Appendix A — Answer Key, Chapter 6: Percent of a Number

SOL 7.CE.2 (d) · Covers textbook Chapter 6 and the companion workbook. Item numbers match the textbook and run continuously from 1 to 95; workbook items are the same problems with the same numbers, so this key serves both books. Reasoning answers show an acceptable response, not the only wording. Money answers are written to two decimal places.

Lesson 6.1 — Benchmark Percentages

Guided practice

1. $43$
2. $30$
3. $7.4$
4. $0.74$
5. $10\%$ of 60 is $6$, so $5\%$ of 60 is $3$.
6. $50\%$ of 200 is $100$ and $25\%$ of 200 is $50$, so $75\%$ of 200 is $150$.

Independent practice

7. a) $125$ b) $22$ c) $450$ d) $6.2$
8. a) $7$ b) $36$ c) $39$ d) $90$
9. $10\%$ of 96 is $9.6$; doubling gives $20\%$ of 96 = $19.2$.
10. $25\%$ of 160 is $40$; $10\%$ of 380 is $38$. So $25\%$ of 160 is greater.
11. Finding $10\%$ means dividing by 10, which shifts the decimal point one place to the left. So $730$ becomes $73$. The answer is $73$.

| Percent | $50\%$ | $25\%$ | $10\%$ | $5\%$ | $1\%$ | $75\%$ |
|---|---|---|---|---|---|---|
| Amount of 600 | $300$ | $150$ | $60$ | $30$ | $6$ | $450$ |

13. $10\%$ tip: $4.00. $20\%$ tip: $8.00.
14. Because $5\%$ is half of $10\%$ as a percent, and taking half of a percent takes half of the amount it produces. In symbols, $5\% = \tfrac{1}{2} \times 10\%$, so $5\%$ of $n$ is half of $10\%$ of $n$ for every number $n$.

Exit ticket 6.1

15. $46$
16. $83$
17. $8.3$
18. The answer is $12$. Way 1: divide 48 by 4, since $25\%$ is one quarter. Way 2: halve 48 to get 24 ($50\%$), then halve again to get 12.

Lesson 6.2 — Estimating a Percent of a Number

Estimates below are the values produced by the benchmark named in each problem. Exact values are given where they help a teacher judge closeness.

Guided practice

19. $50\%$ of 80 = $40$, so about $40$. (Exact: $40.56$.)
20. $25\%$ of 40 = $10$, so about $10$. (Exact: $9.84$.)
21. $10\%$ of 90 = $9$, so about $9$. (Exact: $9.79$.)
22. $75\%$ of 60 = $45$, so about $45$. (Exact: $46.36$.)
23. $10\%$ of 400 = $40$, so about $40$. (Exact: $37.08$.)
24. $50\%$ of 300 = $150$, so about $150$. (Exact: $146.4$.)

Independent practice

25. a) about $30$ b) about $20$ c) about $5$ d) about $60$
26. $90$. Benchmark: $30\%$ of 300, since $10\%$ of 300 is 30 and three of those is 90. (Exact: $92.07$.)
27. $24$. Benchmark: $5\%$ of 400 is 20, and $6\%$ is a little more than $5\%$, so the answer is a little more than 20. Of the listed values, 24 is the closest. (Exact: $23.76$.)
28. About $200$. The estimate is higher than the exact answer, because the percent was rounded up from $48\%$ to $50\%$ and the whole was rounded up from 398 to 400. Both changes make the result larger, so the estimate must be too big. (Exact: $191.04$.)
29. $25\%$ of 60 is $15$ and $10\%$ of 60 is $6$, so about $21$. (Exact: $21.35$.)
30. $20\%$ of $60 is $12, so about $12. (Exact: $11.80.)
31. Friendly whole: 400 students. $25\%$ of 400 = $100$, so about 100 students. (Exact: $103$.)
32. Her estimate is higher than the exact answer. She used $10\%$, but the true percent is $9\%$, which is less than $10\%$, so the exact amount must be less than 24. (Exact: $21.6$.)

Exit ticket 6.2

33. $50\%$ of 80 = $40$, so about $40$. (Exact: $39.78$.)
34. $25\%$ of 120 = $30$, so about $30$. (Exact: $29.04$.)
35. $50$. ($10\%$ of 500 is 50, and $11\%$ of 508 is $55.88$.)
36. Taking the same percent of a larger whole always produces a larger amount. If the percent stays fixed and the whole grows, every piece of it grows too, so the estimate lands above the exact answer.

Lesson 6.3 — Finding an Exact Percent of a Number

Guided practice

37. $9$
38. $10\%$ of 60 is $6$ and $5\%$ of 60 is $3$, so $15\%$ of 60 is $9$.
39. $8\% = 0.08$, and $0.08 \times 250 = 20$.
40. $36$
41. $25\%$ of 140 is $35$ and $10\%$ of 140 is $14$, so $35\%$ of 140 is $49$.
42. $99$

Independent practice

43. a) $21$ b) $36$ c) $26$ d) $9$
44. a) $66$ b) $51$ c) $72$ d) $17$
45. $50\%$ of 240 = $120$, $10\%$ of 240 = $24$, $5\%$ of 240 = $12$. Then $65\% = 50\% + 10\% + 5\%$, so the total is $120 + 24 + 12 = 156$.
46. $0.18 \times 950 = 171$
47. $\dfrac{x}{175} = \dfrac{24}{100}$. Cross products: $100x = 24 \times 175 = 4200$, so $x = 42$.
48. $3$
49. $0.45 \times 640 = 288$ blue T-shirts.
50. $25\%$ means one quarter, and one quarter of a number is that number divided by 4. So $25\%$ of 84 and $84 \div 4$ are the same computation written two ways; both give $21$.

Exit ticket 6.3

51. $26$
52. $33$
53. $65$
54. $21$ both ways. Way 1 (decimal): $0.35 \times 60 = 21$. Way 2 (benchmarks): $25\%$ of 60 is $15$ and $10\%$ of 60 is $6$, and $15 + 6 = 21$.

Lesson 6.4 — Percent in Context

Guided practice

55. $3.60
56. $11.00 ($10\%$ of 55 is $5.50; doubled, $11.00.)
57. $12.00
58. $16.00
59. $100$ students

Independent practice

60. $12.00 ($10\%$ of 80 is $8.00, $5\%$ is $4.00, and $\$8.00 + \$4.00 = \$12.00$.)
61. $10.50
62. Discount $42.00; sale price $78.00
63. $26$ free throws
64. $100$ megabytes
65. Tip $8.64; total $56.64
66. Estimate: $20\%$ of 350 = $70$, so about 70 seats. Exact: $0.22 \times 350 = 77$ seats.
67. Taking $20\%$ off means keeping the other $80\%$, because the whole price is $100\%$ and $100\% - 20\% = 80\%$. Both computations: $20\%$ of $60 is $12, so $\$60 - \$12 = \$48$; and $80\%$ of $60 is $0.80 \times 60 = \$48$.

Exit ticket 6.4

68. $13.00
69. $15.00
70. $42.00
71. Tip: $6.00. Strategy: $10\%$ of $40 is $4.00; half of that is $5\%$, which is $2.00; adding gives $15\%$ = $6.00.

Chapter 6 Review

Part A — Benchmark percentages (7.CE.2d)

72. $74$
73. $80$
74. $7.6$
75. $0.76$
76. $9$
77. $66$

Part B — Estimating a percent of a number (7.CE.2d)

78. $50\%$ of 120 = $60$, so about $60$. (Exact: $59.78$.)
79. $25\%$ of 80 = $20$, so about $20$. (Exact: $20.54$.)
80. $10\%$ of 300 = $30$, so about $30$. (Exact: $27.45$.)
81. $75\%$ of 40 = $30$, so about $30$. (Exact: $30.34$.)
82. $100$. Benchmark: $10\%$ of 300 is 30, so $30\%$ of 300 is 90 and $35\%$ of 300 is 105. Since $34\%$ sits between those, the closest listed value is 100. (Exact: $101.32$.)
83. A high estimate. $10\%$ of 480 is $48$, but $9\%$ of 480 is $43.2$, and $9\%$ is less than $10\%$.

Part C — Determining an exact percent (7.CE.2d)

84. $45$
85. $51$
86. $52$
87. $15$
88. $84$
89. $4$

Part D — Percent in context (7.CE.2d)

90. $10.80
91. Tax $2.70; total $47.70
92. Discount $38.00; sale price $57.00

Part E — Mixed application and reasoning

93. $25\%$ of 260 = $65$ and $10\%$ of 260 = $26$, so $35\%$ of 260 = $65 + 26 = 91$. (Check: $0.35 \times 260 = 91$.)
94. Estimate: $50\%$ of 250 = $125$. Exact: $0.48 \times 250 = 120$. The estimate is high because $48\%$ was rounded up to $50\%$ while the whole was left alone, so the friendly problem takes a slightly larger share of the same 250.
95. Finding $10\%$ means dividing by 10, and finding $1\%$ means dividing by 100. In our place-value system, dividing by 10 shifts the decimal point one place to the left and dividing by 100 shifts it two places. So $10\%$ of 640 is $64$ and $1\%$ of 640 is $6.4$.

Workbook-only items

Page 2, fill in the blanks. $\%$ means per hundred. $35\%$ means 35 out of every 100. $35\% = \tfrac{35}{100} = 0.35$. Each small square is 1% of the grid. You must also know the whole.

Page 2, three ways table. $50\% = \tfrac{50}{100} = 0.5$; $25\% = \tfrac{25}{100} = 0.25$; $10\% = \tfrac{10}{100} = 0.1$; $1\% = \tfrac{1}{100} = 0.01$; $75\% = \tfrac{75}{100} = 0.75$.

Page 3, benchmark table. $50\%$: divide by 2. $25\%$: divide by 4. $75\%$: $50\% +$ 25%. $10\%$: shift one place left. $5\%$: find $10\%$, then halve it. $1\%$: shift two places left. Amounts for a whole of 200: $100\%$ → $200$; $50\%$ → $100$; $25\%$ → $50$; $75\%$ → $150$; $10\%$ → $20$; $5\%$ → $10$; $1\%$ → $2$.

Page 4, decompositions. $35\% = 25\% +$ 10%. $15\% = 10\% +$ 5%. $20\% = 10\% +$ 10%. $60\% = 50\% +$ 10%.

Page 4, over 100 and under 1. $200\%$ of a number is 2 times the number. $150\%$ of $48 = 48 + 24 = 72$. $0.5\%$ is half of $1\%$, so $0.5\%$ of 400 = $2$.

Page 4, double number line. $30\%$ is $18$ students; $70\%$ is $42$ students; $100\%$ is $60$ students.

Page 7, two decisions. Round the percent to a nearby benchmark; round the whole to a friendly number.

Page 7, rounding table. $48\% \to 50\%$; $26\% \to 25\%$; $9\% \to 10\%$; $73\% \to 75\%$; $52\% \to 50\%$; $11\% \to 10\%$.

Page 7, symbol. $\approx$ is read "is approximately equal to."

Page 7, high or low. Percent up → higher. Percent down → lower. Whole up → higher. Whole down → lower.

Page 11, four methods. Method 1: $18\% = 0.18$ and $0.18 \times 250 = 45$. Method 2: $45$. Method 3: $100x = 4500$, so $x = 45$. Method 4: $10\% = 25$, $5\% = 12.5$, $1\% = 2.5$, and $25 + 12.5 + 2.5 + 2.5 + 2.5 = 45$.

Page 11, decimal table. $45\% = 0.45$; $8\% = 0.08$; $120\% = 1.2$; $0.5\% = 0.005$.

Page 12, check yourself. Every answer in items 37–41 is smaller than its whole, because each percent is less than $100\%$. Item 42 is larger than its whole ($99 > 90$) because $110\%$ is more than $100\%$.

Page 15, matching. tip → a percent of a bill added for service. sales tax → a percent of a price added by law. discount → an amount subtracted from a price. sale price → the original price minus the discount.

Page 15, five steps. 1. Name the whole. 2. Name the percent. 3. Estimate with a benchmark. 4. Compute exactly. 5. Answer the question that was asked.

Page 15, benchmark tip. $10\%$ of $40 = $4.00. $20\%$: find $10\%$ and double it → $8.00. $15\%$: $10\%$ = $4.00 plus $5\%$ = $2.00 → $6.00.

Page 16, careful. Item 58 asks for the discount, which is $16.00. (The sale price would be $48.00, but that is not what was asked.)