31 Rates, Ratios, and Proportions

Please put the power point into slideshow before starting the

guided notes.

31 Rates, Ratios, and Proportions

Warm UpSolve each equation. Check your answer.1. 6x = 36

2. 3. 5m = 18

Simplify.

4. 2.3 – 3.6 / 4 – 1.7 5. -0.4 + 1.3 * 4

0.5 – 5.1 / 3

6

48

3.6

-0.3 -4

31 Rates, Ratios, and Proportions

ratio proportionrate cross productsUnit Price unit rate

Vocabulary

Learning Targets• Write and use ratios, rates, and unit rates.• Write and solve proportions.

31 Rates, Ratios, and Proportions

A ratio is a comparison of two quantities by division. The ratio of a to b can be written a:b or , where b ≠ 0. Ratios that name the same comparison are said to be equivalent.

A statement that two ratios are equivalent, such as , is called a proportion.

31 Rates, Ratios, and Proportions

A rate is a ratio of two quantities with different units,

such as Rates are usually written as unit rates.

A unit rate is a rate with a second quantity of 1 unit,

such as or 17 mi/gal. You can convert any rate to

a unit rate.

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

31 Rates, Ratios, and Proportions

http://my.hrw.com/math11/math06_07/nsmedia/lesson_videos/alg1/player.html?contentSrc=7464/7464.xml

Click on the link below . . . (you must be in slideshow for it to work and be sure to have your headphones)

31 Rates, Ratios, and Proportions

Example 1: Finding Unit Rates

Raulf Laue of Germany flipped a pancake 416 times in 120 seconds to set the world record. Find the unit rate. Round your answer to the nearest hundredth.

The unit rate is about 3.47 flips/s.

Write a proportion to find an equivalent ratio with a second quantity of 1.

Divide 416 by 120 on the left side to find x.

31 Rates, Ratios, and Proportions

On Your Own

Cory earns $52.50 in 7 hours. Find the unit rate.

31 Rates, Ratios, and Proportions

http://my.hrw.com/math06_07/nsmedia/lesson_videos/msm3/player.html?contentSrc=7314/7314.xml

Click on the link below . . . (you must be in slideshow for it to work and be sure to have your headphones)

31 Rates, Ratios, and Proportions

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 1: Finding Unit Prices to Compare Costs

$2.19

15= $0.15

=$2.78

20 $0.14

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

price for jar

number of ounces

price for jar

number of ounces

Divide the price by the number of ounces.

31 Rates, Ratios, and Proportions

On Your Own

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

price for package

number of balls =

price for package

number of balls=

The better buy is the

31 Rates, Ratios, and Proportions

http://my.hrw.com/math06_07/nsmedia/lesson_videos/msm3/player.html?contentSrc=7315/7315.xml

Click on the link below . . . (you must be in slideshow for it to work and be sure to have your headphones)

31 Rates, Ratios, and Proportions

Example 2: Converting Rates

Serena ran a race at a rate of 10 kilometers per hour. What was her speed in kilometers per minute? Round your answer to the nearest hundredth.

The rate is about 0.17 kilometer per minute.

To convert the second quantity in a rate, multiply by a conversion factor with that unit in the first quantity.

31 Rates, Ratios, and Proportions

Step 1 Convert the speed to feet per hour.

A cyclist travels 56 miles in 4 hours. What is the cyclist’s speed in feet per second? Round your answer to the nearest tenth, and show that your answer is reasonable.

The speed is 73,920 feet per hour.

31 Rates, Ratios, and Proportions

Step 2 Convert the speed to feet per minute.

To convert the second quantity in a rate, multiply by a conversion factor with that unit in the first quantity.

The speed is 1232 feet per minute.

Step 3 Convert the speed to feet per second.

The speed is approximately 20.5 feet per second.

To convert the second quantity in a rate, multiply by a conversion factor with that unit in the first quantity.

31 Rates, Ratios, and Proportions

On Your Own

An engineer opens a valve that drains 60 gallons of water per minute from a tank. How many quarts were drained per second?

31 Rates, Ratios, and Proportions

In the proportion , the products a • d and

b • c are called cross products. You can solve

a proportion for a missing value by using the

Cross Products property.

Cross Products Property

WORDS NUMBERS ALGEBRA

In a proportion, cross products are equal.

2 • 6 = 3 • 4

If and b ≠ 0

and d ≠ 0then ad = bc.

31 Rates, Ratios, and Proportions

Example 3: Solving Proportions

Solve each proportion.

3(m) = 5(9)

3m = 45

m = 15

Use cross products.

Divide both sides by 3.

Use cross products.

6(7) = 2(y – 3)

42 = 2y – 6+6 +648 = 2y

24 = y

A. B.

Add 6 to both sides.Divide both sides by 2.

31 Rates, Ratios, and Proportions

http://my.hrw.com/math11/math06_07/nsmedia/lesson_videos/alg1/player.html?contentSrc=7466/7466.xml

Click on the link below . . . (you must be in slideshow for it to work and be sure to have your headphones)

31 Rates, Ratios, and Proportions

On your Own

Solve each proportion.

A. B.

31 Rates, Ratios, and Proportions

Example 5: Application

A contractor has a blueprint for a house drawn to the scale 1 in: 3 ft.

A wall on the blueprint is 6.5 inches long. How long is the actual wall?

blueprint 1 in. actual 3 ft.

x • 1 = 3(6.5)

x = 19.5The actual length of the wall is 19.5 feet.

Write the scale as a fraction.

Let x be the actual length.

Use the cross products to solve.

31 Rates, Ratios, and Proportions

On Your Own

A scale model of a human heart is 16 ft. long. The scale is 32:1. How many inches long is the actual heart it represents?

model 32 in. actual 1 in.

Write the scale as a fraction.

Use the cross products to solve.

Since x is multiplied by 32, divide both sides by 32 to undo the multiplication.

Let x be the actual length.Convert 16 ft to inches.