⎞⎟ ⎟ = 3.6 t 31. 3600 kg = (3600 kg)⎜⎜ ⎜⎝1000 kg ⎠⎟⎟

⎛1000 kg ⎞⎛ ⎟⎟⎜1000 g ⎞⎟⎟ = 4 300 000 g 32. 4.3 t = (4.3 t )⎜⎜ ⎜ ⎜⎝ 1 t ⎠⎟⎟⎜⎝ 1 kg ⎠⎟⎟

9 33. 24° C = (24) + 32 = 75.2° F 5 5 35. −6° F = (−6 − 32) = −21. 1 ≈ −21.1° C 9

5 34. 68° F = (68 − 32) = 20° C 9 9 36. 39° C = (39) + 32 = 102.2° F 5

37. l = 4 cm, w = 1.6 cm; A = lw = 4 (1.6) = 6.4 cm 2 38. r = 1.5 cm; A = π r 2 ≈ 3.14 (1.5) = 7.065 ≈ 7.07 cm 2 2

39. a) V = πr 2 h = π (1)(1) ≈ 3.14 m3

⎛1000 kg ⎞

(3.14 m3 )⎜⎜⎜⎝ 1 m3 ⎠⎟⎟⎟ ≈ 3140 kg

40. a) A = lw = (33.7)(26.7) = 899.79 cm 2 ⎛ 1 m2 ⎞⎟ ⎟ = 0.089979 m 2 b) 899.79 cm 2 = (899.79 cm 2 )⎜⎜ 2⎟ ⎜⎝10,000 cm ⎠⎟

41. a) V = lwh = (80)(40)(30) = 96 000 cm3 ⎛ 1 m3 ⎞⎟ ⎜ ⎟⎟ = 0.096 m3 b) 96 000 cm3 = (96 000 cm3 )⎜⎜ 3 ⎜⎜⎝ (100) cm3 ⎠⎟⎟ ⎛ 1 mA ⎞⎟ = 96 000 mA c) 96 000 cm3 = (96 000 cm3 )⎜⎜ ⎜⎝1 cm3 ⎠⎟⎟

⎛ 1 kA ⎞⎟ = 0.096 kA 0.096 m 3 = (0.096 m 3 )⎜⎜ ⎜⎝1 m3 ⎠⎟⎟

42. Since 1 km = 100×1 dam, 1 km 2 = 1002 ×1 dam 2 = 10 000 dam 2 . Thus, 1 square kilometer is 10,000 times larger than a square dekameter. ⎛ 1 in. ⎞⎟ ⎛ 1 lb ⎞⎟ = 933.858... ≈ 33.86 in. 43. (86 cm)⎜⎜ ⎟ = 233.3 ≈ 233.33 lb 44. (105 kg)⎜⎜ ⎜⎝ 2.54 cm ⎠⎟⎟ ⎜⎝ 0.45 kg ⎠⎟⎟ ⎛ 0.9 m ⎞⎟ ⎟ = 74.7 m 45. (83 yd)⎜⎜ ⎜⎝ 1 yd ⎠⎟⎟

⎛ 1 yd ⎞⎟ 46. (100 m)⎜⎜ = 111. 1 ≈ 111.11 yd ⎜⎝ 0.9 m ⎠⎟⎟

⎛1.6 km ⎞⎟ = 72 kph 47. ( 45 mi)⎜⎜ ⎜⎝ 1 mi ⎠⎟⎟

⎛ 1 qt ⎞⎟ = 44.210... ≈ 44.21 qt 48. (42 A )⎜⎜ ⎜⎝ 0.95 A ⎠⎟⎟

⎛ 3.8 A ⎞⎟ ⎟ = 76 A 49. ( 20 gal)⎜⎜ ⎜⎝ 1 gal ⎠⎟⎟

50.

⎛ 1 in.2 ⎞

51.

(96 cm 2 )⎜⎜⎜ 6.5 cm2 ⎟⎟⎟⎟ = 14.76923077 ≈ 14.77 in.2

53.

(15 yd3 )⎜⎜⎜ 1 yd 3

⎝

⎠

⎛ 0.729 m3 ⎞⎟ ⎟⎟ = 10.9 m 3 ⎝ ⎠⎟

⎛ 1 yd 3 ⎞

(60 m3 )⎜⎜⎜ 0.76 m3 ⎟⎟⎟⎟ = 78.947... ≈ 78.95 yd 3 ⎝

⎠

⎛ 0.95 A ⎞⎟ ⎟ = 3.8 A 52. ( 4 qt )⎜⎜ ⎜⎝ 1 qt ⎠⎟⎟ ⎛1.6 km ⎞⎟ 54. (62 mi)⎜⎜ = 99.2 km ⎜⎝ 1 mi ⎠⎟⎟

⎛1.6 km ⎞⎟ 55. (241 mi)⎜⎜ ⎟ = 385.6 km ⎝⎜ 1 mi ⎠⎟

⎛ 1 yd 3 ⎞⎟ ⎟ ≈ 429,322.37 yd 3 56. (326,285 m3 )⎜⎜ ⎜⎝ 0.76 m3 ⎠⎟⎟

57. a) 700 (1.5 kg) = 1050 kg ⎛ 1 lb ⎞⎟ ⎟ = 2333.3 ≈ 2333.33 lb b) 1050 kg = (1050 kg)⎜⎜ ⎜⎝ 0.45 kg ⎠⎟⎟

CHAPTER TEST

297

⎛ 0.09 m 2 ⎞⎟ ⎟ = 32.4 m 2 58. A = lw = ( 24)(15) = 360 ft 2 ; 360 ft 2 = (360 ft 2 )⎜⎜ ⎜⎝ 1 ft 2 ⎠⎟⎟ ⎛ 3.8 A ⎞⎟⎛ 1 kA ⎞ ⎟ = 190 kA ⎟⎜ 59. a) (50,000 gal)⎜⎜ ⎜⎝ 1 gal ⎠⎟⎟⎜⎝⎜1000 A ⎠⎟⎟

⎛1000 A ⎞⎛ 1 kg ⎞⎟ b) (190 kA)⎜⎜ ⎟⎟⎟⎜⎜⎜ ⎟ = 190 000 kg ⎜⎝ 1 kA ⎠⎝ 1 A ⎠⎟

⎛1.6 km ⎞⎟ 60. a) 65 mi/hr = (65 mi/hr )⎜⎜ = 104 km/hr ⎜⎝ 1 mi ⎠⎟⎟ ⎛1000 m ⎞⎟ ⎟ = 104,000 m/hr b) 104 km/hr = (104 km/hr )⎜⎜ ⎜⎝ 1 km ⎠⎟⎟ ⎛ 1 mA ⎞⎛ ⎟⎟⎜⎜ 1 A ⎞⎟⎟ = 252 A 61. a) V = lwh = (90)(70)(40) = 252 000 cm 3 ; 252 000 cm 3 = (252 000 cm3 )⎜⎜ ⎟⎜1000 mA ⎠⎟ ⎜⎝1 cm3 ⎠⎝ ⎛1 kg ⎞⎟ = 252 kg b) 252 A = (252 A)⎜⎜ ⎜⎝ 1 A ⎠⎟⎟ ⎛ 1 lb ⎞⎟ $3.50 ⎟ = 2.2 lb; 62. 1 kg = (1 kg)⎜⎜ = $1.575 ≈ $1.58 per pound ⎜⎝ 0.45 kg ⎠⎟⎟ 2.2

Chapter Test 1. 759 cA = 0.759 daA 2. 273 hm = 2,730,000 m

⎛100 dam ⎞⎟ = 100 dam or 100 times greater 3. 1 km = (1 km)⎜⎜ ⎜⎝ 1 km ⎠⎟⎟ ⎛ 1 km ⎞⎟ = 2.4 km 4. 400 (6) = 2400 m; (2400 m)⎜⎜ ⎜⎝1000 m ⎠⎟⎟

5. b

6. a 8. b

7. c 9. b

⎛1002 cm 2 ⎞⎟ ⎟⎟ = 10 000 cm 2 or 10,000 times greater 10. 1 m 2 = (1 m 2 )⎜⎜⎜ 2 ⎝ 1m ⎠⎟ ⎛10002 mm3 ⎞⎟ ⎟⎟ = 1 000 000 cm3 or 1,000,000 times greater 11. 1 m3 = (1 m3 )⎜⎜ ⎜⎝ 1 m3 ⎠⎟ ⎛ 28 g ⎞⎟ = 6300 g 12. 225 oz = (225 oz )⎜⎜ ⎜⎝ 1 oz ⎠⎟⎟ ⎛ 1 yd ⎞⎛ ⎟⎟⎜ 3 ft ⎞⎟⎟ = 564.3 ft 13. 169.29 m = (169.29 m)⎜⎜ ⎟⎜⎜1 yd ⎟⎟ ⎜⎝ 0.9 m ⎠⎟⎜ ⎝ ⎠ ⎛ 1 mi 2 ⎞⎟ ⎟ 14. 53,321 km 2 = (53,321 km 2 )⎜⎜ ⎜⎝ 2.6 km 2 ⎠⎟⎟

5 15. 10° F = (10 − 32) = −12.2 ≈ −12.22° C 9

≈ 20,508.1 mi 2

9 16. 20° C = ( 20) + 32 = 68° F 5