Evaluating Algebraic Expressions

5-2 Rates and Unit Rates

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California StandardsCalifornia Standards

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Evaluating Algebraic Expressions

5-2 Rates and Unit Rates

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

Evaluating Algebraic Expressions

5-2 Rates and Unit Rates

California Standards

MG1.3 Use measures expressed as rates (e.g., speed, density) and measures expressed as products (e.g., person-days) to solve problems; check the units of the solutions; and use dimensional analysis to check the reasonableness of the answer.

Evaluating Algebraic Expressions

5-2 Rates and Unit Rates

Ratio: 903

Rate: 90 miles3 hours

Read as “90 miles per 3 hours.”

A rate is a comparison of two quantities measured in different units.

Evaluating Algebraic Expressions

5-2 Rates and Unit Rates

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

Evaluating Algebraic Expressions

5-2 Rates and Unit RatesAdditional 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

Evaluating Algebraic Expressions

5-2 Rates and Unit RatesCheck 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

Evaluating Algebraic Expressions

5-2 Rates and Unit RatesAdditional 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 the rate.

Divide to find kilograms per 1 m3.

44,800 kg ÷ 55 m3 ÷ 5

8,960 kg1 m3

Evaluating Algebraic Expressions

5-2 Rates and Unit RatesAdditional 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 the rate.

Multiply to find kilograms per 1 m3.

9650 kg • 20.5 m3 • 2

19,300 kg1 m3

Evaluating Algebraic Expressions

5-2 Rates and Unit RatesCheck 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?

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

18,128 kg4 m3

Write the rate. Divide to find kilograms per 1 m3.

18,128 kg ÷ 44 m3 ÷ 4

4,532 kg1 m3

Evaluating Algebraic Expressions

5-2 Rates and Unit RatesCheck It Out! Example 2B

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

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

3540 kg0.25 m3

Write the rate.

Multiply to find kilograms per 1 m3.

3540 kg • 40.25 m3 • 4

14,160 kg1 m3

Evaluating Algebraic Expressions

5-2 Rates and Unit Rates

Estimate each unit rate.

Additional Example 3A: Estimating Unit Rates

Choose a number close to 468 that is divisible by 91.

468 students to 91 computers

468 students to 91 computers is approximately 5 students per computer.

468 students91 computers

455 students91 computers

5 students1 computer

Divide to find students per computer.

Evaluating Algebraic Expressions

5-2 Rates and Unit Rates

Estimate each unit rate.

Additional Example 3B: Estimating Unit Rates

Choose a number close to 313 that is divisible by 8.

313 feet in 8 seconds

313 feet to 8 seconds is approximately 40 feet per second.

313 feet8 seconds

320 feet8 seconds

40 feet1 second

Divide to find feet per second.

Evaluating Algebraic Expressions

5-2 Rates and Unit Rates

Estimate each unit rate.

Check It Out! Example 3A

Choose a number close to 583 that is divisible by 85.

583 soccer players to 85 soccer balls.

583 soccer players to 85 soccer balls is approximately 7 players per soccer ball.

583 players85 soccer balls

595 players85 soccer balls

7 players1 soccer ball

Divide to find players per soccer ball.

Evaluating Algebraic Expressions

5-2 Rates and Unit Rates

Estimate each unit rate.

Check It Out! Example 3B

Choose a number close to 271 that is divisible by 3.

271 yards in 3 hours

271 yards to 3 hours is approximately 90 yards per hour.

271 yards3 hours

270 yards3 hours

90 yards1 hour

Divide to find yards per hour.

Evaluating Algebraic Expressions

5-2 Rates and Unit Rates

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

Evaluating Algebraic Expressions

5-2 Rates and Unit Rates

Pens can be purchased in a 5-pack for $1.95 or a 15-pack for $6.20. Which pack has the lower unit price?

Additional Example 4A: Finding Unit Prices to Compare Costs

Divide the price by the number of pens.

price for packagenumber of pens

=$1.955

= $0.39

price for packagenumber of pens

= $6.2015

$0.41

The 5-pack for $1.95 has the lower unit price.

Evaluating Algebraic Expressions

5-2 Rates and Unit Rates

Jamie can buy a 15 oz jar of peanut butter for $2.19 or a 20 oz jar for $2.78. Which jar has the lower unit price?

Additional Example 4B: Finding Unit Prices to Compare Costs

$2.1915

= $0.15

= $2.7820

$0.14

The 20 oz jar for $2.78 has the lower unit price.

price for jarnumber of ounces

price for jarnumber of ounces

Divide the price by the number of ounces.

Evaluating Algebraic Expressions

5-2 Rates and Unit Rates

Golf balls can be purchased in a 3-pack for $4.95 or a 12-pack for $18.95. Which pack has the lower unit price?

Check It Out! Example 4A

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 12-pack for $18.95 has the lower unit price.

Evaluating Algebraic Expressions

5-2 Rates and Unit RatesCheck It Out! Example 4B

John can buy a 24 oz bottle of ketchup for $2.19 or a 36 oz bottle for $3.79. Which bottle has the lower unit price?

$2.1924

= $0.09

= $3.7936

$0.11

The 24 oz jar for $2.19 has the lower unit price.

price for bottlenumber of ounces

price for bottlenumber of ounces

Divide the price by the number of ounces.

Evaluating Algebraic Expressions

5-2 Rates and Unit RatesLesson Quiz: Part I

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

Estimate each unit rate.

3. $2.22 for 6 stamps

4. 8 heartbeats in 6 seconds

$0.37 per stamp

≈ 6.94 g/cm3

1.3 beats/s

12

Evaluating Algebraic Expressions

5-2 Rates and Unit RatesLesson Quiz: Part II

Find each unit price. Then tell which has the

lower unit price.

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

for $9.24

6. 4 pens for $5.16 or a ten-pack for $12.90.

a dozen

They cost the same.