,SOLUTION MANUAL FOR Survey of Mathematics
with Applications, A, 12th edition Angel
Notes
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, Section 1.1: Inductive and Deductive Reasoning 1
Chapter One: Critical Thinking Skills
Section 1.1: Inductive and Deductive Reasoning
1. Natural 13.
2. Divisible 14.
3. Counterexample
15.
4. Hypothesis
16.
5. Inductive
17. 9, 11, 13 (Add 2 to previous number.)
6. Deductive
18. 16, 19, 22 (Add 3 to the previous number.)
7. Deductive
19. 16, –32, 64 (Multiply previous number by –2.)
8. Inductive
20. -243, 729, -2187 (Multiply by –3.)
9. 5 × 5 = 25
1 1 1
10. 19 × 10 = 190 21. , , (Increase denominator value by 1.)
5 6 7
9 11 13
11. 1 5
(1+4)
10
(4+6)
10
(4+6)
5
(1+4)
1 22.
10
, ,
12 14
(Increase denominator and
numerator values by 2.)
12. 100,000 = 105
23. 34, 55, 89 (Each number in the sequence is the sum of the previous two numbers.)
24. 76, 123, 199 (Each number in the sequence is the sum of the previous two numbers.)
25. The product of two odd numbers is an odd number.
26. The product of an even number and an odd number is an even number.
27. The sum of two even numbers is an even number
28. The sum of two odd numbers is an even number.
29. a) Answers will vary.
b) Products involving 10 and natural numbers will have a ones digit 0.
30. a) Answers will vary.
b) The sum of the digits is 9.
c) The sum of the digits in the product when a one- or two-digit number is multiplied by 9 is divisible by 9.
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.
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.
33. Blue: 1, 5, 7, 10, 12; Purple: 2, 4, 6, 9, 11; 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
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,2 Chapter 1: Critical Thinking Skills
35. a) $3700
b) We are using observation of specific cases to make a prediction.
36. a) $415
b) We are using observation of specific cases to make a prediction.
37. a) You should obtain the original number.
b) You should obtain the original number.
c) Conjecture: The result is always the original number.
38. 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.
39. a) You should obtain the number 5.
b) You should obtain the number 5.
c) Conjecture: The result is always the number 5.
40. a) You should obtain the number 8.
b) You should obtain the number 8.
c) Conjecture: The result is always the number 8.
3𝑛+6 3𝑛 6
41. 𝑛 → 3𝑛 → 3𝑛 + 6 → = + = 𝑛+2 →𝑛+2−2 =𝑛
3 3 3
4𝑛+6 4𝑛 6
42. 𝑛 → 4𝑛 → 4𝑛 + 6 → = + = 2𝑛 + 3 → 2𝑛 + 3 − 3 = 2𝑛
2 2 2
2𝑛+10 2𝑛 10
43. 𝑛 → 𝑛 + 1 → 𝑛 + (𝑛 + 1) = 2𝑛 + 1 → 2𝑛 + 1 + 9 = 2𝑛 + 10 → = + =𝑛+5→ 𝑛+5−𝑛 =
2 2 2
5
2𝑛+16 2𝑛 16
44. 𝑛 → 𝑛 + 5 → 𝑛 + (𝑛 + 5) = 2𝑛 + 5 → 2𝑛 + 5 + 11 = 2𝑛 + 16 → = + =𝑛+8→ 𝑛+8−
2 2 2
𝑛=8
45. 3 × 5 = 15 is a counterexample.
46. 10 + 11 + 12 = 33, which is not a three-digit number.
5
47. Two is a counting number. The sum of 2 and 3 is 5. Five divided by two is , which is not an even number.
2
48. 900 is a three-digit number. The product of 900 and 900 is 810,000, which is not a five-digit number.
49. One and two are counting numbers. The difference of 1 and 2 is 1 − 2 = −1, which is not a counting number.
50. The sum of the odd numbers 1 and 5 is 6, which is not divisible by 4.
51. 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°.
52. 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°.
53. Inductive reasoning: a general conclusion is obtained from observation of specific cases.
54. Inductive reasoning: a general conclusion is obtained from observation of specific cases.
55. 129; The numbers in the positions of each inner 4 × 4 square is determined as follows.
𝑎 𝑏
𝑐 𝑎+𝑏+𝑐
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, Section 1.2: Estimation Techniques 3
56. 1881, 8008, 8118 (They look the same when looked at in a mirror.)
57. c
Section 1.2: Estimation Techniques
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. Estimation 10. 400.15 − 297.87 ≈ 400 − 300 = 100
2. Equal 11. 22% 9116 20% 9000 = 0.20 9000
3. 39.7 + 60.3 + 18 + 1.8 =1800
≈ 40 + 60 + 20 + 0 = 120
12. 11% of 8221 10% of 8000 = 0.10 8000
4. 86 + 47.2 + 289.8 + 532.4 + 12.8 = 800
≈ 90 + 50 + 290 + 530 + 10 = 970
$91.35 $90
5. 197,500 ÷ 4.063 ≈ 200,000 ÷ 4.000 = 13. ≈ = $30
3 3
50,000 $210 $200
14. ≈ = $25
1,032,645 1,000,000 8 8
6. ≈ = 20
49,827 50,000
15. 12 months × $47 ≈ 12 × $50 = $600
7. 1776 × 0.0098 ≈ 1800 × 0.01 = 18 16. 19 gallons × $3.11 ≈ 20 × $3 = $60
8. 0.63 × 1523 ≈ 0.6 × 1500 = 900
9. 23.97 − 7.05 ≈ 24 − 7 = 17
17. $7.99 + $4.23 + $16.82 + $3.51 + $20.12 ≈ $8 + $4 + $17 + $4 + $20 = $53
18. $1.29 + $6.86 + $12.43 + $25.62 + $8.99 ≈ $1 + $7 + $12 + $26 + $9 = $55
19. 95lb + 127lb + 210lb ≈ 100 + 100 + 200 = 400lb
$400 $400
20. ≈ = 16
$23 $25
21. 20% of $49.47 ≈ 20% of $50 = 0.20 × $50 = $10
22. 15% of $160.36 ≈ 15% of $160 = 0.15 × $160 = $24
400
23. 396 miles with 20.1 gallons ≈ = 20 miles per gallon
20
200
24. 207 miles with 3.8 gallons ≈ = 50 miles per gallon
4
25. $22 × 18 ≈ $20 × 20 = $400
26. $29 × 42 ≈ $30 × 40 = $1200
27. $4.95 × 6 × 52 ≈ $5 × 6 × 50 = $1500
28. $22 × 28 × 48 ≈ $20 × 30 × 50 = $30,000
29. $595 + $289 + $120 + $110 + 230 ≈ $600 + $300 + $100 + $100 + $200 = $1300
30. Team A: 189 + 172 + 191 ≈ 190 + 170 + 190 = 550
Team B: 183 + 229 + 167 ≈ 180 + 230 + 170 = 580
580 − 550 = 30 lb
31. 14 min, 55 sec/mi × 3.1 mi ≈ 15 min/mi × 3 mi = 45 min;
260 min 1
32. 10 min, 5 sec/mi × 26.2 mi ≈ 10 min/mi × 26 mi = 260 min; ≈ 4 hours = 4 hr 20 min
60 min 3
599 Mexican pesos 600
33. ≈ = $30
20.14 Mexican pesos/U.S. dollars 20
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,4 Chapter 1: Critical Thinking Skills
1605 Egyptian pounds 1600
34. ≈ = $80
19.15 Egyptian pounds/U.S. dollars 20
35. $1619 + $980 + $140 × 2 + $70 × 3 = $1600 + $1000 + $300 + $200 = $3100
36. $973 + 6 ( $61) + 6 ( $97 ) + 6 ( $200 ) $970 + 6 ( $60 ) + 6 ( $100 ) + 6 ( $200 )
= $970 + $360 + $600 + $1200
= $3130
37. 90 miles
38. 75 miles
39. a) 23% of 700 ≈ 25% of 700 = 0.25 × 700 = 175
b) 12% of 700 ≈ 10% of 700 = 0.10 × 700 = 70
c) 21% of 700 ≈ 20% of 700 = 0.20 × 700 = 140
40. a) 18% of 2987 ≈ 18% of 3000 = 0.18 × 3000 = 540
b) 42% of 2987 ≈ 42% of 3000 = 0.42 × 3000 = 1260
c) 7% of 2987 ≈ 7% of 3000 = 0.07 × 3000 = 210
41. a) 5 million
b) 98 million
c) 98 million − 37 million = 61 million
d) 19 million + 79 million + 84 million + 65 million + 33 million = 280 million
42. a) 19%
b) 25%
c) 20% of 179 lb ≈ 20% of 180 = 0.2 × 180 = 36 lb
43. a) 85%
b) 68% − 53% = 15%
c) 85% of 70 million acres = 59,500,000 acres
d) No, since we are not given the area of each state.
44. a) 2(410) + 4(545) ≈ 2(400) + 4(500) = 800 + 2000 = 2800 calories
b) Running: 4(920) ≈ 4(900) = 3600 calories
Walking: 4(330) ≈ 4(300) = 1200 calories
Difference: 3600 − 1200 = 2400 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
45. 20 53. 9 square units
46. 25 54. 12 square units
47. 120 bananas 55. 160 feet
240
48. 100 grapes 56. 4(60) = 240 in. or ≈ 20 ft
12
49. 150°
57. Answers will vary.
50. 315°
51. 10%
52. 25%
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, Section 1.3: Problem-Solving Procedures 5
$24.99 $25
58. Cost of one vitamin C tablet: ≈ = $0.05
500 500
$47.99 $48
Cost of one fish oil tablet: ≈ = $0.40
120 120
$17.99 $18
Cost of one vitamin D tablet: ≈ = $0.10
180 180
$19.99 $20
Cost of one multivitamin tablet: ≈ = $0.25
80 80
Roberto’s daily cost: 5 × $0.05 + 2 × $0.40 + 1 × $0.10 + 1 × $0.25 = $1.40
59. There are 118 ridges around the edge.
60. a) Answers will vary.
b) 11.6 days. There are 24 ⋅ 60 ⋅ 60 = 86,400 seconds in a day, and 1,000,000/86,400 = 11.5740.
61. Answers will vary.
62. Answers will vary.
63. Answers will vary.
Section 1.3: Problem-Solving Procedures
1 in. 4.25 in. 6 ft person x ft turbine
1. = 4. =
12 mi x mi 2 ft shadow 95 ft shadow
x = 12 ( 4.25 ) 2 x = 6 ( 95 )
x = 51 mi 570
x= = 285 ft
2
1 in. 22.4 in.
2. =
2.5 yd x yd 5. 4 × $113 = $452
9.25% of $452 = 0.0925 × $452 = $41.81
1x = 2.5 ( 22.4 )
$452 + $41.81 = $493.81
x = 56 yd
6. 2 × $187 = $374
3 ft x ft 9.5% of $374 = 0.095 × $374 = $35.53
3. = $374 + $35.53 = $409.53
1.2 ft 15.36 ft
3 (15.36 ) = 1.2 x
46.03 1.2 x
=
1.2 1.2
46.03
x= = 38.4 ft
1.2
165 ft 2
7. a) = 13.75, so 14 bags.
12 ft 2
b) $3.97 × 14 = $55.58
c) 7% of $55.58 = 0.07 × $55.58 = $3.8906; $55.58 + $3.89 = $59.47
184 ft 2
8. a) = 61.33, so 62 bags.
3 ft 2
b) $6.98 × 62 = $432.76
c) 9% of $432.76 = 0.09 × $432.76 = $38.95; $432.76 + $38.95 = $471.71
9. a) Transportation: 16.0% of $2600 = 0.160 × $2600 = $416.00
Food: 11.9% of $2600 = 0.119 × $2600 = $309.40
$416.00 − $309.40 = $106.60
b) Housing: 34.9% of $12600 = 0.349 × $2600 = $907.40
Healthcare: 8.4% of $12600 = 0.084 × $2600 = $218.40
$907.40 − $218.40 = $689.00
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,6 Chapter 1: Critical Thinking Skills
10. a) Business: 19% of 2,038,431 = 0.19 × 2,038,431 = 387,302
Health Professions: 12.6% of 2,038,431 = 0.126 × 2,038,431 = 256,842
387,302 − 256,842 = 130,460 degrees
b) Engineering: 6.3% of 2,038,431 = 0.063 × 2,038,431 = 128,421
Psychology: 5.9% of 2,038,431 = 0.059 × 2,038,431 = 120,267
128,421 − 120,267 = 8154 degrees
$100
11. a) = 4.76, so 4 units of 24-pack are affordable 4 × $21 = $84
$21
$100 − $84 = $16, so 1 unit of 12-pack is affordable
number of golf balls = 4 × 24 + 12 = 96 + 12 = 108
b) $84 + $14 = $98
$87
12. a) = 6.21, so 6 large pizzas are affordable 6 × $14 = $84
$14
$87 − $84 = $3, so medium pizza is not affordable
6 × (12 slices) = 72 slices
b) $84
13. $250 + $130(18) = $250 + $2340 = $2590
Savings: $2590 − $2500 = $90
14. $560 + $55(48) = $560 + $2640 = $3200
Savings: $3200 − $2800 = $400
15. 15-year mortgage: $887.63(12)(15) = $159,773.40
30-year mortgage: $572.90(12)(30) = $206,244.00
Savings: $206,244.00 − $159,773.40 = $46,470.60
16. 4 years: $795 per month × 4 years = $795 per month × 4(12 months) = $795 × 48 = $38,160
5 years: $650 per month × 5 years = $650 per month × 5(12 months) = $650 × 60 = $39,000
Savings: $39,000 − $38,160 = $840
17. Points needed for 90 average: 90(5) = 450 points
Carissa’s points so far: 88 + 96 + 91 + 83 = 358 points
Grade needed on fifth test: 450 − 358 = 92
18. 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
460 1400
19. a) = 18.4 min c) = 40 min
25 35
1550 2200
b) = 31 min d) = 27.5 min
50 80
460+350 1550+350
20. a) = 23.14 min c) = 76 min
35 25
1400+2200 1550+2200
b) = 72 min d) = 46.88 min
50 80
433.1 mi
21. a) 102,910.2 mi − 102,477.1 mi = 433.1 mi; ≈ 30.5 mpg
14.2 gal
2(115 miles)
b) = 7.54 gal
30.5 mpg
c) (7.54 gal)($3.29 per gallon) = $24.81
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, Section 1.3: Problem-Solving Procedures 7
297.6 mi
22. a) 72,555.4 mi − 72,257.8 mi = 297.6 mi; ≈ 15.5 mpg
19.2 gal
2(124 miles)
b) = 16 gal
15.5 mpg
c) (16 gal)($4.23 per gallon) = $67.68
23. By mail: ($52.80 + $5.60 + $8.56) × 4 = $66.96 × 4 = $267.84
Tire store: $324 + 0.08 × $324 = $324 + $25.92 = $349.92
Savings: $349.92 − $267.84 = $82.08
24. 15,000 ft −3000 ft = 12,000 ft decrease in elevation. Temperature increases 2.4°𝐹 for every 1000 ft decrease in
elevation.
12 × 2.4°𝐹 = 28.8°𝐹 and −6°𝐹 + 28.8°𝐹 = 22.8°F; The precipitation at the airport will be snow.
25. a) 7.5% of $56.28 is 0.075($56.28) = $4.22; $56.28 + $4.22 = $60.50
b) 20% of $56.28 is 0.20($56.28) = $11.26;
c) $60.50 + $11.26 = $71.76
26. a) $175(3) = $525
b) 8.5% of $525 is 0.085($525) = $44.63;
c) $75(3) = $225
d) $525 + $44.63 + $225 = $794.63
525,600 oz
27. a) 1 × 60 × 24 × 365 = 525,600 oz and = 4106.25 gal
128
4106.25
b) × $11.20 = 4.10625 × $11.20 = $45.99
1000
28. a) 0.1 × 60 × 60 × 24 × 365 = 3,153,600 cm3
Volume of basin (30 cm)(20 cm)(20 cm) 12,000 cm3
b) = = ≈ 1.4; It would take about 1.4 days.
Daily water loss 0.1×60×60×24 8640 cm3
20,000 20,000
29. a) − ≈ 961.538 − 925.926 = 35.612 ≈ 35.61 gal
20.8 21.6
b) 35.61 × $3.00 = $106.83
c) 140,000,000 × 35.61 = 4,985,400,000 gal
30. a) Parking: $30 × 6 = $180
Tax: 9.5% of $180 = 0.095 × $180 = $17.10
Total: $180 + $17.10 = $197.10
b) Parking: $17.95 × 6 = $107.70
Tax: 8.75% of $107.70 = 0.0875 × $107.70 ≈ $9.42
Total: $107.70 + $9.42 = $117.12
c) She can save $197.10 − $117.12 = $79.98.
31. Cost after 1 year: $4.29 + 0.05($4.29) = $4.29 + $0.21 = $4.50
Cost after second year: $4.50 + 0.05($4.50) = $4.50 + $0.23 = $4.73
32. Value after first year: $1000 + 0.10($1000) = $1000 + $100 = $1100
Value after second year: $1100 − 0.10($1100) = $1100 − $110 = $990
$990 is less than the initial investment of $1000.
33. After paying the $100 deductible, Yungchen must pay 20% of the cost of x-rays.
First x-ray: $100 + 0.20($620) = $100 + $124 = $224
Second x-ray: 0.20($980) = $196
Total: $224 + $196 = $420
Copyright © 2024 Pearson Education, Inc.
, 8 Chapter 1: Critical Thinking Skills
34. a) $130 + $215 + $55 = $400
b) $25 + $215(100% − 90%) + $55(100% − 80%) = $25 + $215(10%) + $55(20%) =
$25 + $215(0.1) + $55(0.2) = $25 + $21.5 + $11.0 = $57.5
$425
35. a) = 1.77; 1 box of 20
$240
$425 − $240 = $185; 1 box of 12
1 box of 20 and 1 box of 12.
b) $240 + $180 = $420
$1100
36. a) = 4.58; 4 boxes of 20 can be purchased, the remaining amount is:
$240
$1100 − $240(4) = $1100 − $960 = $140, which is not enough for box of 12;
In case of purchasing 3 boxes of 20, remaining amount is:
$380
$1100 − $240(3) = $1100 − $720 = $380; = 2.11; 2 boxes of 12.
180
3 boxes of 20 and 2 boxes of 12.
b) $240(3) + $180(2) = $1080
37. 1 ft 2 = 12 in. × 12 in. = 144 square inches
38. 1 ft 3 = 12 in. × 12 in. × 12 in. = 1728 cubic inches
39. Area of original rectangle: 𝑙𝑤
Area of new rectangle: (2𝑙)(2𝑤) = 4(𝑙𝑤)
Thus, if the length and width of a rectangle are doubled, the area is 4 times as large.
400ft 2
40. 20ft × 20ft =400ft 2 ; 5ft × 5ft = 25 ft 2 ; = 16squares
25ft 2
41. Volume of original cube: 𝑙𝑤ℎ
Volume of new cube: (2𝑙)(2𝑤)(2ℎ) = 8(𝑙𝑤ℎ)
Thus, if the length, width, and height of a cube are doubled, the volume is 8 times as large.
42. 11 ft is one-sixth of the pole, so the length is 6 × 11 ft = 66 ft.
4
43. a) Chili powder: (3 tsp) × = 2 tsp
6
1 4 1
Cayenne pepper: ( tsp) = tsp
4 6 6
3 tsp 5 tsp 5 tsp(6 servings)
b) = ; 𝑥= = 10 servings
6 servings 𝑥 servings 3 tsp
3 3
44. a) Chopped garlic: (1 tbs) × = tbs
4 4
3 15 3
Chopped red onion: (5 tbs) = tbs = 3 tbs
4 4 4
1 tbs 𝑥 tbs 1 tbs (6 avocados) 6 1
b) Chopped garlic: = ; 𝑥= = tbs = 1 tbs
4 avocados 6 avocados 4 avocados 4 2
5 tbs 𝑥 tbs 5 tbs (6 avocados) 30 1
Chopped red onion: = ; 𝑥= = tbs = 7 tbs
4 avocados 6 avocados 4 avocados 4 2
45. Left side Right side
1(−6) = −6 1(2) = 2
2(−2) = −4 1(3) = 3
(6) = 6
−6 + (−4) = −10 2 + 3 + 6 = 11
Place it at –1, so the left side would total −10 + (−1) = −1.
Copyright © 2024 Pearson Education, Inc.
with Applications, A, 12th edition Angel
Notes
1- The file is chapter after chapter.
2- We have shown you few pages sample.
3- The file contains all Appendix and Excel sheet
if it exists.
4- We have all what you need, we make update
at every time. There are many new editions
waiting you.
5- If you think you purchased the wrong file You
can contact us at every time, we can replace it
with true one.
Our email:
, Section 1.1: Inductive and Deductive Reasoning 1
Chapter One: Critical Thinking Skills
Section 1.1: Inductive and Deductive Reasoning
1. Natural 13.
2. Divisible 14.
3. Counterexample
15.
4. Hypothesis
16.
5. Inductive
17. 9, 11, 13 (Add 2 to previous number.)
6. Deductive
18. 16, 19, 22 (Add 3 to the previous number.)
7. Deductive
19. 16, –32, 64 (Multiply previous number by –2.)
8. Inductive
20. -243, 729, -2187 (Multiply by –3.)
9. 5 × 5 = 25
1 1 1
10. 19 × 10 = 190 21. , , (Increase denominator value by 1.)
5 6 7
9 11 13
11. 1 5
(1+4)
10
(4+6)
10
(4+6)
5
(1+4)
1 22.
10
, ,
12 14
(Increase denominator and
numerator values by 2.)
12. 100,000 = 105
23. 34, 55, 89 (Each number in the sequence is the sum of the previous two numbers.)
24. 76, 123, 199 (Each number in the sequence is the sum of the previous two numbers.)
25. The product of two odd numbers is an odd number.
26. The product of an even number and an odd number is an even number.
27. The sum of two even numbers is an even number
28. The sum of two odd numbers is an even number.
29. a) Answers will vary.
b) Products involving 10 and natural numbers will have a ones digit 0.
30. a) Answers will vary.
b) The sum of the digits is 9.
c) The sum of the digits in the product when a one- or two-digit number is multiplied by 9 is divisible by 9.
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.
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.
33. Blue: 1, 5, 7, 10, 12; Purple: 2, 4, 6, 9, 11; 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
Copyright © 2024 Pearson Education, Inc.
,2 Chapter 1: Critical Thinking Skills
35. a) $3700
b) We are using observation of specific cases to make a prediction.
36. a) $415
b) We are using observation of specific cases to make a prediction.
37. a) You should obtain the original number.
b) You should obtain the original number.
c) Conjecture: The result is always the original number.
38. 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.
39. a) You should obtain the number 5.
b) You should obtain the number 5.
c) Conjecture: The result is always the number 5.
40. a) You should obtain the number 8.
b) You should obtain the number 8.
c) Conjecture: The result is always the number 8.
3𝑛+6 3𝑛 6
41. 𝑛 → 3𝑛 → 3𝑛 + 6 → = + = 𝑛+2 →𝑛+2−2 =𝑛
3 3 3
4𝑛+6 4𝑛 6
42. 𝑛 → 4𝑛 → 4𝑛 + 6 → = + = 2𝑛 + 3 → 2𝑛 + 3 − 3 = 2𝑛
2 2 2
2𝑛+10 2𝑛 10
43. 𝑛 → 𝑛 + 1 → 𝑛 + (𝑛 + 1) = 2𝑛 + 1 → 2𝑛 + 1 + 9 = 2𝑛 + 10 → = + =𝑛+5→ 𝑛+5−𝑛 =
2 2 2
5
2𝑛+16 2𝑛 16
44. 𝑛 → 𝑛 + 5 → 𝑛 + (𝑛 + 5) = 2𝑛 + 5 → 2𝑛 + 5 + 11 = 2𝑛 + 16 → = + =𝑛+8→ 𝑛+8−
2 2 2
𝑛=8
45. 3 × 5 = 15 is a counterexample.
46. 10 + 11 + 12 = 33, which is not a three-digit number.
5
47. Two is a counting number. The sum of 2 and 3 is 5. Five divided by two is , which is not an even number.
2
48. 900 is a three-digit number. The product of 900 and 900 is 810,000, which is not a five-digit number.
49. One and two are counting numbers. The difference of 1 and 2 is 1 − 2 = −1, which is not a counting number.
50. The sum of the odd numbers 1 and 5 is 6, which is not divisible by 4.
51. 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°.
52. 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°.
53. Inductive reasoning: a general conclusion is obtained from observation of specific cases.
54. Inductive reasoning: a general conclusion is obtained from observation of specific cases.
55. 129; The numbers in the positions of each inner 4 × 4 square is determined as follows.
𝑎 𝑏
𝑐 𝑎+𝑏+𝑐
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, Section 1.2: Estimation Techniques 3
56. 1881, 8008, 8118 (They look the same when looked at in a mirror.)
57. c
Section 1.2: Estimation Techniques
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. Estimation 10. 400.15 − 297.87 ≈ 400 − 300 = 100
2. Equal 11. 22% 9116 20% 9000 = 0.20 9000
3. 39.7 + 60.3 + 18 + 1.8 =1800
≈ 40 + 60 + 20 + 0 = 120
12. 11% of 8221 10% of 8000 = 0.10 8000
4. 86 + 47.2 + 289.8 + 532.4 + 12.8 = 800
≈ 90 + 50 + 290 + 530 + 10 = 970
$91.35 $90
5. 197,500 ÷ 4.063 ≈ 200,000 ÷ 4.000 = 13. ≈ = $30
3 3
50,000 $210 $200
14. ≈ = $25
1,032,645 1,000,000 8 8
6. ≈ = 20
49,827 50,000
15. 12 months × $47 ≈ 12 × $50 = $600
7. 1776 × 0.0098 ≈ 1800 × 0.01 = 18 16. 19 gallons × $3.11 ≈ 20 × $3 = $60
8. 0.63 × 1523 ≈ 0.6 × 1500 = 900
9. 23.97 − 7.05 ≈ 24 − 7 = 17
17. $7.99 + $4.23 + $16.82 + $3.51 + $20.12 ≈ $8 + $4 + $17 + $4 + $20 = $53
18. $1.29 + $6.86 + $12.43 + $25.62 + $8.99 ≈ $1 + $7 + $12 + $26 + $9 = $55
19. 95lb + 127lb + 210lb ≈ 100 + 100 + 200 = 400lb
$400 $400
20. ≈ = 16
$23 $25
21. 20% of $49.47 ≈ 20% of $50 = 0.20 × $50 = $10
22. 15% of $160.36 ≈ 15% of $160 = 0.15 × $160 = $24
400
23. 396 miles with 20.1 gallons ≈ = 20 miles per gallon
20
200
24. 207 miles with 3.8 gallons ≈ = 50 miles per gallon
4
25. $22 × 18 ≈ $20 × 20 = $400
26. $29 × 42 ≈ $30 × 40 = $1200
27. $4.95 × 6 × 52 ≈ $5 × 6 × 50 = $1500
28. $22 × 28 × 48 ≈ $20 × 30 × 50 = $30,000
29. $595 + $289 + $120 + $110 + 230 ≈ $600 + $300 + $100 + $100 + $200 = $1300
30. Team A: 189 + 172 + 191 ≈ 190 + 170 + 190 = 550
Team B: 183 + 229 + 167 ≈ 180 + 230 + 170 = 580
580 − 550 = 30 lb
31. 14 min, 55 sec/mi × 3.1 mi ≈ 15 min/mi × 3 mi = 45 min;
260 min 1
32. 10 min, 5 sec/mi × 26.2 mi ≈ 10 min/mi × 26 mi = 260 min; ≈ 4 hours = 4 hr 20 min
60 min 3
599 Mexican pesos 600
33. ≈ = $30
20.14 Mexican pesos/U.S. dollars 20
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,4 Chapter 1: Critical Thinking Skills
1605 Egyptian pounds 1600
34. ≈ = $80
19.15 Egyptian pounds/U.S. dollars 20
35. $1619 + $980 + $140 × 2 + $70 × 3 = $1600 + $1000 + $300 + $200 = $3100
36. $973 + 6 ( $61) + 6 ( $97 ) + 6 ( $200 ) $970 + 6 ( $60 ) + 6 ( $100 ) + 6 ( $200 )
= $970 + $360 + $600 + $1200
= $3130
37. 90 miles
38. 75 miles
39. a) 23% of 700 ≈ 25% of 700 = 0.25 × 700 = 175
b) 12% of 700 ≈ 10% of 700 = 0.10 × 700 = 70
c) 21% of 700 ≈ 20% of 700 = 0.20 × 700 = 140
40. a) 18% of 2987 ≈ 18% of 3000 = 0.18 × 3000 = 540
b) 42% of 2987 ≈ 42% of 3000 = 0.42 × 3000 = 1260
c) 7% of 2987 ≈ 7% of 3000 = 0.07 × 3000 = 210
41. a) 5 million
b) 98 million
c) 98 million − 37 million = 61 million
d) 19 million + 79 million + 84 million + 65 million + 33 million = 280 million
42. a) 19%
b) 25%
c) 20% of 179 lb ≈ 20% of 180 = 0.2 × 180 = 36 lb
43. a) 85%
b) 68% − 53% = 15%
c) 85% of 70 million acres = 59,500,000 acres
d) No, since we are not given the area of each state.
44. a) 2(410) + 4(545) ≈ 2(400) + 4(500) = 800 + 2000 = 2800 calories
b) Running: 4(920) ≈ 4(900) = 3600 calories
Walking: 4(330) ≈ 4(300) = 1200 calories
Difference: 3600 − 1200 = 2400 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
45. 20 53. 9 square units
46. 25 54. 12 square units
47. 120 bananas 55. 160 feet
240
48. 100 grapes 56. 4(60) = 240 in. or ≈ 20 ft
12
49. 150°
57. Answers will vary.
50. 315°
51. 10%
52. 25%
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, Section 1.3: Problem-Solving Procedures 5
$24.99 $25
58. Cost of one vitamin C tablet: ≈ = $0.05
500 500
$47.99 $48
Cost of one fish oil tablet: ≈ = $0.40
120 120
$17.99 $18
Cost of one vitamin D tablet: ≈ = $0.10
180 180
$19.99 $20
Cost of one multivitamin tablet: ≈ = $0.25
80 80
Roberto’s daily cost: 5 × $0.05 + 2 × $0.40 + 1 × $0.10 + 1 × $0.25 = $1.40
59. There are 118 ridges around the edge.
60. a) Answers will vary.
b) 11.6 days. There are 24 ⋅ 60 ⋅ 60 = 86,400 seconds in a day, and 1,000,000/86,400 = 11.5740.
61. Answers will vary.
62. Answers will vary.
63. Answers will vary.
Section 1.3: Problem-Solving Procedures
1 in. 4.25 in. 6 ft person x ft turbine
1. = 4. =
12 mi x mi 2 ft shadow 95 ft shadow
x = 12 ( 4.25 ) 2 x = 6 ( 95 )
x = 51 mi 570
x= = 285 ft
2
1 in. 22.4 in.
2. =
2.5 yd x yd 5. 4 × $113 = $452
9.25% of $452 = 0.0925 × $452 = $41.81
1x = 2.5 ( 22.4 )
$452 + $41.81 = $493.81
x = 56 yd
6. 2 × $187 = $374
3 ft x ft 9.5% of $374 = 0.095 × $374 = $35.53
3. = $374 + $35.53 = $409.53
1.2 ft 15.36 ft
3 (15.36 ) = 1.2 x
46.03 1.2 x
=
1.2 1.2
46.03
x= = 38.4 ft
1.2
165 ft 2
7. a) = 13.75, so 14 bags.
12 ft 2
b) $3.97 × 14 = $55.58
c) 7% of $55.58 = 0.07 × $55.58 = $3.8906; $55.58 + $3.89 = $59.47
184 ft 2
8. a) = 61.33, so 62 bags.
3 ft 2
b) $6.98 × 62 = $432.76
c) 9% of $432.76 = 0.09 × $432.76 = $38.95; $432.76 + $38.95 = $471.71
9. a) Transportation: 16.0% of $2600 = 0.160 × $2600 = $416.00
Food: 11.9% of $2600 = 0.119 × $2600 = $309.40
$416.00 − $309.40 = $106.60
b) Housing: 34.9% of $12600 = 0.349 × $2600 = $907.40
Healthcare: 8.4% of $12600 = 0.084 × $2600 = $218.40
$907.40 − $218.40 = $689.00
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,6 Chapter 1: Critical Thinking Skills
10. a) Business: 19% of 2,038,431 = 0.19 × 2,038,431 = 387,302
Health Professions: 12.6% of 2,038,431 = 0.126 × 2,038,431 = 256,842
387,302 − 256,842 = 130,460 degrees
b) Engineering: 6.3% of 2,038,431 = 0.063 × 2,038,431 = 128,421
Psychology: 5.9% of 2,038,431 = 0.059 × 2,038,431 = 120,267
128,421 − 120,267 = 8154 degrees
$100
11. a) = 4.76, so 4 units of 24-pack are affordable 4 × $21 = $84
$21
$100 − $84 = $16, so 1 unit of 12-pack is affordable
number of golf balls = 4 × 24 + 12 = 96 + 12 = 108
b) $84 + $14 = $98
$87
12. a) = 6.21, so 6 large pizzas are affordable 6 × $14 = $84
$14
$87 − $84 = $3, so medium pizza is not affordable
6 × (12 slices) = 72 slices
b) $84
13. $250 + $130(18) = $250 + $2340 = $2590
Savings: $2590 − $2500 = $90
14. $560 + $55(48) = $560 + $2640 = $3200
Savings: $3200 − $2800 = $400
15. 15-year mortgage: $887.63(12)(15) = $159,773.40
30-year mortgage: $572.90(12)(30) = $206,244.00
Savings: $206,244.00 − $159,773.40 = $46,470.60
16. 4 years: $795 per month × 4 years = $795 per month × 4(12 months) = $795 × 48 = $38,160
5 years: $650 per month × 5 years = $650 per month × 5(12 months) = $650 × 60 = $39,000
Savings: $39,000 − $38,160 = $840
17. Points needed for 90 average: 90(5) = 450 points
Carissa’s points so far: 88 + 96 + 91 + 83 = 358 points
Grade needed on fifth test: 450 − 358 = 92
18. 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
460 1400
19. a) = 18.4 min c) = 40 min
25 35
1550 2200
b) = 31 min d) = 27.5 min
50 80
460+350 1550+350
20. a) = 23.14 min c) = 76 min
35 25
1400+2200 1550+2200
b) = 72 min d) = 46.88 min
50 80
433.1 mi
21. a) 102,910.2 mi − 102,477.1 mi = 433.1 mi; ≈ 30.5 mpg
14.2 gal
2(115 miles)
b) = 7.54 gal
30.5 mpg
c) (7.54 gal)($3.29 per gallon) = $24.81
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, Section 1.3: Problem-Solving Procedures 7
297.6 mi
22. a) 72,555.4 mi − 72,257.8 mi = 297.6 mi; ≈ 15.5 mpg
19.2 gal
2(124 miles)
b) = 16 gal
15.5 mpg
c) (16 gal)($4.23 per gallon) = $67.68
23. By mail: ($52.80 + $5.60 + $8.56) × 4 = $66.96 × 4 = $267.84
Tire store: $324 + 0.08 × $324 = $324 + $25.92 = $349.92
Savings: $349.92 − $267.84 = $82.08
24. 15,000 ft −3000 ft = 12,000 ft decrease in elevation. Temperature increases 2.4°𝐹 for every 1000 ft decrease in
elevation.
12 × 2.4°𝐹 = 28.8°𝐹 and −6°𝐹 + 28.8°𝐹 = 22.8°F; The precipitation at the airport will be snow.
25. a) 7.5% of $56.28 is 0.075($56.28) = $4.22; $56.28 + $4.22 = $60.50
b) 20% of $56.28 is 0.20($56.28) = $11.26;
c) $60.50 + $11.26 = $71.76
26. a) $175(3) = $525
b) 8.5% of $525 is 0.085($525) = $44.63;
c) $75(3) = $225
d) $525 + $44.63 + $225 = $794.63
525,600 oz
27. a) 1 × 60 × 24 × 365 = 525,600 oz and = 4106.25 gal
128
4106.25
b) × $11.20 = 4.10625 × $11.20 = $45.99
1000
28. a) 0.1 × 60 × 60 × 24 × 365 = 3,153,600 cm3
Volume of basin (30 cm)(20 cm)(20 cm) 12,000 cm3
b) = = ≈ 1.4; It would take about 1.4 days.
Daily water loss 0.1×60×60×24 8640 cm3
20,000 20,000
29. a) − ≈ 961.538 − 925.926 = 35.612 ≈ 35.61 gal
20.8 21.6
b) 35.61 × $3.00 = $106.83
c) 140,000,000 × 35.61 = 4,985,400,000 gal
30. a) Parking: $30 × 6 = $180
Tax: 9.5% of $180 = 0.095 × $180 = $17.10
Total: $180 + $17.10 = $197.10
b) Parking: $17.95 × 6 = $107.70
Tax: 8.75% of $107.70 = 0.0875 × $107.70 ≈ $9.42
Total: $107.70 + $9.42 = $117.12
c) She can save $197.10 − $117.12 = $79.98.
31. Cost after 1 year: $4.29 + 0.05($4.29) = $4.29 + $0.21 = $4.50
Cost after second year: $4.50 + 0.05($4.50) = $4.50 + $0.23 = $4.73
32. Value after first year: $1000 + 0.10($1000) = $1000 + $100 = $1100
Value after second year: $1100 − 0.10($1100) = $1100 − $110 = $990
$990 is less than the initial investment of $1000.
33. After paying the $100 deductible, Yungchen must pay 20% of the cost of x-rays.
First x-ray: $100 + 0.20($620) = $100 + $124 = $224
Second x-ray: 0.20($980) = $196
Total: $224 + $196 = $420
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, 8 Chapter 1: Critical Thinking Skills
34. a) $130 + $215 + $55 = $400
b) $25 + $215(100% − 90%) + $55(100% − 80%) = $25 + $215(10%) + $55(20%) =
$25 + $215(0.1) + $55(0.2) = $25 + $21.5 + $11.0 = $57.5
$425
35. a) = 1.77; 1 box of 20
$240
$425 − $240 = $185; 1 box of 12
1 box of 20 and 1 box of 12.
b) $240 + $180 = $420
$1100
36. a) = 4.58; 4 boxes of 20 can be purchased, the remaining amount is:
$240
$1100 − $240(4) = $1100 − $960 = $140, which is not enough for box of 12;
In case of purchasing 3 boxes of 20, remaining amount is:
$380
$1100 − $240(3) = $1100 − $720 = $380; = 2.11; 2 boxes of 12.
180
3 boxes of 20 and 2 boxes of 12.
b) $240(3) + $180(2) = $1080
37. 1 ft 2 = 12 in. × 12 in. = 144 square inches
38. 1 ft 3 = 12 in. × 12 in. × 12 in. = 1728 cubic inches
39. Area of original rectangle: 𝑙𝑤
Area of new rectangle: (2𝑙)(2𝑤) = 4(𝑙𝑤)
Thus, if the length and width of a rectangle are doubled, the area is 4 times as large.
400ft 2
40. 20ft × 20ft =400ft 2 ; 5ft × 5ft = 25 ft 2 ; = 16squares
25ft 2
41. Volume of original cube: 𝑙𝑤ℎ
Volume of new cube: (2𝑙)(2𝑤)(2ℎ) = 8(𝑙𝑤ℎ)
Thus, if the length, width, and height of a cube are doubled, the volume is 8 times as large.
42. 11 ft is one-sixth of the pole, so the length is 6 × 11 ft = 66 ft.
4
43. a) Chili powder: (3 tsp) × = 2 tsp
6
1 4 1
Cayenne pepper: ( tsp) = tsp
4 6 6
3 tsp 5 tsp 5 tsp(6 servings)
b) = ; 𝑥= = 10 servings
6 servings 𝑥 servings 3 tsp
3 3
44. a) Chopped garlic: (1 tbs) × = tbs
4 4
3 15 3
Chopped red onion: (5 tbs) = tbs = 3 tbs
4 4 4
1 tbs 𝑥 tbs 1 tbs (6 avocados) 6 1
b) Chopped garlic: = ; 𝑥= = tbs = 1 tbs
4 avocados 6 avocados 4 avocados 4 2
5 tbs 𝑥 tbs 5 tbs (6 avocados) 30 1
Chopped red onion: = ; 𝑥= = tbs = 7 tbs
4 avocados 6 avocados 4 avocados 4 2
45. Left side Right side
1(−6) = −6 1(2) = 2
2(−2) = −4 1(3) = 3
(6) = 6
−6 + (−4) = −10 2 + 3 + 6 = 11
Place it at –1, so the left side would total −10 + (−1) = −1.
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