warm up divide. round answers to the nearest tenth. 1. 2. 3. 4

Ratios, Rates, and Unit Rates5-2

Warm UpDivide. Round answers to the nearest tenth.

1. 2.

3. 4.

23.3 3.5

23.8 23.9

420 18

73 21

380 16

430 18


Learn to work with rates and ratios.


Vocabulary

rateunit rateunit price


Ratio: 903

Rate: 90 miles3 hours

Read as “90 miles per 3 hours.”

A rate is a comparison of two quantities that have different units.


Unit rates are rates in which the second quantity is 1.

unit rate: 30 miles,1 hour

or 30 mi/h

The ratio 903

can be simplified by dividing:

903

= 301


Additional Example 1: Finding Unit Rates

Geoff can type 30 words in half a minute. How many words can he type in 1 minute?

Write a rate.

=

Geoff can type 60 words in one minute.

Multiply to find words per minute.

60 words 1 minute

30 words minute

12

30 words • 2 minute • 212


Check It Out: Example 1

Penelope can type 90 words in 2 minutes. How many words can she type in 1 minute?

90 words 2 minutes

Write a rate.

=

Penelope can type 45 words in one minute.

90 words ÷ 2 2 minutes ÷ 2

Divide to find words per minute.

45 words 1 minute


Additional Example 2A: Chemistry Application

Five cubic meters of copper has a mass of 44,800 kilograms. What is the density of copper?

Copper has a density of 8,960 kg/m3.

44,800 kg5 m3

Write a rate.

Divide to find kilograms per 1 m3.

44,800 kg ÷ 55 m3 ÷ 5

8,960 kg1 m3


Additional Example 2B: Chemistry Application

A piece of gold with a volume of 0.5 cubic meters weighs 9650 kilograms. What is the density of gold?

Gold has a density of 19,300 kg/m3.

9650 kg0.5 m3

Write a rate.

Multiply to find kilograms per 1 m3.

9650 kg • 20.5 m3 • 2

19,300 kg1 m3


Check It Out: Example 2A

Four cubic meters of precious metal has a mass of 18,128 kilograms. What is the density of the precious metal?

Precious metal has a density of 4,532 kg/m3.

18,128 kg4 m3

Write a rate.

Divide to find kilograms per 1 m3.

18,128 kg ÷ 44 m3 ÷ 4

4,532 kg1 m3


Check It Out: Example 2B

A piece of gem stone with a volume of 0.25 cubic meters weighs 3540 kilograms. What is the density of the gem stone?

The gem stone has a density of 14,160 kg/m3.

3540 kg0.25 m3

Write a rate.

Multiply to find kilograms per 1 m3.

3540 kg • 40.25 m3 • 4

14,160 kg1 m3


A driver is competing in a 500-mile auto race.

Additional Example 3A: Application

Find the ratio of distance to time.

In the first 2 hours of the race, the driver travels 356 miles. What is the driver's average speed?

The driver's average speed is 178 mi/h.

=356 mi

2 h

= 178 mi/h

Substitute 356 for d and 2 hours for t.

dt

r =

Divide to find the unit rate.


A driver is competing in a 500-mile auto race.

Additional Example 3B: Application

Use the formula d = rt.

The driver estimates that he will finish the race in less than 3 hours. If the driver keeps traveling at the same average speed, is his estimate reasonable? Explain.

500 = 178t Substitute 500 for d and 178 for r.

Determine how long the trip will take.

_ ___ 178 178 Divide both sides by 178.

Simplify.2.8 ≈ t

Yes; at an average speed of 178 mi/h, the race will take about 2.8 hours.


Helpful Hint

The formula r = is equivalent to d= rt,

as shown below.

r =

r ▪ t = ▪ t

rt = d

d t

d t

d t


A cyclist is competing in a 70-mile bike race.

Check It Out: Example 3A

Find the ratio of distance to time.

In the first 2 hours of the race, the cyclist travels 14 miles. What is the cyclist's average speed?

The cyclist's average speed is 7 mi/h.

=14 mi2 h

= 7 mi/h

Substitute 14 for d and 2 hours for t.

dt

r =

Divide to find the unit rate.


Check It Out: Example 3B

Use the formula d = rt.

The cyclist estimates that he will finish the race in less than 8 hours. If the cyclist keeps traveling at the same average speed, is the estimate reasonable? Explain.

70 = 7t Substitute 70 for d and 7 for r.

Determine how long the trip will take.

_ ___ 7 7 Divide both sides by 7.

Simplify.10 = t

No; at an average speed of 7 mi/h, the race will take about 10 hours.

A cyclist is competing in a 70-mile bike race.


Unit price is a unit rate used to compare price per item.


Jamie can buy a 15-oz jar of peanut butter for $2.19 or a 20-oz jar for $2.78. Which is the better buy?

Additional Example 4: Finding Unit Prices to Compare Costs

$2.1915

= $0.15

= $2.7820

$0.14

The better buy is the 20-oz jar for $2.78.

price for jarnumber of ounces

price for jarnumber of ounces

Divide the price by the number of ounces.


Golf balls can be purchased in a 3-pack for $4.95 or a 12-pack for $18.95. Which is the better buy?

Check It Out: Example 4

Divide the price by the number of balls.

price for packagenumber of balls

$4.953

= $1.65

price for packagenumber of balls

= $18.9512

$1.58

The better buy is the 12-pack for $18.95.


Standard Lesson Quiz

Lesson Quizzes

Lesson Quiz for Student Response Systems


Lesson Quiz: Part I

1. Meka can make 6 bracelets per half hour. How many bracelets can she make in 1 hour?

2. A penny has a mass of 2.5 g and a volume of

approximately 0.360 cm3. What is the

approximate density of a penny?

3. Melissa is driving to her grandmother's house.

In the first 3 hours of the drive, she travels 159

miles. What is Melissa's average speed?

≈ 6.94 g/cm3

53 mi/h

12


Lesson Quiz: Part II

Determine the better buy.

5. A half dozen carnations for $4.75 or a dozen for

$9.24 a dozen


1. John can walk 16 miles in 4 hours. How many

miles can he walk in one hour?

A. 16 miles

B. 8 miles

C. 4 miles

D. 2 miles



2. Estimate the unit rate.

272 sailors in 17 ships

A. 12 sailors per ship

B. 14 sailors per ship

C. 16 sailors per ship

D. 18 sailors per ship



3. Which of the following would be a better buy than purchasing 4 mangoes for $16?

A. 2 mangoes for $10

B. half a dozen mangoes for $25

C. 8 mangoes for $ 28

D. one dozen mangoes for $54


warm up divide. round answers to the nearest tenth. 1. 2. 3. 4

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