Appendix A — Answer Key, Chapter 5: Ratio Tables and Proportions

SOL 7.CE.2 (a–c) · Covers textbook Chapter 5 and the companion workbook. Item numbers match the textbook; workbook items are the same problems, so this key serves both. Proportion answers show the setup and the solving step, not just the result. Reasoning answers show an acceptable response, not the only wording.

Lesson 5.1 — Ratio Tables and Missing Values

Guided practice

1. Unit rate $9 per ticket.

   Tickets	1	2	3	5	10
   Cost ($)	9	18	27	45	90

2. $24 \div 3 = 8$ pages per minute.

   Minutes	1	3	6	9	12
   Pages	8	24	48	72	96

3. $5$. Check: $4 \times 5 = 20$ hours and $26 \times 5 = 130$ miles.

4. $21 \div 7 = \$3$ per pound, so 4 pounds cost $4 \times 3 = \$12$.

5. $(6, 24)$ comes from $(3, 12)$ by multiplying both entries by 2, so the ratio $\tfrac{3}{12} = \tfrac{6}{24} = \tfrac14$ is unchanged. To get $(6, 15)$ you would have to multiply the top by 2 and the bottom by only 1.25, which changes the ratio: $\tfrac{6}{15} = \tfrac25$, not $\tfrac14$.

Independent practice

6. Unit rate 35 square feet per gallon.

   Gallons	1	2	3	5	8
   Square feet	35	70	105	175	280

7. $4.50 \div 6 = \$0.75$ per bottle.

   Bottles	1	4	6	10	12
   Cost ($)	0.75	3.00	4.50	7.50	9.00

8. Divide both entries by 2, then by 3, then by 6 from the first column.

   Sandwiches	12	6	4	2
   People	18	9	6	3

9. The wrong entry is 28, under 12 bags. The other columns all give $10 \div 4 = 2.5$, $20 \div 8 = 2.5$, and $40 \div 16 = 2.5$ apples per bag, but $28 \div 12 \approx 2.33$. The correct value is $12 \times 2.5 = 30$.

10. $210 \div 3 = 70$ beats per minute.

    Minutes	1	3	5	10
    Beats	70	210	350	700

11. $180 \div 6 = 30$ miles per gallon.

    Gallons	1	6	11
    Miles	30	180	330

    The car can travel 330 miles on 11 gallons.

12. Devon's columns are $(2, 6)$, $(5, 9)$, and $(8, 12)$. The ratios are $6 \div 2 = 3$, $9 \div 5 = 1.8$, and $12 \div 8 = 1.5$, so the comparison changes from column to column and the table is not a ratio table. Ratios are preserved by multiplying, not by adding. The correct third column, scaling $(2, 6)$ by 4, is $(8, 24)$.

Exit ticket 5.1

13. $10 \div 4 = \$2.50$ per notebook.

    Notebooks	1	4	6
    Cost ($)	2.50	10.00	15.00

14. $4$, since $3 \times 4 = 12$ and $21 \times 4 = 84$.

15. $8 \div 2 = 4$ liters per minute, so $7 \times 4 = 28$ liters.

16. The ratio between the two quantities. Equivalently, the unit rate — how much of the second quantity goes with one of the first — is the same in every column.

Lesson 5.2 — Writing a Proportion

Guided practice

17. $\dfrac{3 \text{ apples}}{2.10 \text{ dollars}} = \dfrac{8 \text{ apples}}{c \text{ dollars}}$. The flipped version $\dfrac{2.10}{3} = \dfrac{c}{8}$ is equally correct.
18. $\dfrac{5 \text{ minutes}}{350 \text{ meters}} = \dfrac{12 \text{ minutes}}{d \text{ meters}}$, or the flipped version $\dfrac{350}{5} = \dfrac{d}{12}$.
19. $4 \times 21 = 84$ and $6 \times 14 = 84$. The cross products are equal, so yes, it is a proportion.
20. $3 \times 30 = 90$ and $8 \times 12 = 96$. Since $90 \neq 96$, it is not a proportion.
21. $t$ is the number of tiles needed to cover 40 square feet.

Independent practice

22. a) $6 \times 15 = 90$ and $9 \times 10 = 90$ — yes. b) $5 \times 35 = 175$ and $12 \times 15 = 180$ — no. c) $7 \times 12 = 84$ and $4 \times 21 = 84$ — yes.
23. $\dfrac{9 \text{ tiles}}{12 \text{ sq ft}} = \dfrac{t}{40 \text{ sq ft}}$
24. $\dfrac{4 \text{ pounds}}{10 \text{ days}} = \dfrac{15 \text{ pounds}}{d}$ and $\dfrac{10 \text{ days}}{4 \text{ pounds}} = \dfrac{d}{15 \text{ pounds}}$. Flipping both ratios at once keeps the equation true.
25. $\dfrac{2 \text{ inches}}{35 \text{ miles}} = \dfrac{7 \text{ inches}}{m \text{ miles}}$
26. The units do not line up. On the left, hours are on top and miles on the bottom; on the right, miles are on top. A correct setup is $\dfrac{6 \text{ hours}}{270 \text{ miles}} = \dfrac{h \text{ hours}}{450 \text{ miles}}$.
27. $\dfrac{15 \text{ ounces}}{3.60 \text{ dollars}} = \dfrac{25 \text{ ounces}}{c \text{ dollars}}$. A real store often prices the larger box at less per ounce to encourage buying the bigger size, so the actual price is usually lower than the proportional prediction.
28. The second equation is the first with both ratios flipped, and flipping both sides of a true equation keeps it true. Every pair of quantities compared is the same pair, just written the other way around. They give the same value, $x = 12$.

Exit ticket 5.2

29. $8 \times 35 = 280$ and $14 \times 20 = 280$. Equal, so yes.
30. $\dfrac{7 \text{ buses}}{336 \text{ students}} = \dfrac{10 \text{ buses}}{s \text{ students}}$
31. b) $\dfrac{3}{240} = \dfrac{g}{400}$, which keeps gallons on top and square feet on the bottom on both sides.
32. A proportion says two ratios are equal, and a ratio only means something once you know which quantity is being compared to which. If gallons are on top on one side and on the bottom on the other, the two fractions are comparing different things, so setting them equal describes a relationship that is not the one in the problem. The arithmetic still produces a number, which is what makes the error dangerous.

Lesson 5.3 — Solving a Proportion

Guided practice

33. $\dfrac{x}{4} = \dfrac{15}{12}$ → $12x = 4 \times 15 = 60$ → $x = 60 \div 12 = 5$. Check: $5 \times 12 = 60$ and $4 \times 15 = 60$.
34. $\dfrac{3}{7} = \dfrac{12}{y}$ → $3y = 7 \times 12 = 84$ → $y = 84 \div 3 = 28$.
35. $\dfrac{n}{9} = \dfrac{8}{6}$ → $6n = 9 \times 8 = 72$ → $n = 72 \div 6 = 12$.
36. $\dfrac{5}{8} = \dfrac{m}{40}$ → $8m = 5 \times 40 = 200$ → $m = 200 \div 8 = 25$.
37. $\dfrac{2.5}{1} = \dfrac{w}{6}$ → $w = 2.5 \times 6 = 15$.

Independent practice

38. a) $\dfrac{6}{x} = \dfrac{9}{15}$ → $9x = 6 \times 15 = 90$ → $x = 10$. b) $\dfrac{a}{12} = \dfrac{7}{4}$ → $4a = 12 \times 7 = 84$ → $a = 21$. c) $\dfrac{14}{21} = \dfrac{10}{b}$ → $14b = 21 \times 10 = 210$ → $b = 15$.
39. By scaling: $45 \div 5 = 9$, so multiply the numerator by 9 as well: $t = 4 \times 9 = 36$. By cross multiplication: $5t = 4 \times 45 = 180$, so $t = 180 \div 5 = 36$.
40. $\dfrac{9}{x} = \dfrac{6}{8}$ → $6x = 9 \times 8 = 72$ → $x = 12$.
41. $\dfrac{5}{7} = \dfrac{9}{k}$ → $5k = 7 \times 9 = 63$ → $k = 63 \div 5 = 12.6$ exactly.
42. $\dfrac{3 \text{ pounds}}{5.40 \text{ dollars}} = \dfrac{7 \text{ pounds}}{c}$ → $3c = 5.40 \times 7 = 37.80$ → $c = 37.80 \div 3 = 12.60$. The grapes cost $12.60. Check with the unit price: $5.40 \div 3 = \$1.80$ per pound and $7 \times 1.80 = \$12.60$.
43. $\dfrac{2 \text{ minutes}}{84 \text{ words}} = \dfrac{5 \text{ minutes}}{w}$ → $2w = 84 \times 5 = 420$ → $w = 210$ words.
44. Begin with $\dfrac{a}{b} = \dfrac{c}{d}$, where $b$ and $d$ are not zero. Multiplying both sides of a true equation by the same nonzero number keeps it true, so multiply both sides by $b \times d$. On the left, $\tfrac{a}{b} \times b \times d$ cancels the $b$'s and leaves $a \times d$. On the right, $\tfrac{c}{d} \times b \times d$ cancels the $d$'s and leaves $b \times c$. That gives $a \times d = b \times c$, which is exactly the pair of cross products. So cross multiplication is not a separate rule; it is one legal multiplication with the cancelling already carried out.

Exit ticket 5.3

45. $\dfrac{x}{6} = \dfrac{20}{15}$ → $15x = 6 \times 20 = 120$ → $x = 8$.
46. $\dfrac{7}{4} = \dfrac{35}{y}$ → $7y = 4 \times 35 = 140$ → $y = 20$.
47. $\dfrac{5 \text{ tickets}}{62.50 \text{ dollars}} = \dfrac{8 \text{ tickets}}{c}$ → $5c = 62.50 \times 8 = 500$ → $c = 100$. Eight tickets cost $100.
48. Any of the following. Scaling: $20 \div 4 = 5$, so $x = 3 \times 5 = 15$. Unit rate: $3 \div 4 = 0.75$, so $x = 0.75 \times 20 = 15$. Equivalent fractions: $\tfrac34 = \tfrac{15}{20}$ because both simplify to $\tfrac34$. Each gives $x = 15$.

Lesson 5.4 — Converting Units with a Given Conversion Factor

Guided practice

49. $\dfrac{1 \text{ mile}}{5{,}280 \text{ feet}} = \dfrac{3 \text{ miles}}{f}$ → $f = 5{,}280 \times 3 = 15{,}840$ feet.
50. $\dfrac{1 \text{ kg}}{2.2 \text{ lb}} = \dfrac{15 \text{ kg}}{p}$ → $p = 2.2 \times 15 = 33$ pounds.
51. $\dfrac{1 \text{ gallon}}{4 \text{ quarts}} = \dfrac{g}{26 \text{ quarts}}$ → $4g = 26$ → $g = 6.5$ gallons.
52. $\dfrac{1 \text{ inch}}{2.54 \text{ cm}} = \dfrac{12 \text{ inches}}{c}$ → $c = 2.54 \times 12 = 30.48$ centimeters.
53. $\dfrac{1 \text{ hour}}{3{,}600 \text{ s}} = \dfrac{2.5 \text{ hours}}{s}$ → $s = 3{,}600 \times 2.5 = 9{,}000$ seconds.

Independent practice

54. a) $3y = 21$, so $y = 7$ yards. (Feet to yards is small to big, so the number gets smaller.) b) $16 \times 5.5 = 88$ ounces. c) $2{,}450 \div 1{,}000 = 2.45$ liters.
55. $1.609 \times 8 = 12.872$, which rounds to $12.9$ kilometers.
56. $60 \div 8 = 7.5$ cups.
57. $100 \times 3.4 = 340$ centimeters.
58. $453.6 \times 2.5 = 1{,}134$ grams.
59. $1.08 \times 250 = 270$, so the student receives $270.
60. Feet are larger than inches, so converting inches to feet must give a smaller number, not a much larger one. Getting 576 from 48 means the classmate multiplied by 12 when the setup called for dividing. The correct answer is $48 \div 12 = 4$ feet.

Exit ticket 5.4

61. $12 \times 7.5 = 90$ inches.
62. $4{,}750 \div 1{,}000 = 4.75$ kilograms.
63. $3.785 \times 6 = 22.71$, which rounds to $22.7$ liters.
64. Comparing the sizes of the two units tells you in advance which direction the number must move: more of a smaller unit, fewer of a larger one. If your answer moves the other way, you multiplied where you should have divided, which means the two ratios were written with the units in mismatched positions. It is a one-second check that catches the most common setup error.

Lesson 5.5 — Proportional Reasoning in Context

Guided practice

65. $3.00 \div 12 = \$0.25$ per ounce; $4.60 \div 20 = \$0.23$ per ounce. The 20-ounce jar is the better buy.
66. $\dfrac{6 \text{ servings}}{2.5 \text{ cups}} = \dfrac{15 \text{ servings}}{b}$ → $6b = 2.5 \times 15 = 37.5$ → $b = 6.25$ cups.
67. $\dfrac{1 \text{ inch}}{12 \text{ feet}} = \dfrac{4.5 \text{ inches}}{L}$ → $L = 12 \times 4.5 = 54$ feet.
68. $150 \div 3 = 50$ miles per hour, so $7 \times 50 = 350$ miles.
69. $1 + 3 = 4$ equal parts; one part is $24 \div 4 = 6$ stickers. The shares are $1 \times 6 = 6$ and $3 \times 6 = 18$. Check: $6 + 18 = 24$.

Independent practice

70. A: $2.50 \div 10 = \$0.25$ per ounce. B: $3.84 \div 16 = \$0.24$ per ounce. C: $5.75 \div 25 = \$0.23$ per ounce. Least to greatest cost per ounce: C ($0.23), B ($0.24), A ($0.25).

71. $\dfrac{8 \text{ gallons}}{2{,}000 \text{ sq ft}} = \dfrac{x}{4{,}500 \text{ sq ft}}$ → $2{,}000x = 8 \times 4{,}500 = 36{,}000$ → $x = 18$ gallons.

72. $3 + 5 = 8$ equal parts; one part is $40 \div 8 = 5$ marbles. Blue: $3 \times 5 = 15$. Green: $5 \times 5 = 25$. The ratio holds because $15 + 25 = 40$ and $\tfrac{15}{25}$ simplifies to $\tfrac{3}{5}$.

73. Unit rate $0.12 per page.

    Pages	10	25	100
    Cost ($)	1.20	3.00	12.00

    The graph would be a straight line passing through the origin, because zero pages cost zero dollars and every additional page adds the same $0.12.

74. $42 \div 3 = 14$ kilometers per hour, so in 5 hours the cyclist rides $5 \times 14 = 70$ kilometers. Converting, $70 \times 0.621 = 43.47$, which rounds to $43.5$ miles.

75. $18 \div 2 = 9$ inches per year, so in 8 years the tree grows $8 \times 9 = 72$ inches. Converting, $72 \div 12 = 6$ feet.

76. For 10 GB the cost is $20 + 10 \times 0.10 = \$21$. For 20 GB it is $20 + 20 \times 0.10 = \$22$. Doubling the data would have to double the cost for the relationship to be proportional, and $2 \times 21 = 42$, not 22. The $20 monthly base charge is the problem: it is paid once regardless of data used, so it does not scale with the gigabytes. Graphed, the line would cross the vertical axis at 20 instead of passing through the origin.

Exit ticket 5.5

77. $4.50 \div 6 = \$0.75$ per juice; $7.00 \div 10 = \$0.70$ per juice. The 10-pack is the better buy.
78. $\dfrac{4 \text{ servings}}{3 \text{ cups}} = \dfrac{10 \text{ servings}}{r}$ → $4r = 3 \times 10 = 30$ → $r = 7.5$ cups.
79. $\dfrac{1 \text{ inch}}{8 \text{ feet}} = \dfrac{2.5 \text{ inches}}{L}$ → $L = 8 \times 2.5 = 20$ feet.
80. The $40 visit fee is charged once no matter how long the work takes. A 2-hour visit costs $40 + 24 = \$64$ and a 4-hour visit costs $40 + 48 = \$88$, but a proportional relationship would require the 4-hour cost to be $2 \times 64 = \$128$. Only the hourly part scales; the flat fee does not, so the total is not proportional to the hours.

Chapter 5 Review

Part A — Ratio tables and missing values (7.CE.2a)

81. Unit rate $2.75 per muffin.

    Muffins	1	4	6	12
    Cost ($)	2.75	11.00	16.50	33.00

82. $96 \div 4 = 24$ envelopes per minute.

    Minutes	1	4	7	10
    Envelopes	24	96	168	240

83. The wrong entry is 88, under 8 hours. Every other column gives 12 pages per hour ($24 \div 2$, $60 \div 5$, $120 \div 10$), but $88 \div 8 = 11$. The correct value is $8 \times 12 = 96$.

84. $3$, since $6 \times 3 = 18$ and $15 \times 3 = 45$.

Part B — Writing and solving proportions (7.CE.2b)

85. $9 \times 35 = 315$ and $15 \times 21 = 315$. The cross products are equal, so yes.
86. a) $\dfrac{x}{8} = \dfrac{21}{24}$ → $24x = 8 \times 21 = 168$ → $x = 7$. b) $\dfrac{5}{6} = \dfrac{n}{42}$ → $6n = 5 \times 42 = 210$ → $n = 35$. c) $\dfrac{12}{p} = \dfrac{8}{10}$ → $8p = 12 \times 10 = 120$ → $p = 15$.
87. $\dfrac{4 \text{ pounds}}{7.00 \text{ dollars}} = \dfrac{10 \text{ pounds}}{c}$ → $4c = 7.00 \times 10 = 70$ → $c = 17.50$. Ten pounds cost $17.50.
88. On the left, cups are on top and servings on the bottom. On the right, $s$ is a number of servings and it sits on top, while the 10 cups sit on the bottom — the two units are swapped. A correct proportion is $\dfrac{4 \text{ cups}}{6 \text{ servings}} = \dfrac{10 \text{ cups}}{s \text{ servings}}$, which gives $4s = 60$ and $s = 15$ servings.
89. $\dfrac{7}{2} = \dfrac{x}{9}$ → $2x = 7 \times 9 = 63$ → $x = 31.5$ exactly.

Part C — Converting units with a given factor (7.CE.2c)

90. $16 \times 3.75 = 60$ ounces.
91. $3.28 \times 25 = 82$ feet.
92. $34 \div 4 = 8.5$ gallons.
93. $0.621 \times 12 = 7.452$, which rounds to $7.5$ miles.

Part D — Mixed application and reasoning

94. $4.05 \div 9 = \$0.45$ per ounce; $6.60 \div 15 = \$0.44$ per ounce. The 15-ounce bag is the better buy.
95. Two hours is $2 \times 60 = 120$ minutes. Then $\dfrac{3 \text{ miles}}{50 \text{ minutes}} = \dfrac{m}{120 \text{ minutes}}$ → $50m = 3 \times 120 = 360$ → $m = 7.2$ miles.
96. The second service is proportional: the cost is always $9 times the number of months, so zero months costs zero dollars and doubling the months doubles the cost. Graphed, it is a straight line through the origin with a rise of 9 dollars per month. The first service is not proportional, because the $10 signup fee is paid once regardless of how many months you subscribe. One month costs $18 and two months cost $26, not $36. Graphed, it is still a straight line, but it crosses the vertical axis at 10 rather than passing through the origin.

Workbook-only items

Page 2, fill in the blanks. A ratio compares two quantities. In a proportional relationship, one quantity is always the same fixed number of times the other. To move between columns you multiply or divide both entries by the same number. You may not add the same number to both entries: adding 4 to both entries of $(1, 9)$ gives $(5, 13)$, but $9 \times 5 = 45$, so the ratio changed. A unit rate is a rate whose second quantity is 1. One bottle costs $4.50 \div 6 = \$0.75$.

Page 7, label the units. The blank is dollars.

Page 7, correct or incorrect. $\tfrac{3}{5.40} = \tfrac{7}{c}$ — correct. $\tfrac{5.40}{3} = \tfrac{c}{7}$ — correct (both ratios flipped). $\tfrac{3}{5.40} = \tfrac{c}{7}$ — incorrect (pounds and dollars swapped on the right).

Page 8, why it works. Multiply both sides by $b$ $\times$ $d$. The left leaves $a \times d$; the right leaves $b \times c$. So $a \times d =$ $b$ $\times$ $c$.

Page 12, four strategies for $\tfrac{x}{4} = \tfrac{15}{12}$. Equivalent fractions: $\tfrac{15}{12} = \tfrac54$, so $x = 5$. Scaling: $12 \div 4 = 3$ and $15 \div 3 = 5$, so $x = 5$. Unit rate: $15 \div 12 = 1.25$ and $4 \times 1.25 = 5$. Cross multiply: $12x = 60$, so $x = 5$.

Page 12, four steps. 1. Multiply each numerator by the other denominator. 2. Set the two products equal. 3. Divide both sides by the number in front of the variable. 4. Check your answer by comparing cross products.

Page 16, direction check. Big unit to small unit gives more of the smaller unit. Small unit to big unit gives fewer of the bigger unit. A conversion factor is a ratio.