Appendix A — Answer Key, Chapter 3: Percent Applications: Discounts, Markups, Tax, and Tip

SOL 8.CE.1 · Covers textbook Chapter 3 and the companion workbook. Item numbers match the textbook; workbook items are the same problems, so this key serves both. Item numbers run continuously from 1 to 132. Reasoning answers show an acceptable response, not the only wording.

Conventions used throughout this key: every dollar amount is rounded to the nearest cent, rounding a half cent up, and rounded once at the end of the computation. Where a discount, markup, tax, or tip can be done two ways, the second route is shown as a check. Percent changes are exact in every item of this chapter; none required rounding.

Lesson 3.1 — Estimating and Finding a Percent of a Price

Guided practice

1. $\$4.60$. Move the decimal point in $46.00$ one place left.
2. $\$2.30$. Half of the $10\%$ amount, $\$4.60$.
3. $\$0.84$. Move the decimal point in $84.00$ two places left.
4. $\$9.20$. $36.80 \div 4 = 9.20$, or halve twice: $18.40$, then $9.20$.
5. About $\$8.00$. $10\%$ of $\$40.00$ is $\$4.00$, so $20\%$ is about $\$8.00$. (The exact value is $0.20 \times 38.75 = 7.75$, so this estimate is high because the price was rounded up.)
6. $\$7.80$. $10\%$ is $\$5.20$ and $5\%$ is $\$2.60$, and $5.20 + 2.60 = 7.80$. Check: $0.15 \times 52.00 = 7.80$.

Independent practice

7. a) $\$13.00$ b) $\$7.35$ c) $\$7.20$ d) $\$2.00$ Work: a) $0.20 \times 65 = 13$ b) $0.30 \times 24.50 = 7.35$ c) $0.06 \times 120 = 7.20$ d) $0.025 \times 80 = 2$
8. $\$2.12$. $5.3\% = 0.053$, and $0.053 \times 40.00 = 2.12$.
9. $\$11.43$. $0.18 \times 63.50 = 11.43$ exactly, with nothing to round.
10. a) $\$7.31$ b) $\$13.00$ c) $\$0.44$ In b) the third decimal is a 5, so the cents round up from $12.99$ to $13.00$.
11. Estimate about $\$12.00$; exact $\$11.91$; off by $\$0.09$. Rounding $\$79.40$ up to $\$80.00$ gives $10\%$ of $\$8.00$ plus $5\%$ of $\$4.00$. Exactly, $0.15 \times 79.40 = 11.91$. The estimate is high because the price was rounded up.
12. $1\%$ means one hundredth, so finding $1\%$ of a price means dividing by 100. Dividing by 100 shifts every digit two place values to the right, which is the same as moving the decimal point two places to the left.
13. $\$22.20$. $0.30 \times 74.00 = 22.20$.
14. The student multiplied by the number 8 instead of by the percent $8\%$. A percent must be written as a decimal first: $8\% = 0.08$, so $0.08 \times 45.00 = 3.60$. The answer is $\$3.60$. A quick check: $8\%$ is a small part of the price, so the answer has to be far less than $\$45.00$, never more.

Exit ticket 3.1

15. $\$2.74$
16. $\$3.10$
17. $\$11.40$. $0.12 \times 95.00 = 11.40$.
18. Estimate about $\$8.00$, and the estimate is low. Rounding $\$41.20$ down to $\$40.00$ made the price smaller than it really is, so $20\%$ of it is smaller than the true $20\%$. The exact value is $0.20 \times 41.20 = 8.24$.

Lesson 3.2 — Discounts and Sale Prices

Guided practice

19. $\$12.00$. $0.20 \times 60.00 = 12.00$.
20. $\$48.00$. $60.00 - 12.00 = 48.00$. Check: $0.80 \times 60.00 = 48.00$.
21. $\$40.50$. Discount $0.10 \times 45.00 = 4.50$, so $45.00 - 4.50 = 40.50$. Check: $0.90 \times 45.00 = 40.50$.
22. $\$24.00$. $100\% - 25\% = 75\%$, and $0.75 \times 32.00 = 24.00$.
23. Discount $\$18.00$; sale price $\$102.00$. $0.15 \times 120.00 = 18.00$ and $120.00 - 18.00 = 102.00$. Check: $0.85 \times 120.00 = 102.00$.
24. About $\$30.00$. $25\%$ of $\$40.00$ is $\$10.00$, and $40.00 - 10.00 = 30.00$. (Exactly, the sale price is $\$29.85$, so the estimate is high by $\$0.15$.)

Independent practice

25. a) discount $\$28.00$, sale price $\$52.00$ b) discount $\$4.90$, sale price $\$19.60$ c) discount $\$9.30$, sale price $\$9.30$ d) discount $\$60.00$, sale price $\$89.99$ Work: a) $0.35 \times 80 = 28$; $80 - 28 = 52$. b) $0.20 \times 24.50 = 4.90$; $24.50 - 4.90 = 19.60$. c) $0.50 \times 18.60 = 9.30$; half off means the discount and the sale price are equal. d) $0.40 \times 149.99 = 59.996$, which rounds to $\$60.00$; $149.99 - 60.00 = 89.99$. Check with the complement: $0.60 \times 149.99 = 89.994$, which also rounds to $\$89.99$.
26. $\$50.40$. $0.70 \times 72.00 = 50.40$. Two-step check: discount $0.30 \times 72.00 = 21.60$ and $72.00 - 21.60 = 50.40$.
27. Discount $\$8.46$; sale price $\$47.94$. $0.15 \times 56.40 = 8.46$ and $56.40 - 8.46 = 47.94$. Check: $0.85 \times 56.40 = 47.94$.
28. $\$54.45$. Discount $0.45 \times 99.00 = 44.55$, and $99.00 - 44.55 = 54.45$. Check: $0.55 \times 99.00 = 54.45$.
29. If $20\%$ of the price comes off, then $100\% - 20\% = 80\%$ of the price stays on, and the part that stays on is exactly what the shopper pays. Since $80\% = 0.80$, multiplying the original price by $0.80$ gives the sale price in one step instead of finding the discount and subtracting.
30. Saved $\$51.00$; the coat costs $\$34.00$. $0.60 \times 85.00 = 51.00$ and $85.00 - 51.00 = 34.00$. Check: $0.40 \times 85.00 = 34.00$.
31. The student found the discount amount, not the sale price: $0.30 \times 48.00 = 14.40$ is the money saved. The sale price still has to be found by subtracting: $48.00 - 14.40 = 33.60$, so the sale price is $\$33.60$. A useful check is that a sale price must be less than the original but not tiny — $\$14.40$ is only about $30\%$ of the price, which is a signal that it is the discount.
32. Estimate about $\$48.00$; exact sale price $\$49.40$; the estimate is low. Rounding $\$61.75$ down to $\$60.00$ shrank the price, which shrank the sale price too. Exactly: $0.20 \times 61.75 = 12.35$ and $61.75 - 12.35 = 49.40$, or $0.80 \times 61.75 = 49.40$.

Exit ticket 3.2

33. $\$30.00$. $0.75 \times 40.00 = 30.00$.
34. $\$6.50$. $0.10 \times 65.00 = 6.50$. (The sale price would be $\$58.50$, but the item asked only for the discount.)
35. $\$18.40$. Half of $\$36.80$.
36. Because the price is $100\%$ of itself, and the discount and the sale price together make up that whole. Subtracting the discount removes the $r\%$ that comes off; multiplying by the complement keeps the $(100 - r)\%$ that stays on. Removing part of a whole and keeping the rest of that whole describe the same amount, so $P - rP$ and $(1 - r)P$ are equal for every price $P$.

Lesson 3.3 — Markups and Selling Prices

Guided practice

37. $\$10.00$. $0.25 \times 40.00 = 10.00$.
38. $\$50.00$. $40.00 + 10.00 = 50.00$. Check: $1.25 \times 40.00 = 50.00$.
39. $\$27.00$. Markup $0.50 \times 18.00 = 9.00$, and $18.00 + 9.00 = 27.00$. Check: $1.50 \times 18.00 = 27.00$.
40. $\$68.20$. $1.10 \times 62.00 = 68.20$. Two-step check: $0.10 \times 62.00 = 6.20$ and $62.00 + 6.20 = 68.20$.
41. Markup $\$34.00$; selling price $\$119.00$. $0.40 \times 85.00 = 34.00$ and $85.00 + 34.00 = 119.00$. Check: $1.40 \times 85.00 = 119.00$.
42. About $\$39.00$. $30\%$ of $\$30.00$ is $\$9.00$, and $30.00 + 9.00 = 39.00$. (Exactly, the selling price is $\$38.48$, so the estimate is high because the cost was rounded up.)

Independent practice

43. a) markup $\$3.60$, selling price $\$27.60$ b) markup $\$30.00$, selling price $\$180.00$ c) markup $\$5.70$, selling price $\$15.20$ d) markup $\$18.90$, selling price $\$66.15$ Work: a) $0.15 \times 24 = 3.60$; $24 + 3.60 = 27.60$. b) $0.20 \times 150 = 30$; $150 + 30 = 180$. c) $0.60 \times 9.50 = 5.70$; $9.50 + 5.70 = 15.20$. d) $0.40 \times 47.25 = 18.90$; $47.25 + 18.90 = 66.15$.
44. $\$49.68$. Markup $0.35 \times 36.80 = 12.88$, and $36.80 + 12.88 = 49.68$. Check: $1.35 \times 36.80 = 49.68$.
45. Markup $\$50.40$; selling price $\$162.40$. $0.45 \times 112.00 = 50.40$ and $112.00 + 50.40 = 162.40$. Check: $1.45 \times 112.00 = 162.40$.
46. $\$22.63$. Markup $0.55 \times 14.60 = 8.03$, and $14.60 + 8.03 = 22.63$. Check: $1.55 \times 14.60 = 22.63$.
47. A $25\%$ markup keeps the whole cost, which is $100\%$, and adds another $25\%$ on top, so the selling price is $125\%$ of the cost, and $125\% = 1.25$. A markup multiplier is always greater than 1 because a markup adds to the cost: the selling price is $(1 + r) \times \text{cost}$, and $1 + r > 1$ for any positive rate $r$.
48. Markup $\$10.50$; selling price $\$18.90$. $1.25 \times 8.40 = 10.50$ and $8.40 + 10.50 = 18.90$. Check: $2.25 \times 8.40 = 18.90$. Because the rate is over $100\%$, the markup is larger than the cost, so the cake sells for more than twice what its ingredients cost.
49. The student found the markup amount, not the selling price: $0.30 \times 50.00 = 15.00$. The markup still has to be added: $50.00 + 15.00 = 65.00$, so the selling price is $\$65.00$. A selling price must always be greater than the cost, and $\$15.00$ is less than $\$50.00$, which flags the error immediately.
50. Estimate about $\$100.00$; exact selling price $\$98.00$. Rounding $\$78.40$ up to $\$80.00$ gives a markup of $\$20.00$ and a price of about $\$100.00$. Exactly: $0.25 \times 78.40 = 19.60$ and $78.40 + 19.60 = 98.00$, or $1.25 \times 78.40 = 98.00$.

Exit ticket 3.3

51. $\$26.00$. $1.30 \times 20.00 = 26.00$.
52. $\$16.00$. $0.25 \times 64.00 = 16.00$. (The selling price would be $\$80.00$, but the item asked only for the markup.)
53. $\$54.60$. Markup $0.20 \times 45.50 = 9.10$, and $45.50 + 9.10 = 54.60$. Check: $1.20 \times 45.50 = 54.60$.
54. A discount is subtracted from a price and a markup is added to a cost. So a discount multiplier is $1 - r$, which is less than 1 and makes the result smaller than the original, while a markup multiplier is $1 + r$, which is greater than 1 and makes the result larger.

Lesson 3.4 — Sales Tax and the Total

Guided practice

55. $\$2.12$. $0.053 \times 40.00 = 2.12$.
56. $\$42.12$. $40.00 + 2.12 = 42.12$. Check: $1.053 \times 40.00 = 42.12$.
57. Tax $\$1.50$; total $\$26.50$. $0.06 \times 25.00 = 1.50$ and $25.00 + 1.50 = 26.50$.
58. $\$18.95$. $1.053 \times 18.00 = 18.954$, which rounds to $\$18.95$. Two-step check: $0.053 \times 18.00 = 0.954$, which rounds to $\$0.95$, and $18.00 + 0.95 = 18.95$.
59. $\$63.63$. $1\%$ of $\$63.00$ is $\$0.63$, and $63.00 + 0.63 = 63.63$.
60. About $\$63.00$. $10\%$ of $\$60.00$ is $\$6.00$, so $5\%$ is $\$3.00$, and $60.00 + 3.00 = 63.00$. (The exact total is $\$62.97$, so this estimate is 3 cents high.)

Independent practice

61. a) tax $\$1.70$, total $\$33.70$ b) tax $\$4.50$, total $\$79.50$ c) tax $\$6.39$, total $\$126.89$ d) tax $\$1.05$, total $\$16.04$ Work: a) $0.053 \times 32 = 1.696 \rightarrow 1.70$; $1.053 \times 32 = 33.696 \rightarrow 33.70$. b) $0.06 \times 75 = 4.50$; $75 + 4.50 = 79.50$. c) $0.053 \times 120.50 = 6.3865 \rightarrow 6.39$; $1.053 \times 120.50 = 126.8865 \rightarrow 126.89$. d) $0.07 \times 14.99 = 1.0493 \rightarrow 1.05$; $1.07 \times 14.99 = 16.0393 \rightarrow 16.04$.
62. $\$261.14$. $1.053 \times 248.00 = 261.144$, which rounds to $\$261.14$. Two-step check: tax $0.053 \times 248 = 13.144 \rightarrow \$13.14$, and $248.00 + 13.14 = 261.14$.
63. Tax $\$0.53$; total $\$9.28$. $0.06 \times 8.75 = 0.525$; the third decimal is a 5, so the cents round up to $\$0.53$. Then $8.75 + 0.53 = 9.28$. Check: $1.06 \times 8.75 = 9.275 \rightarrow \$9.28$.
64. $\$55.18$. $1.053 \times 52.40 = 55.1772$, which rounds to $\$55.18$. Two-step check: tax $0.053 \times 52.40 = 2.7772 \rightarrow \$2.78$, and $52.40 + 2.78 = 55.18$.
65. The price itself is $100\%$ of the price, and the tax adds another $5.3\%$ of that same price on top. Since the total is the price plus the tax, it is $100\% + 5.3\% = 105.3\%$ of the price, so the total can be found in one step as $1.053 \times \text{price}$.
66. Subtotal $\$43.00$; tax $\$2.58$; total $\$45.58$. $2 \times 21.50 = 43.00$, then $0.06 \times 43.00 = 2.58$, then $43.00 + 2.58 = 45.58$. Check: $1.06 \times 43.00 = 45.58$. Tax is charged on the subtotal, so the tickets are added up before the tax is applied.
67. The student found the tax, not the total. The question asks what the purchase costs altogether, so the tax has to be added to the price: $40.00 + 2.12 = 42.12$, and the total is $\$42.12$. Every tax problem in this chapter asks for the resulting total.
68. Estimate about $\$84.00$; exact total $\$83.82$; the estimate is high by $\$0.18$. Estimating with $\$80.00$ and $5\%$ gives a tax of $\$4.00$ and a total of $\$84.00$. Exactly: $0.053 \times 79.60 = 4.2188 \rightarrow \$4.22$, and $79.60 + 4.22 = 83.82$. Two errors partly cancel here: rounding the price up pushed the estimate high, while using $5\%$ instead of $5.3\%$ pushed it low.

Exit ticket 3.4

69. Tax $\$1.59$; total $\$31.59$. $0.053 \times 30.00 = 1.59$ exactly, and $30.00 + 1.59 = 31.59$.
70. $\$47.70$. $1.06 \times 45.00 = 47.70$.
71. $\$13.48$. $1.053 \times 12.80 = 13.4784$, which rounds to $\$13.48$. Two-step check: tax $0.053 \times 12.80 = 0.6784 \rightarrow \$0.68$, and $12.80 + 0.68 = 13.48$.
72. $5\%$ is close to $5.3\%$ — the two rates differ by only three tenths of a percent, which is 3 cents on every $\$10.00$ — and $5\%$ is easy in your head because it is half of $10\%$. Because $5\%$ is less than $5.3\%$, an estimate that uses $5\%$ always gives a tax that is a little low. (The estimated total can still land high overall if the price was rounded up, as in item 68.)

Lesson 3.5 — Tip and the Total

Guided practice

73. Tip $\$6.00$; total $\$46.00$. $10\%$ is $\$4.00$ and $5\%$ is $\$2.00$, so $15\%$ is $\$6.00$; then $40.00 + 6.00 = 46.00$.
74. Tip $\$7.00$; total $\$42.00$. $0.20 \times 35.00 = 7.00$ and $35.00 + 7.00 = 42.00$. Check: $1.20 \times 35.00 = 42.00$.
75. Tip $\$9.00$; total $\$59.00$. $0.18 \times 50.00 = 9.00$ and $50.00 + 9.00 = 59.00$.
76. Tip $\$9.30$; total $\$71.30$. $10\%$ is $\$6.20$ and $5\%$ is $\$3.10$, so the tip is $\$9.30$; then $62.00 + 9.30 = 71.30$.
77. Tip $\$4.92$; total $\$29.52$. $0.20 \times 24.60 = 4.92$ and $24.60 + 4.92 = 29.52$. Check: $1.20 \times 24.60 = 29.52$.
78. About $\$6.00$. $10\%$ of $\$40.00$ is $\$4.00$ and $5\%$ is $\$2.00$. (The exact tip is $0.15 \times 38.40 = 5.76$, so the estimate is high by $\$0.24$.)

Independent practice

79. a) tip $\$8.40$, total $\$64.40$ b) tip $\$16.60$, total $\$99.60$ c) tip $\$4.95$, total $\$32.45$ d) tip $\$7.01$, total $\$53.76$ Work: a) $0.15 \times 56 = 8.40$; $56 + 8.40 = 64.40$. b) $0.20 \times 83 = 16.60$; $83 + 16.60 = 99.60$. c) $0.18 \times 27.50 = 4.95$; $27.50 + 4.95 = 32.45$. d) $0.15 \times 46.75 = 7.0125 \rightarrow \$7.01$; $46.75 + 7.01 = 53.76$. Check on d): $1.15 \times 46.75 = 53.7625 \rightarrow \$53.76$.
80. $\$87.36$. $1.20 \times 72.80 = 87.36$. Two-step check: tip $0.20 \times 72.80 = 14.56$, and $72.80 + 14.56 = 87.36$.
81. Tip $\$3.60$; total $\$23.59$. $0.18 \times 19.99 = 3.5982$, which rounds to $\$3.60$, and $19.99 + 3.60 = 23.59$. Check: $1.18 \times 19.99 = 23.5882 \rightarrow \$23.59$.
82. $\$139.24$. Tip $0.18 \times 118.00 = 21.24$, and $118.00 + 21.24 = 139.24$. Check: $1.18 \times 118.00 = 139.24$.
83. Find $10\%$ by moving the decimal point one place left, then add half of that amount. It works because half of $10\%$ is $5\%$, and $10\% + 5\% = 15\%$. So the two easy pieces add up to exactly the tip you want, with no multiplication needed.
84. Tax $\$3.84$; tip $\$12.80$; total $\$80.64$. $0.06 \times 64.00 = 3.84$ and $0.20 \times 64.00 = 12.80$, so $64.00 + 3.84 + 12.80 = 80.64$. Both percents are taken of the same pre-tax bill of $\$64.00$; neither is applied to the other's result.
85. The $\$9.00$ is the tip, not the payment. The diner still owes the food: the total is $45.00 + 9.00 = 54.00$, so $\$54.00$ should be paid. A tip is added to the bill, so the amount paid must always be larger than the bill.
86. Estimate about $\$60.00$; exact total $\$56.76$; off by $\$3.24$. Rounding $\$47.30$ up to $\$50.00$ moved the bill by $\$2.70$, and then $20\%$ of that extra amount added another $\$0.54$ to the error. Exactly: $0.20 \times 47.30 = 9.46$, and $47.30 + 9.46 = 56.76$, or $1.20 \times 47.30 = 56.76$. This is the estimate in the chapter that lands furthest from the exact answer, and it shows why an estimate is a check rather than an answer.

Exit ticket 3.5

87. Tip $\$3.00$; total $\$23.00$. $10\%$ is $\$2.00$ and $5\%$ is $\$1.00$.
88. Tip $\$8.30$; total $\$49.80$. $0.20 \times 41.50 = 8.30$ and $41.50 + 8.30 = 49.80$. Check: $1.20 \times 41.50 = 49.80$.
89. Tip $\$6.48$; total $\$42.48$. $0.18 \times 36.00 = 6.48$ and $36.00 + 6.48 = 42.48$.
90. Because the bill is $100\%$ of itself and the tip adds more on top of it. A total of bill plus tip is $(100 + r)\%$ of the bill, and $100 + r$ is greater than 100 for any positive tip rate. A total that came out at $100\%$ of the bill or less would mean no tip was left at all.

Lesson 3.6 — Percent Increase and Percent Decrease

Guided practice

91. Increase $\$6.00$; $15\%$ increase. $46 - 40 = 6$ and $\frac{6}{40} = 0.15 = 15\%$.
92. Decrease $\$8.00$; $16\%$ decrease. $50 - 42 = 8$ and $\frac{8}{50} = 0.16 = 16\%$.
93. $20\%$ increase. $30 - 25 = 5$ and $\frac{5}{25} = 0.20$.
94. $25\%$ decrease. $80 - 60 = 20$ and $\frac{20}{80} = 0.25$.
95. $12.5\%$ increase. $13.50 - 12.00 = 1.50$ and $\frac{1.50}{12.00} = 0.125$.
96. A decrease of $25\%$. The amount went down, and $36 - 27 = 9$ with $\frac{9}{36} = 0.25$.

Independent practice

97. a) $30\%$ increase b) $20\%$ decrease c) $12.5\%$ increase d) $12.5\%$ decrease Work: a) change $6$, $\frac{6}{20} = 0.30$. b) change $15$, $\frac{15}{75} = 0.20$. c) change $6$, $\frac{6}{48} = 0.125$. d) change $1.20$, $\frac{1.20}{9.60} = 0.125$. Note that c) and d) have the same percent size in opposite directions, which is a coincidence of these numbers, not a rule.
98. $6\%$ increase. $901 - 850 = 51$ and $\frac{51}{850} = 0.06$.
99. $10\%$ decrease. $240 - 216 = 24$ and $\frac{24}{240} = 0.10$.
100. $10\%$ increase. $49.50 - 45.00 = 4.50$ and $\frac{4.50}{45.00} = 0.10$.
101. $15\%$ decrease. $180 - 153 = 27$ and $\frac{27}{180} = 0.15$.
102. A percent change measures the change against where the quantity started, because "increase" and "decrease" describe movement away from the original amount. With $\$40.00$ rising to $\$50.00$, the change is $\$10.00$. Dividing by the original gives $\frac{10}{40} = 0.25$, a $25\%$ increase, which is the correct answer. Dividing by the new amount gives $\frac{10}{50} = 0.20$, or $20\%$ — a true statement about what fraction of the new price the raise accounts for, but not the percent increase that was asked for.
103. $12\%$ increase. $1400 - 1250 = 150$ and $\frac{150}{1250} = 0.12$. The method is unchanged even though the quantity is tons rather than dollars.
104. The student divided by the new amount, $\$75.00$, instead of the original amount, $\$60.00$. The amount of change, $75 - 60 = 15$, was correct. Dividing correctly gives $\frac{15}{60} = 0.25$, so it is a $25\%$ increase.

Exit ticket 3.6

105. $16\%$ increase. $58 - 50 = 8$ and $\frac{8}{50} = 0.16$.
106. $20\%$ decrease. $120 - 96 = 24$ and $\frac{24}{120} = 0.20$.
107. $15\%$ increase. $18.40 - 16.00 = 2.40$ and $\frac{2.40}{16.00} = 0.15$.
108. "Decreased by $20\%$" means $20\%$ of the original was taken away, so the new price is the remaining $80\%$ of the original. "The new price is $20\%$ of the original" means only a fifth of the original price is left, which is a decrease of $80\%$. On a $\$50.00$ price the first describes a new price of $\$40.00$ and the second describes a new price of $\$10.00$ — very different outcomes from very similar wording.

Chapter 3 Review

Part A — One discount or one markup and the resulting price (8.CE.1a)

109. Discount $\$19.00$; sale price $\$76.00$. $0.20 \times 95.00 = 19.00$ and $95.00 - 19.00 = 76.00$. Check: $0.80 \times 95.00 = 76.00$.
110. $\$31.95$. Discount $0.25 \times 42.60 = 10.65$, and $42.60 - 10.65 = 31.95$. Check: $0.75 \times 42.60 = 31.95$.
111. $\$72.80$. Markup $0.30 \times 56.00 = 16.80$, and $56.00 + 16.80 = 72.80$. Check: $1.30 \times 56.00 = 72.80$.
112. $\$33.93$. Markup $0.45 \times 23.40 = 10.53$, and $23.40 + 10.53 = 33.93$. Check: $1.45 \times 23.40 = 33.93$.
113. Estimate about $\$72.00$; exact sale price $\$71.10$. Rounding $\$118.50$ up to $\$120.00$ gives a discount of $0.40 \times 120 = 48$ and a sale price of about $\$72.00$. Exactly: $0.40 \times 118.50 = 47.40$ and $118.50 - 47.40 = 71.10$, or $0.60 \times 118.50 = 71.10$. The estimate is high because the price was rounded up.
114. Saved $\$10.20$; the coat costs $\$57.80$. $0.15 \times 68.00 = 10.20$ and $68.00 - 10.20 = 57.80$. Check: $0.85 \times 68.00 = 57.80$.

Part B — Sales tax, tip, and the resulting total (8.CE.1b)

115. $\$67.39$. $1.053 \times 64.00 = 67.392$, which rounds to $\$67.39$. Two-step check: tax $0.053 \times 64 = 3.392 \rightarrow \$3.39$, and $64.00 + 3.39 = 67.39$.
116. Tax $\$1.88$; total $\$33.13$. $0.06 \times 31.25 = 1.875$; the third decimal is a 5, so the cents round up to $\$1.88$. Then $31.25 + 1.88 = 33.13$. Check: $1.06 \times 31.25 = 33.125 \rightarrow \$33.13$.
117. $\$63.72$. Tip $0.18 \times 54.00 = 9.72$, and $54.00 + 9.72 = 63.72$. Check: $1.18 \times 54.00 = 63.72$.
118. $\$106.08$. Tip $0.20 \times 88.40 = 17.68$, and $88.40 + 17.68 = 106.08$. Check: $1.20 \times 88.40 = 106.08$.
119. $\$28.18$. Tax $0.01 \times 27.90 = 0.279$, which rounds to $\$0.28$, and $27.90 + 0.28 = 28.18$. Check: $1.01 \times 27.90 = 28.179 \rightarrow \$28.18$.
120. Tax $\$2.74$; tip $\$6.84$; total $\$55.18$. $0.06 \times 45.60 = 2.736$, which rounds to $\$2.74$; $0.15 \times 45.60 = 6.84$ exactly. Then $45.60 + 2.74 + 6.84 = 55.18$. Both percents are taken of the same pre-tax bill.

Part C — Percent increase and percent decrease (8.CE.1c)

121. $15\%$ increase. $92 - 80 = 12$ and $\frac{12}{80} = 0.15$.
122. $20\%$ decrease. $150 - 120 = 30$ and $\frac{30}{150} = 0.20$.
123. $15\%$ increase. $20.70 - 18.00 = 2.70$ and $\frac{2.70}{18.00} = 0.15$.
124. $15\%$ decrease. $320 - 272 = 48$ and $\frac{48}{320} = 0.15$.
125. $12.5\%$ increase. $72 - 64 = 8$ and $\frac{8}{64} = 0.125$.
126. $15\%$ decrease. $96.00 - 81.60 = 14.40$ and $\frac{14.40}{96.00} = 0.15$.

Part D — Mixed application and reasoning

127. Sale price $\$40.50$; total $\$42.65$. First the one discount: $0.25 \times 54.00 = 13.50$ and $54.00 - 13.50 = 40.50$, or $0.75 \times 54.00 = 40.50$. Then the tax on the sale price: $1.053 \times 40.50 = 42.6465$, which rounds to $\$42.65$. Two-step check on the tax: $0.053 \times 40.50 = 2.14665 \rightarrow \$2.15$, and $40.50 + 2.15 = 42.65$. Note that only one discount was applied; the tax is not a second discount or markup.
128. Tip $\$15.70$; total $\$94.20$. $0.20 \times 78.50 = 15.70$ and $78.50 + 15.70 = 94.20$. Check: $1.20 \times 78.50 = 94.20$.
129. A discount takes $35\%$ away, so what is left is $100\% - 35\% = 65\%$ of the price, and $65\% = 0.65$. A markup adds $35\%$ on, so the result is $100\% + 35\% = 135\%$ of the cost, and $135\% = 1.35$. In symbols the two multipliers are $1 - r$ and $1 + r$ for the same rate $r$: one is below 1 and shrinks the amount, the other is above 1 and grows it.
130. The student divided by the new amount, $\$68.00$, instead of the original amount, $\$80.00$. The change of $80 - 68 = 12$ was right. Dividing correctly gives $\frac{12}{80} = 0.15$, so it is a $15\%$ decrease. A quick sanity check: $\$12.00$ is clearly less than a fifth of $\$80.00$, so the answer has to be under $20\%$.
131. LESS than $\$20.00$ is the wrong choice — the item costs more than $\$20.00$. Reasoning without exact computation: round the price to $\$30.00$. Then $30\%$ off is a discount of about $\$9.00$, leaving about $\$21.00$, which is above $\$20.00$. Since the true price of $\$29.90$ is slightly below $\$30.00$, the true discount is slightly below $\$9.00$ too, but not by nearly enough to pull the sale price under $\$20.00$. Exactly: $0.30 \times 29.90 = 8.97$ and $29.90 - 8.97 = 20.93$, or $0.70 \times 29.90 = 20.93$, so the sale price is $\$20.93$.
132. No, they are not equal. Going from $\$40.00$ to $\$50.00$ is a $\frac{10}{40} = 0.25 = 25\%$ increase. Going from $\$50.00$ to $\$40.00$ is a $\frac{10}{50} = 0.20 = 20\%$ decrease. The amount of change is $\$10.00$ in both cases, but the original amount — the denominator — is $\$40.00$ in the first and $\$50.00$ in the second. Dividing the same change by a larger number gives a smaller percent, so a rise and the matching fall are never the same percent.

Every item in this chapter applies exactly one discount or one markup, as 8.CE.1a requires; no item chains two discounts or combines a discount with a markup. Every tax item and every tip item asks for the resulting total, as 8.CE.1b requires. All dollar answers above are rounded to the nearest cent.