Appendix A — Answer Key, Chapter 8: Ratios

SOL 6.PFA.1 · Covers textbook Chapter 8 and the companion workbook. Item numbers match the textbook; workbook items that repeat textbook problems share the same answers, and workbook-only items are keyed at the end. Ratios are given in the notation the item asks for; an equivalent notation is acceptable unless the item names one. Reasoning and situation-writing answers show an acceptable response, not the only wording.

Lesson 8.1 — What a Ratio Is

Guided practice

1. $6$ to $9$
2. $9$ to $6$
3. $2$ to $5$
4. "Three more oranges than apples" tells how far apart the two counts are; you find it by subtracting. "The ratio of apples to oranges is 4 to 7" tells how the counts pair up — 4 apples for every 7 oranges — and it stays true for a bigger bowl with 8 apples and 14 oranges, where the difference would be 6 instead of 3.
5. $30$ to $5$

Independent practice

6. a) $7$ to $4$ b) $4$ to $7$
7. $11$ to $5$
8. $3$ to $4$
9. For every 9 tomato plants in the garden there are 2 pepper plants.
10. Possible quantities: riders and miles, or riders and trips, or miles and trips. Sample ratio: riders to miles is $48$ to $6$.
11. The cook reversed the two amounts, using 5 cups of concentrate and 2 cups of water instead of 5 cups of water and 2 cups of concentrate. The drink will taste far too strong, because there is more concentrate than water instead of the other way around.
12. The order tells you which quantity each number counts. If a shelter has 3 cats for every 8 dogs, then $3$ to $8$ is correct and $8$ to $3$ would claim there are 8 cats for every 3 dogs — a completely different shelter. Same numbers, different meaning.

Exit ticket 8.1

1. $9$ to $12$
2. $12$ to $9$
3. $8$ to $20$
4. The first number always counts the first quantity named. Switching the order switches which quantity each number describes, so it describes a different relationship.

Lesson 8.2 — Three Ways to Write a Ratio

Guided practice

1. $\frac{6}{11}$, $6:11$, 6 to 11
2. $\frac{12}{5}$
3. $8:3$ and 8 to 3
4. $2:5$
5. The 7 counts cats and the 10 counts dogs.

Independent practice

6. a) $\frac{9}{13}$, $9:13$, 9 to 13 b) $\frac{4}{1}$, $4:1$, 4 to 1
7. a) $\frac{6}{7}$ and 6 to 7 b) $11:4$ and 11 to 4 c) $\frac{15}{8}$ and $15:8$
8. a) $3:8$ b) $8:3$ c) $3:11$ (there are $3 + 8 = 11$ writing tools)
9. $\frac{5}{7}$ compares blue tiles to yellow tiles, a part-to-part comparison. $\frac{5}{12}$ compares blue tiles to all 12 tiles, a part-to-whole comparison, since $5 + 7 = 12$.
10. $\frac{180}{3}$, $180:3$, 180 to 3
11. Colon form: $1:3$. In words: for every 1 part of cement, use 3 parts of sand. With 2 parts of cement you need 6 parts of sand, since $1 \times 2 = 2$ and $3 \times 2 = 6$.
12. The 9 counts cashews, not the whole mix. The mix has $2 + 9 = 11$ parts, so almonds are $\frac{2}{11}$ of the mix.

Exit ticket 8.2

1. $\frac{6}{5}$, $6:5$, 6 to 5
2. $10:3$
3. $\frac{4}{10}$ (there are $4 + 6 = 10$ baked goods)
4. $\frac{4}{6}$ compares tarts to cookies, two parts of the tray. $\frac{4}{10}$ compares tarts to all 10 baked goods. The first term is the same, but the second term counts something different.

Lesson 8.3 — Part-to-Part, Part-to-Whole, Whole-to-Whole

Guided practice

1. $5:7$
2. $5 + 7 = 12$ juice boxes
3. $7:12$
4. $9:15$, a whole-to-whole ratio
5. Add the parts. The whole is the sum of all the parts of the group.

Independent practice

6. a) $18:12$ b) $12:30$, since the choir has $18 + 12 = 30$ members c) sopranos to altos is part to part; altos to the whole choir is part to whole
7. Cheese slices: $8 - 3 = 5$. Pepperoni to cheese is $3:5$ (part to part). Pepperoni to all slices is $3:8$ (part to whole).
8. a) part to part b) part to whole c) whole to whole
9. $29 - 11 = 18$ students did not choose soccer.
10. There are $25 + 15 = 40$ vehicles. Cars to trucks, $25:15$ (part to part). Cars to all vehicles, $25:40$ (part to whole). Trucks to all vehicles, $15:40$ (part to whole).
11. The mix has $6 + 4 = 10$ cups. Part to part: pretzels to popcorn, $6:4$. Part to whole: pretzels to mix, $6:10$; popcorn to mix, $4:10$. To answer "what portion of the mix is popcorn," use $4:10$, because the question compares the popcorn part to the entire mix, not to the pretzels.
12. The $5:4$ ratio compares two parts to each other, so it does not include the whole. The club has $5 + 4 = 9$ members for every 5 sixth graders, so sixth graders to all members is $5:9$.

Exit ticket 8.3

1. $7:5$
2. $5:12$, since the drawer has $7 + 5 = 12$ socks
3. whole to whole
4. A part-to-part ratio compares two pieces of the same group to each other. A part-to-whole ratio compares one piece to the entire group, so the second term includes the first.

Lesson 8.4 — Equivalent Ratios and Ratio Tables

Guided practice

1. $4:18$ and $6:27$ (any multiple of both terms by the same number)
2. $20$, since $3 \times 4 = 12$ and $5 \times 4 = 20$
3. Yes. Divide both terms by 5: $10 \div 5 = 2$ and $15 \div 5 = 3$, giving $2:3$.

5. 10 cups of water, since $1 \times 5 = 5$ and $2 \times 5 = 10$

Independent practice

6. a) $10:12$, $15:18$, $20:24$ b) $14:4$, $21:6$, $28:8$
7. a) $12$, since $9 \times 3 = 27$ and $4 \times 3 = 12$ b) $5$, since $8 \div 4 = 2$ and $20 \div 4 = 5$ c) $5$, since $15 \div 3 = 5$ and $21 \div 3 = 7$
8. Equivalent to $6:8$: $3:4$, $9:12$, and $18:24$. Not equivalent: $8:10$, which reduces to $4:5$.

| First quantity | 9 | 18 | 27 | 36 | 45 |
|---|---|---|---|---|---|
| Second quantity | 4 | 8 | 12 | 16 | 20 |

The fifth column is the ratio $45:20$.

| Laps | 3 | 6 | 9 | 12 | 15 |
|---|---|---|---|---|---|
| Cost (dollars) | 7 | 14 | 21 | 28 | 35 |

15 laps cost 35 dollars. With 28 dollars you can buy 12 laps.

12. Adding 2 to each term does not preserve a ratio; only multiplying or dividing both terms by the same number does. $4:6$ reduces to $2:3$, while $6:8$ reduces to $3:4$, so they are not equivalent. A ratio equivalent to $4:6$ with first term 6 is $6:9$, since $4 \times 1.5 = 6$ and $6 \times 1.5 = 9$. (Checking: $6:9$ reduces to $2:3$.)

Exit ticket 8.4

1. $10:6$ and $15:9$ (any correct multiples)
2. $35$, since $2 \times 5 = 10$ and $7 \times 5 = 35$

4. A ratio is a comparison by division, so it is preserved when both terms are scaled by the same factor. Adding the same amount to each term changes how the terms compare: $1:2$ says the second amount is twice the first, but adding 2 to each gives $3:4$, where the second amount is no longer double.

Lesson 8.5 — Writing a Situation for a Ratio

Situations vary. A response is correct if it names two quantities, keeps the order, makes the requested type of comparison, and lets a reader rebuild the original ratio.

Guided practice

1. Sample: For every 4 blue chairs in the room, there are 3 red chairs.
2. Sample: 2 out of every 5 students in the class walk to school.
3. Sample: The car travels 60 miles every 1 hour.
4. Sample: For every 8 forks in the drawer, there are 3 knives.
5. It sounds as though there are two separate groups of students, 3 and 10, for a total of 13. In a part-to-whole sentence the second number must include the first. Clear version: 3 out of every 10 students ride the bus.

Independent practice

6. a) Sample: For every 7 cars in the lot, there are 2 motorcycles. b) Sample: 5 out of every 8 seats on the bus are filled. c) Sample: For every 9 minutes of the show, there are 4 minutes of commercials.
7. Sample: 3 out of every 10 books on the shelf are mysteries. The 3 counts mystery books; the 10 counts all the books, including the mysteries.
8. Sample: My class collected 24 cans and the class next door collected 18 cans.
9. Sample: The bakery sells 15 rolls for 3 dollars. In simplest terms the ratio is $5:1$, since $15 \div 3 = 5$ and $3 \div 3 = 1$ — 5 rolls per dollar.
10. Part to part sample: For every 2 goldfish in the tank, there are 9 guppies (11 fish in all). Part to whole sample: 2 out of every 9 fish in the tank are goldfish (7 guppies for every 2 goldfish). The symbols are identical, but in the first situation the 9 counts only guppies and in the second the 9 counts every fish, so the two situations describe different tanks.
11. Notations: $\frac{1}{12}$, $1:12$, 1 to 12. With 36 campers the camp needs 3 counselors, since $1 \times 3 = 3$ and $12 \times 3 = 36$. Sample situation: When 36 campers attend, the camp needs 3 counselors, keeping 1 counselor for every 12 campers.
12. In $\frac{3}{7}$ read as a part-to-whole ratio, the 7 counts all the children and already includes the 3 boys. Devin counted the 7 as girls in addition to the boys. The correct number of girls is $7 - 3 = 4$.

Exit ticket 8.5

1. Sample: For every 6 apples in the crate, there are 5 oranges.
2. Sample: 3 out of every 8 students in the class play a sport.
3. Sample: The machine fills 45 bottles every 1 minute.
4. The numbers alone do not say what is being compared. Naming both quantities tells the reader what each term counts and which comparison — part to part, part to whole, or whole to whole — the ratio makes.

Chapter 8 Review

Part A — Representing a relationship with a ratio (6.PFA.1a)

1. $8:5$
2. $150:5$
3. Subtraction tells how much more one amount is than the other, a single difference. A ratio tells how the amounts pair up, "for every," and stays the same when both amounts are scaled together.

Part B — The three notations (6.PFA.1b)

4. $\frac{11}{4}$, $11:4$, 11 to 4
5. $9:5$ and 9 to 5
6. $\frac{7}{12}$
7. $\frac{5}{9}$ compares red chips to blue chips (part to part). $\frac{5}{14}$ compares red chips to all $5 + 9 = 14$ chips (part to whole).

Part C — Types of comparison (6.PFA.1c)

8. a) $13:12$ b) $12:25$, since the class has $13 + 12 = 25$ students c) boys to girls is part to part; girls to the whole class is part to whole
9. a) part to whole b) part to part c) whole to whole
10. $20 - 9 = 11$ beads are not green.

Part D — Tables of equivalent ratios (6.PFA.1e, 6.PFA.1f)

11. $16:6$, $24:9$, $32:12$
12. a) $16$, since $5 \times 4 = 20$ and $4 \times 4 = 16$ b) $5$, since $6 \div 3 = 2$ and $15 \div 3 = 5$

| First quantity | 7 | 14 | 21 | 28 | 35 |
|---|---|---|---|---|---|
| Second quantity | 2 | 4 | 6 | 8 | 10 |

| Apples | 4 | 8 | 12 | 16 | 20 |
|---|---|---|---|---|---|
| Cost (dollars) | 3 | 6 | 9 | 12 | 15 |

20 apples cost 15 dollars.

Part E — Writing a situation for a ratio (6.PFA.1d)

15. Sample: For every 9 sixth graders on the field trip, there are 2 chaperones.
16. Sample: 5 out of every 12 books in the classroom library are nonfiction.

Part F — Mixed application and reasoning

17. The mix has $5 + 3 = 8$ cups. Part to part: $5:3$. Oats to mix: $5:8$. Cherries to mix: $3:8$.

    Oats (cups)	5	10	15	20
    Cherries (cups)	3	6	9	12

18. Marco added 2 to each amount instead of multiplying each by 2. Doubling $3:2$ gives $6:4$, so he needs 6 cups of broth and 4 cups of rice.

19. $\frac{2}{3}$ is part to whole when the 3 counts everything: "2 of the 3 slices are cheese," so cheese is two-thirds of the pizza. It is part to part when the 3 counts something else: "there are 2 cats for every 3 dogs," where the group holds 5 animals and cats are two-fifths of them. The notation is the same; the labels tell you which comparison is meant.

20. Yes, the mixes match. $12 \div 6 = 2$ and $18 \div 6 = 3$, so $12:18 = 2:3$. Also $10 \div 5 = 2$ and $15 \div 5 = 3$, so $10:15 = 2:3$. Both reduce to $2:3$, so they are equivalent ratios.

Workbook-only items

Page 2, fill in the blanks. A ratio compares two quantities by division. The first quantity named in the question is written first. Swapping the order changes what the ratio says.

Page 2, table. 6 to 9; 9 to 6; 2 to 5; 30 to 5; 11 to 5; 3 to 4

Page 3, tray. Muffins to rolls: 5 to 8. Rolls to muffins: 8 to 5.

Page 3, explain. The numbers are the same but they count different things. In $3:8$ the 3 counts the first quantity named; in $8:3$ the 8 does. Reversing the order describes a different situation.

Page 3, apply it. The cook reversed the amounts. The drink will be far too strong, since there is more concentrate than water.

Page 5, notation table.

Page 5, rewrite. $6:7$ → $\frac{6}{7}$ and 6 to 7. $\frac{11}{4}$ → $11:4$ and 11 to 4. 15 to 8 → $\frac{15}{8}$ and $15:8$.

Page 6, fraction form. $\frac{5}{7}$ compares blue to yellow tiles. $\frac{5}{12}$ compares blue to all the tiles ($5 + 7 = 12$).

Page 6, backpack. Pens to pencils $3:8$; pencils to pens $8:3$; pens to all writing tools $3:11$.

Page 6, think. The 9 counts cashews, not the whole mix. Almonds are $\frac{2}{11}$ of the mix.

Page 8, class table.

Page 8, fill in. To find the whole from the parts, you add the parts.

Page 9, sort. 4 red to 9 blue: part to part. 4 red to 13 pens: part to whole. 13 pens to 20 pens: whole to whole. 18 sopranos to 12 altos: part to part. 12 altos to the 30-member choir: part to whole. 45 points to 36 points: whole to whole.

Page 9, pizza. Pepperoni to cheese $3:5$; pepperoni to all slices $3:8$.

Page 9, survey. $29 - 11 = 18$ students.

Page 9, explain. $5:4$ compares two parts, so it leaves out the whole. There are $5 + 4 = 9$ members for every 5 sixth graders, so the correct part-to-whole ratio is $5:9$.

Page 11, three equivalents. $5:6$ → $10:12$, $15:18$, $20:24$. $7:2$ → $14:4$, $21:6$, $28:8$. $2:9$ → $4:18$, $6:27$, $8:36$. (Other correct multiples are fine.)

Page 11, fill in. a) $12$ b) $5$ c) $5$ d) $20$ e) $35$ f) $16$

Page 11, circle. Equivalent to $6:8$: $3:4$, $9:12$, $18:24$. Not equivalent: $8:10$ (reduces to $4:5$) and $12:14$ (reduces to $6:7$).

Page 12, table 1 — paint.

Page 12, table 2 — ratio $9:4$.

Page 12, table 3 — go-karts.

15 laps cost 35 dollars; 28 dollars buys 12 laps.

Page 12, find the error. Adding the same number to both terms does not keep a ratio. $4:6$ reduces to $2:3$ but $6:8$ reduces to $3:4$. The ratio equivalent to $4:6$ with first term 6 is $6:9$.

Page 14, three things. A good situation names both quantities, keeps the order, and uses the phrase "for every."

Page 14, write a situation. Samples: $4:3$ — for every 4 red tulips there are 3 white tulips. $\frac{2}{5}$ — 2 out of every 5 students bring lunch from home. $60:1$ — the machine seals 60 boxes every 1 minute. $7:2$ — for every 7 cars in the lot there are 2 motorcycles. $\frac{5}{8}$ — 5 out of every 8 seats on the bus are filled. 9 to 4 — for every 9 minutes of the show there are 4 minutes of commercials.

Page 15, two situations for $2:9$. Part to part sample: for every 2 goldfish there are 9 guppies, so the tank holds 11 fish. Part to whole sample: 2 out of every 9 fish are goldfish, so 7 of the 9 are guppies. The symbols match, but the 9 counts guppies in the first and all the fish in the second.

Page 15, rewrite. "There are 8 and 3 of something" → for every 8 forks in the drawer there are 3 knives. "For every 3 students there are 10 students" → 3 out of every 10 students ride the bus.

Page 15, find the error. The 7 in $\frac{3}{7}$ counts all the children and already includes the 3 boys. Girls: $7 - 3 = 4$.

Pages 17–18, Chapter 8 review. Same items and answers as the textbook Chapter 8 Review above.