Appendix A — Answer Key, Chapter 9: Proportional Relationships and Unit Rate

SOL 6.PFA.2 · Covers textbook Chapter 9 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. Every unit rate in this chapter is positive. Reasoning answers show an acceptable response, not the only wording.

Lesson 9.1 — Unit Rate

Guided practice

1. $120 \div 4 = 30$ miles per hour
2. $18 \div 6 = 3$ dollars per pen
3. $84 \div 2 = 42$ words per minute
4. $9 \div 2 = 4.5$, and $18 \div 4 = 4.5$, and $27 \div 6 = 4.5$, so 4.5 dollars per hour
5. 5 dollars per pound. It means one pound costs 5 dollars.

Independent practice

6. a) $135 \div 3 = 45$ miles per hour b) $10 \div 5 = 2$ dollars per pound
7. $96 \div 8 = 12$ ounces per bottle
8. $30 \div 4 = 7.5$, and $60 \div 8 = 7.5$, and $90 \div 12 = 7.5$, so 7.50 dollars per ticket
9. 6 pages per hour. The point $(1, 6)$ gives it directly, and $18 \div 3 = 6$ confirms it.
10. $300 \div 12 = 25$ miles per gallon
11. $7.50 \div 3 = 2.50$ dollars per pound; $11.25 \div 5 = 2.25$ dollars per pound. The 5-pound package is the better buy, because 2.25 dollars per pound is less than 2.50 dollars per pound.
12. $180 \div 3 = 60$ tells how many miles are covered in one hour. $3 \div 180 \approx 0.0167$ tells how many hours it takes to cover one mile. The question asks how far in one hour, so the miles must be divided by the hours, giving 60 miles per hour.

Exit ticket 9.1

1. $240 \div 4 = 60$ miles per hour
2. $24 \div 8 = 3$ dollars per notebook
3. $40 \div 5 = 8$ pounds per bag (and $80 \div 10 = 8$)
4. A unit rate tells how much of the second quantity goes with exactly one of the first — the amount "per one."

Lesson 9.2 — Finding Missing Values in a Ratio Table

Guided practice

1. Unit rate $20 \div 5 = 4$ dollars per kilogram; $9 \times 4 = 36$ dollars
2. Unit rate $24 \div 3 = 8$ cans per box; $96 \div 8 = 12$ boxes
3. Unit rate $220 \div 4 = 55$ miles per hour; $7 \times 55 = 385$ miles

5. Find the unit rate from a column that is completely filled in. Then multiply the unit rate by any first-quantity value to get its partner, or divide any second-quantity value by the unit rate to get its partner.

Independent practice

6. Unit rate $6 \div 1 = 6$ dollars per item; $48 \div 6 = 8$ items

7. Unit rate $10 \div 8 = 1.25$ dollars per muffin; $14 \times 1.25 = 17.50$ dollars

8. Unit rate $6 \div 4 = 1.5$ pages per minute; $15 \div 1.5 = 10$ minutes and $12 \times 1.5 = 18$ pages.

   Minutes	4	10	12
   Pages	6	15	18

9. Unit rate $1{,}050 \div 3 = 350$ square feet per gallon; $7 \times 350 = 2{,}450$ square feet

10. Unit rate $6 \div 2 = 3$ servings per cup; $24 \div 3 = 8$ cups and $11 \times 3 = 33$ servings.

    Cups	2	5	8	11
    Servings	6	15	24	33

11. $250 \div 10 = 25$ miles per gallon. On 16 gallons: $16 \times 25 = 400$ miles. For 350 miles: $350 \div 25 = 14$ gallons.

12. Sam divided when he should have multiplied. He was given the first quantity, 12 pounds, so he needed $12 \times 2 = 24$ dollars. The check: 12 pounds is more than 1 pound, so the cost must be more than 2 dollars, and 6 dollars is far too small for a dozen pounds.

Exit ticket 9.2

1. Unit rate $9 \div 6 = 1.50$ dollars per pound; $10 \times 1.50 = 15$ dollars

2. Unit rate $21 \div 3 = 7$ miles per hour.

   Hours	3	5	9
   Miles	21	35	63

3. Unit rate $45 \div 15 = 3$ minutes per problem; $25 \times 3 = 75$ minutes

4. Multiply by the unit rate when you know the first quantity and want the second. Divide by the unit rate when you know the second quantity and want the first.

Lesson 9.3 — Deciding Whether a Relationship Is Proportional

Guided practice

1. Yes. $6 \div 2 = 3$, $15 \div 5 = 3$, $27 \div 9 = 3$. Unit rate 3 miles per hour.
2. No. $4 \div 1 = 4$, $7 \div 2 = 3.5$, $10 \div 3 \approx 3.33$. The quotients disagree.
3. Not proportional. Zero visits still costs the 20 dollar fee, so the graph would not pass through the origin. Checking: 1 visit costs 25 dollars and 2 visits cost 30 dollars, and $25 \div 1 = 25$ but $30 \div 2 = 15$.
4. Proportional. There is one constant rate of 3 dollars per pound and no starting charge, so zero pounds costs zero dollars.
5. Passing through the origin is one of the two conditions for a proportional graph. If the graph is also a straight line, the relationship is proportional.

Independent practice

6. a) Yes. $9 \div 4 = 2.25$, $18 \div 8 = 2.25$, $27 \div 12 = 2.25$. Unit rate 2.25 dollars per item. b) No. $5 \div 2 = 2.5$, $9 \div 4 = 2.25$, $13 \div 6 \approx 2.17$. The quotients disagree. (This table charges 1 dollar plus 2 dollars per item.)
7. Yes. The pool starts empty, so zero minutes means zero gallons, and the rate of 12 gallons per minute never changes.
8. No. Zero hours of work still costs the 45 dollar call fee, so the relationship does not pass through the origin.
9. Yes. The graph is a straight line through the origin, and $6 \div 2 = 3$ and $15 \div 5 = 3$. Unit rate 3.
10. No. It is a straight line, but at a first coordinate of 0 the second coordinate is 4, not 0. The relationship starts with 4 already in place.
11. Plan A is proportional: 10 cents per minute with no fee, so 0 minutes costs 0 dollars. Plan B is not proportional because of the 5 dollar monthly fee. For 200 minutes, Plan A costs $200 \times 0.10 = 20$ dollars and Plan B costs $5 + 200 \times 0.05 = 5 + 10 = 15$ dollars.
12. A straight line only shows that the rate is constant; it does not show that the relationship starts at zero. The line must also pass through the origin. Example: a line through $(0, 6)$ and $(1, 8)$ is perfectly straight, but $6 \div 0$ is not even defined and $8 \div 1 = 8$, so there is no single unit rate.

Exit ticket 9.3

1. Yes. $12 \div 3 = 4$, $24 \div 6 = 4$, $40 \div 10 = 4$. Unit rate 4 dollars per hour.
2. No. $3 \div 1 = 3$ but $5 \div 2 = 2.5$.
3. Not proportional. The 8 dollar charge applies even with zero packages.
4. Because zero of the first quantity must give zero of the second in a proportional relationship. If the graph starts above or below the origin, there is a fixed amount that does not depend on the rate, and the ratios of the pairs will not all be equal.

Lesson 9.4 — Graphing Proportional Relationships

Guided practice

1. $(1, 3)$, $(2, 6)$, $(3, 9)$
2. 9
3. Because zero of the first quantity always gives zero of the second in a proportional relationship, so the pair $(0, 0)$ belongs to it.

5. $20 \div 4 = 5$

Independent practice

6. $(2, 10)$, $(4, 20)$, $(6, 30)$. Unit rate $10 \div 2 = 5$ pounds per bag.
7. Unit rate $21 \div 3 = 7$; at a first coordinate of 5 the second coordinate is $5 \times 7 = 35$.

Ordered pairs: $(1, 2.50)$, $(2, 5)$, $(3, 7.50)$, $(4, 10)$, plus $(0, 0)$. 9. Line B is steeper, because its unit rate of 5 is greater than line A's 3. Steeper means the second quantity grows faster for each unit of the first. 10. Unit rate $24 \div 8 = 3$; when the second coordinate is 45, the first is $45 \div 3 = 15$. 11.

| Minutes | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| Pages | 0 | 15 | 30 | 45 | 60 |

Ordered pairs: $(0, 0)$, $(1, 15)$, $(2, 30)$, $(3, 45)$, $(4, 60)$. For 90 pages: $90 \div 15 = 6$ minutes.

12. The point $(0, 0)$ is missing, and so is the part of the line between the origin and $(2, 6)$. Zero of the first quantity gives zero of the second, so the graph of a proportional relationship starts at the origin; a line that begins at $(2, 6)$ leaves out every smaller pair, such as $(1, 3)$.

Exit ticket 9.4

1. $(1, 8)$, $(2, 16)$, $(3, 24)$
2. 6
3. $7 \times 6 = 42$
4. The graph must be a straight line and it must pass through the origin $(0, 0)$. If both are true, one constant unit rate describes every point.

Lesson 9.5 — Connecting Tables, Graphs, and Verbal Descriptions

Verbal descriptions vary. A response is correct if it names both quantities and states the correct unit rate with the word "per" or an equivalent phrase.

Guided practice

1. $9 \div 1 = 9$ and $18 \div 2 = 9$. Sample: the trip covers 9 miles every hour.

3. Unit rate 12. Sample: the item costs 12 dollars per pound.
4. Car A: $150 \div 3 = 50$ miles per hour. Car B: 45 miles per hour. Car A is faster.
5. It says that one unit of the first quantity goes with $r$ units of the second, so $r$ is the unit rate.

Independent practice

Ordered pairs: $(1, 20)$, $(2, 40)$, $(3, 60)$, $(4, 80)$, $(5, 100)$. 7. $7 \div 2 = 3.5$, $14 \div 4 = 3.5$, $21 \div 6 = 3.5$. Sample: the job pays 3.50 dollars per hour. 8. Unit rate $9 \div 2 = 4.5$.

9. (i) and (ii) describe the same relationship. In (ii), $6 \div 2 = 3$ and $15 \div 5 = 3$, matching the 3 dollars per pound in (i). Representation (iii) has unit rate 4, so it is different.

10. Sample description: a machine produces 6 items per minute.

    Minutes	1	2	3
    Items	6	12	18

    Points: $(1, 6)$, $(2, 12)$, $(3, 18)$. The point $(0, 0)$ is also on the graph.

11. Runner A: $6 \div 30 = 0.2$ kilometers per minute. Runner B: $2.5 \div 10 = 0.25$ kilometers per minute. Runner B is faster. In 60 minutes, A covers $60 \times 0.2 = 12$ kilometers and B covers $60 \times 0.25 = 15$ kilometers.

12. In a table, the unit rate is the second quantity divided by the first in any column. On a graph, it is the vertical coordinate of the point whose horizontal coordinate is 1. They give the same number because each column of the table is a point on the graph, and dividing the coordinates of any point gives the amount that goes with one unit — which is exactly what the point at 1 shows directly.

Exit ticket 9.5

1. $15 \div 3 = 5$ dollars per pound (and $30 \div 6 = 5$)
2. $(1, 24)$
3. Sample: a printer prints 7 pages every minute.
4. Every column of the table is a point on the graph, and every point on the graph can be written as a column of the table. Both show the same pairs of values and the same unit rate, one as numbers and one as a picture.

Chapter 9 Review

Part A — Identifying the unit rate (6.PFA.2a)

1. $21 \div 3 = 7$, $42 \div 6 = 7$, $63 \div 9 = 7$, so 7 dollars per book
2. $156 \div 3 = 52$ miles per hour
3. 11

Part B — Missing values in a ratio table (6.PFA.2b)

4. Unit rate $9 \div 2 = 4.5$. Then $5 \times 4.5 = 22.5$ and $36 \div 4.5 = 8$.

   First quantity	2	5	8
   Second quantity	9	22.5	36

5. Unit rate $14 \div 4 = 3.50$ dollars per pound; $10 \times 3.50 = 35$ dollars

Part C — Deciding whether a relationship is proportional (6.PFA.2c)

6. Yes. $8 \div 2 = 4$, $12 \div 3 = 4$, $20 \div 5 = 4$. Unit rate 4 dollars per item.
7. No. $6 \div 1 = 6$ but $11 \div 2 = 5.5$.
8. Not proportional. The 12 dollar registration fee is charged even for zero hours, so the relationship does not start at the origin.
9. Not proportional. The line is straight but crosses the vertical axis at 2 instead of 0, so zero of the first quantity does not give zero of the second.

Part D — From a context to a table or graph (6.PFA.2d)

10. Unit rate 7 gallons per minute.

    Minutes	0	1	2	3	4	5
    Gallons	0	7	14	21	28	35

    Ordered pairs: $(0, 0)$, $(1, 7)$, $(2, 14)$, $(3, 21)$, $(4, 28)$, $(5, 35)$.

11. Unit rate $36 \div 3 = 12$ dollars per shirt.

    Shirts	1	2	3	4	5
    Cost (dollars)	12	24	36	48	60

Part E — Connecting representations (6.PFA.2e)

12. $13 \div 2 = 6.5$ and $26 \div 4 = 6.5$. Sample description: the trip covers 6.5 miles every hour. The point at a first coordinate of 1 is $(1, 6.5)$.

13. Unit rate $18 \div 4 = 4.5$.

    First	1	2	3
    Second	4.5	9	13.5

14. (i) and (ii) match: $15 \div 3 = 5$ dollars per hour, the same as "5 dollars per hour." Representation (iii) has unit rate $12 \div 2 = 6$, so it describes a different relationship.

Part F — Mixed application and reasoning

15. Unit rate $12.50 \div 5 = 2.50$ dollars per pound. Cost of 8 pounds: $8 \times 2.50 = 20$ dollars. Pounds for 20 dollars: $20 \div 2.50 = 8$ pounds.
16. Plan A is proportional, because there is no fee and zero minutes costs zero dollars. For 200 minutes: Plan A costs $200 \times 0.10 = 20$ dollars; Plan B costs $5 + 200 \times 0.05 = 15$ dollars.
17. Crossing the vertical axis above the origin means that when the first quantity is 0, the second quantity is not 0. Some fixed amount is present before the rate does any work, so the pairs do not all form the same ratio and no single unit rate describes the whole graph.
18. Worker A: $108 \div 9 = 12$ dollars per hour. Worker B: $52 \div 4 = 13$ dollars per hour. Worker B earns more, by 1 dollar per hour.

Workbook-only items

Page 2, fill in the blanks. A rate compares two quantities with different units. A unit rate is a rate whose second quantity is 1. To find a unit rate, divide the first quantity by the second. "Per" means for each one.

Page 2, unit rate table.

Page 3, table 1. Unit rate 4.5 dollars per hour; division $9 \div 2 = 4.5$.

Page 3, table 2. Unit rate 7.50 dollars per ticket; division $30 \div 4 = 7.5$.

Page 3, graphs. The unit rate is the second coordinate of the point where the first coordinate is 1. Cost against pounds through $(1, 5)$: unit rate 5. Pages against hours through $(1, 6)$ and $(3, 18)$: unit rate 6.

Page 3, better buy. $7.50 \div 3 = 2.50$ dollars per pound; $11.25 \div 5 = 2.25$ dollars per pound. The 5-pound package is the better buy.

Page 5, table 1. Unit rate 15 dollars per hour.

Page 5, table 2. Unit rate 31 miles per gallon.

Page 5, table 3. Unit rate 2.5 servings per cup.

Page 5, table 4. Unit rate 6 dollars per item.

Page 6, solve with the unit rate.

1. 4 dollars per kilogram; 36 dollars.
2. 55 miles per hour; 385 miles.
3. 1.25 dollars per muffin; 17.50 dollars.
4. 350 square feet per gallon; 2,450 square feet.
5. 25 miles per gallon; 400 miles; 14 gallons.

Page 6, find the error. Sam divided instead of multiplying. The correct cost is $12 \times 2 = 24$ dollars. Quick check: 12 pounds must cost more than 1 pound does, so an answer of 6 dollars is far too small.

Page 8, the three tests. Table: proportional if all quotients are equal. Context: proportional if there is a constant rate and no starting fee. Graph: a straight line through the origin.

Page 8, decide for each table.

Page 9, contexts.

Page 9, graphs. Through $(0,0)$, $(2,6)$, $(5,15)$: proportional, unit rate 3. Through $(0,4)$ and $(2,10)$: not proportional, because it does not pass through the origin — at a first coordinate of 0 the value is 4.

Page 9, explain. A straight line only shows a constant rate. It must also pass through the origin. A line through $(0, 6)$ and $(1, 8)$ is straight but not proportional.

Page 11, ordered pairs.

1. $(1, 3)$, $(2, 6)$, $(3, 9)$
2. $(2, 10)$, $(4, 20)$, $(6, 30)$; unit rate 5
3. $(1, 8)$, $(2, 16)$, $(3, 24)$

Page 11, blueberries.

Ordered pairs: $(0, 0)$, $(1, 2.50)$, $(2, 5)$, $(3, 7.50)$, $(4, 10)$. Plotted, they lie on a straight line from the origin.

Page 12, fill in. Through $(1, 9)$: unit rate 9. Through $(4, 20)$: unit rate 5. Through $(3, 21)$: unit rate 7, and at $x = 5$ the value is 35. Through $(8, 24)$: unit rate 3, and the first coordinate is 15 when the second is 45.

Page 12, steepness. Line B is steeper. A steeper line has the greater unit rate, so its second quantity grows faster for each unit of the first.

Page 12, printer.

Ordered pairs: $(0, 0)$, $(1, 15)$, $(2, 30)$, $(3, 45)$, $(4, 60)$. 90 pages takes 6 minutes.

Page 14, where is the unit rate. Words: after "per" or "for each." Table: the second quantity divided by the first, in any column. Graph: the vertical coordinate of the point where the horizontal coordinate is 1.

Page 14, conversions.

1. Sample: the trip covers 9 miles every hour.

3. Unit rate 12; sample: the item costs 12 dollars per pound.

4. Unit rate 3.5; sample: the job pays 3.50 dollars per hour.

5. Unit rate 4.5.

   First	1	2	3	4
   Second	4.5	9	13.5	18

Page 15, compare.

1. Car A: 50 miles per hour. Car B: 45 miles per hour. Car A is faster.
2. Runner A: 0.2 kilometers per minute. Runner B: 0.25 kilometers per minute. Runner B is faster. In 60 minutes A covers 12 kilometers and B covers 15 kilometers.

Page 15, which match. Circle (i) and (ii); both have a unit rate of 3 dollars per pound. Item (iii) has unit rate 4, so it does not match.

Page 15, explain. In a table the unit rate is the second quantity divided by the first in any column; on a graph it is the height of the point at a horizontal coordinate of 1. The numbers agree because each table column is a point on the graph, and every point of a proportional relationship gives the same quotient.

Pages 17–18, Chapter 9 review. Same items and answers as the textbook Chapter 9 Review above.