math grade 6 unit 6 rate - rusd math -...

MATH GRADE 6 UNIT 6

RATE

ANSWERS FOR EXERCISES

Copyright © 2014 Pearson Education, Inc. 26

Grade 6 Unit 6: Rate

ANSWERS

ANSWERS

1. $6.25

2. $0.625, or $0.63

3. $5.25

4. $0.3125, or $0.31

5. a. $2.50

b. $13.75

6. a.

b. 1 pint = $0.75 6.5 pints = 6.5 • $0.75 = $4.875, or $4.88

c. See the table in problem 6a.

7. a. $3.00 per dozen ÷ 12 eggs = $0.25 per egg Elijah can buy only 1 egg, because eggs cost $0.25 each.

b. $0.43 + $0.32 = $0.75 Elijah needs $0.43 more, for a total of $0.75

Challenge Problem

8. The car costs about $6.47 per pound. At this rate, a 20-pound wheel costs about $129.44. Most people would not judge a car by dollars per pound. While it can be mathematically correct, the information may not influence a decision.

LESSON 2: PRICE AS A RATE

Amount (pt) 1 2 3 4 5 6

Cost—non-organic ($) $0.75 $1.50 $2.25 $3.00 $3.75 $4.50

Cost—organic ($) $2.25 $4.50 $6.75 $9.00 $11.25 $13.50

Amount (pt) 7 8 9 10 11 12

Cost—non-organic ($) $5.25 $6.00 $6.75 $7.50 $8.25 $9.00

Cost—organic ($) $15.75 $18.00 $20.25 $22.50 $24.75 $27.00



ANSWERS

ANSWERS

1. 22.5 gallons

2. 9.25 gallons

3. 160 miles

4. 75 miles

5. a. Distance (mi) 12.5 25 75 125 200 275 350 425

Gas (gal) 1 2 6 10 16 22 28 34

b. The fuel efficiency is 12.5 miles per gallon. The unit rate is given in the first column of the table.

c. 137.5 miles

6. a. Gas (gal) 1 2 4 8 12 16 22 28

Distance (mi) 12.5 25 50 100 150 200 275 350

b. The fuel efficiency is 1

12.5 = 0.08 gallon per mile.

The unit rate is given in the first column of the table.

c. 8.96 gallons

7. Answers will vary. Here is one example: Most of the time people talk about miles per gallon. Miles per gallon is a number greater than 0. The inverse rate, gallons per mile, is a decimal number less than 1. People usually find it easier to work with numbers greater than 1 when comparing rates.

Challenge Problem

8. $0.20 per mile

LESSON 3: FUEL EFFICIENCY AS A RATE



ANSWERS

ANSWERS

1. City Population—

2010 CensusArea (sq mi)

Population Density (to nearest tenth)

Chicago 2,695,598 227.6 11,843.6 people/sq mi

San Francisco 805,235 46.9 17,169.2 people/sq mi

2. Even though it has a smaller population, San Francisco is more crowded, because its population density is greater than that of Chicago.

3. The units are people per square mile. Other units would depend on the units used for area, such as people per square kilometer, and so on.

4. State Population—

2000 CensusArea (sq mi)

Population Density (to nearest tenth)

California 37,593,222 163,695 229.7 people/sq mi

Georgia 8,186,453 59,425 137.8 people/sq mi

Montana 902,195 147,042 6.1 people/sq mi

New Jersey 8,414,350 8,722 964.7 people/sq mi

5. New Jersey is the most crowded because it has the highest population density, and Montana is the least crowded because it has the lowest population density.

6. Population density would increase if the state’s area were to shrink (assuming the population in the state remains fixed).

7. Population density would decrease if the state’s population were to shrink (assuming the state’s area remains fixed).

Challenge Problem

8. Answers will vary.

LESSON 4: POPULATION DENSITY AS A RATE



ANSWERS

ANSWERS

1. Answers will vary. You might say that a rate is a type of quantity in which the unit is of the form “A per B.” A rate measures the relationship between two aspects of a situation.

Situation Rate Not a Rate Reason

2. Emma earns $6.50 per hour for babysitting.

√ This is a unit price.

3.

1

2

3

4

5

6

Hei

gh

t (f

t)

Alex Bella Craig Desiree

√ Only height is measured.

4.

30 drops per minute

√ The rate is drops per minute.

5. Denzel read 3 books this week to gather information for his science project.

√He could read any number of books in the following weeks.

LESSON 5: WHAT IS A RATE?



ANSWERSLESSON 5: WHAT IS A RATE?

Situation Rate Not a Rate Reason

6.

1 quart (0.946 liter)

√The amount lists the rate of liters per quart.

7. The distance between where you live and your school.

√ Only distance is measured.

8. Tickets to a play cost $9.00 for children and $12.00 for adults.

√The rates are price per ticket.

Challenge Problem

9. Answers will vary. Here are three examples:

• price per pound (e.g., use to find the price of 3 pounds of grapes at $3.29 per pound)

• miles per hour (e.g., use to find speed)

• heart beats per minute (e.g., use to find heart rate)



ANSWERS

ANSWERS

1. 2 meters per second or 120 meters per minute

2. a. 0.125 mile per minute

b. 7.5 miles per hour

c. 0.5 lap per minute

d. 11 feet per second

3. 10 minutes

4. 0.4 hour, or 24 minutes

5. a. t = 2n

b. n = 0.5t

6. a. Distance (mi)

8 16 24 32 40 48 56 64 720

2 4 6 8 10 12 14 16 180

Time (min)

b. His speed is 0.25 mile per minute, or 15 miles per hour. You can find the speed in miles per hour by multiplying the miles per minute by 60.

c. 18 miles

7. a. Time (min) 4 8 12 24 48 60 100

Distance (mi) 1 2 3 6 12 15 25

b. His speed is 4 minutes per mile, or 1

15 hour per mile. You can find the hours

per mile by dividing the minutes per mile by 60. In this case, minutes per mile

probably makes more sense in reporting the speed.

c. 92 minutes

LESSON 6: SPEED AS RATE



ANSWERSLESSON 6: SPEED AS RATE

Challenge Problem

8. The rate is 4 minutes per mile = 1

4 mile per minute. In this case, minutes per mile is a

number greater than 1, so you would discuss minutes per mile rather than miles per

minute. Minutes makes more sense in this situation.



ANSWERS

ANSWERS

1. 7.62 cm

2. 33.02 cm

3. About 16.54 inches

4. About 39.37 inches

5. 4,680 gallons

6. 1 divided by 1.61 is about 0.62 mile in 1 kilometer.

7. a. 297 ft

2.4 acres = 104,544 sq ft

b. 352 feet

8. a. 599 gallon containers

b. 161,011 cubic inches

c. 93 cubic feet

Challenge Problem

9. A rate is one quantity per another quantity. It compares two quantities. A conversion factor compares two quantities.

LESSON 7: CONVERSION FACTORS



ANSWERS

ANSWERS

1. a. 0.5 block per minute

b. 2 minutes

2. This calculation tells their average speed in miles per hour, since Suraj divides the number of miles by the number of hours.

3. This calculation tells their average speed in hours per mile, since Suraj’s brother divides the number of hours by the number of miles.

4. 442 miles ÷ 6.5 hours = 68 miles per hour. 68 gives their average speed in miles per hour because you are dividing miles by hours.

5. 6.5 hours ÷ 442 miles ≈ 0.0147 hour per mile. 0.0147 gives their average speed in hours per mile because you are dividing hours by miles.

6. 20 miles per gallon

7. 0.05 gallon per mile

8. 2.5 gallons per hour

Challenge Problem

9. Days and years both cancel out, so the units would be in dollars.

LESSON 8: RATES AND UNITS



ANSWERS

ANSWERS

1. Train A: 70 miles per hour (fastest)

Train B: 60 miles per hour

Train C: 55 miles per hour (slowest; given)

2.

Time (hr)10

50

0

100

150

200

250

300

350

400

450

500

2 3 4 5 6

Dis

tan

ce (

mi)

y

x

Train A

Train B

Train C

Train A produces the steepest line.

3. Caroline: 6.875 yards per second

Aiko: about 7.333 yards per second

LESSON 13: RATES AND GRAPHS



ANSWERSLESSON 13: RATES AND GRAPHS

4.

Aiko

Caroline

10

50

100

150

200

250

300

350

400

450

20 30 40 50 60 70

y

x00

Time (sec)

Dis

tan

ce (

yd)

5.

AikoCaroline

10

50

100

150

200

250

300

350

400

450

20 30 40 50 60 70

y

x

Time (sec)

Dis

tan

ce (

yd)

00



ANSWERSLESSON 13: RATES AND GRAPHS

6.

Time (sec)

Aiko

Caroline

100

50

0

100

150

200

250

300

350

400

450

20 30 40 50 60

Dis

tan

ce (

yd)

y

x

Challenge Problem

7. Answers will vary. If your graph has average speed on the y-axis, it will be composed of horizontal straight lines, since speed is assumed constant. If your graph has average speed on the x-axis, it will be composed of vertical straight lines.



ANSWERS

ANSWERS

Answers may vary for problems 1–4. Here are examples:

1. d = 3t, where d gives distance in miles and t the time in hours

2. p = 32 a, where p gives the price in dollars and a the number of artichokes

3. d = 7t, where d gives the distance in miles and t the time in hours

4. d = 25g, where d gives the distance in miles and g the number of gallons

5. 2512

pages per second; 1225

seconds per page

6. p = 125m; p = 2512

s

7. m = 1

125 p; s =

1225

p

Challenge Problem

8. Answers will vary.

LESSON 14: RATES AND FORMULAS



ANSWERS

ANSWERS

1. The graph is a straight line and goes through the origin and the point (1, 60).

2. Multiplying any amount of time in minutes by 60 gives the appropriate number of pages.

3. The line must be a straight line that passes through the origin and the point (1, 30).

30

60

90

42 x

y

Time (min)

Nu

mb

er o

f Pa

ges

1 3 5

120

150

00

4. p = 30t

5. The line must be a straight line that passes through the origin and the point (1, 120).

120

0

240

360

42 x

y

Time (min)

Nu

mb

er o

f Pa

ges

10 3 5

480

600

6. p = 120t

LESSON 15: REPRESENTATIONS OF RATES



ANSWERSLESSON 15: REPRESENTATIONS OF RATES

Challenge Problem

7. The three graphs all show the relationship between the number of pages and the time it takes to print them. The three graphs appear to have the same slope when you scale the axes; however, greater rates would produce steeper slopes if you drew all graphs on the same coordinate plane.

math grade 6 unit 6 rate - rusd math -...

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