Chapter 2.4

Rates, Ratios, and

Proportions

Slide 5.1- 2

A ratio compares two quantities.

a. Ratio of amount spent on salad to amount spent on bread.

The ratio of salad to bread is

b. Ratio of fish to bread.

ParallelExample 1

Writing Ratios

Slide 5.1- 3

$8

$7

Numerator(mentioned first)

Denominator(mentioned second)

8

7

$13

$7

13

7

Tamara spent $13 on fish, $8 on salad and $7 on bread. Write each ratio as a fraction.

a. 80 days to 20 days.

Divide the numerator and denominator by 20.

b. 30 ounces of medicine to 140 ounces of medicine

ParallelExample 2

Writing Ratios in Lowest Terms

Slide 5.1- 4

80

20

2080

2 200

4

1

30

140

30

14

10

100

3

14

Write each ratio in lowest terms.

The price of a bag of dog food increased from $22.95 to $25.50. Find the ratio of the increase in price to the original price.

Find the ratio of the increase in price to the original price.

Now write the ratio as a ratio of whole numbers.

ParallelExample 3

Using Decimal Numbers in a Ratio

Slide 5.1- 5

new price – original price = increase

$25.50 $22.95 = $2.55

increase in price

origina

2

l

.55

22.9 pric5 e

2.55 2.55 100

22.95 22.95 100

255

2295

255

2295

255

255

1

9

Write each ratio as a comparison of whole numbers in lowest terms.

a. 4 days to

Write the ratio and divide out common units.

Write as improper fractions.

ParallelExample 4

Using Mixed Numbers in Ratios

Slide 5.1- 6

14 days

4

4

41

17

4 days144 days 1

4

4

4

14

4

474

1

1

4

1

4

1

4

7

16

17

Reciprocals

b.

ParallelExample 4

Using Mixed Numbers in Ratios

Slide 5.1- 7

34

8

935

8 4

35 4

8 9

12

4

3 14 pounds to 2 pounds

8 4

35

8

9

4

4

3589

1

38

4

4

2

35

8

2

4

1

9

35

18

Write each rate as a fraction in lowest terms.

a. 8 gallons of antifreeze for $40.

b. 192 calories in 6 ounces of yogurt

Slide 5.2- 8

ParallelExample 1 Writing Rates in Lowest Terms

8 gallons

40 dollars

8

8

1 gallon

5 dollars

1 692 calories

6 ounce s 6

32 calories

1 ounce

Write each rate as a fraction in lowest terms.

c. 84 hamburgers on 7 grills.

Slide 5.2- 9

ParallelExample 1continued

Writing Rates in Lowest Terms

84 hamburgers

7 gril

7

ls 7

12 hamburgers

1 grill

Slide 5.2- 10

Use per or a slash mark (/) when writing unit rates.

When the denominator of a rate is 1, it is called a unit rate. For example, you earn $16.25 for 1 hour of work.

This unit rate is written:

$16.25 per hour

Find each unit rate.

a.

Slide 5.2- 11

ParallelExample 2 Finding Unit Rates

445.5 miles on 16.5 gallons of gas

445.5 miles

16.5 gallons

16.5

16.5

27 miles

1 gallon

Divide to find the unit rate.

2716.5 445.5

445.5 miles

16.5 gallons

The unit rate is 27 miles per gallon or 27 miles/gallon.

Find each unit rate.

b.

Slide 5.2- 12

ParallelExample 2continued

Finding Unit Rates

413 feet in 14 seconds

413 feet

14 secondsDivide to find the unit rate.

29.514 413.0

The unit rate is 29.5 feet/second.

A local store charges the following prices for jars of jelly.

Slide 5.2- 13

ParallelExample 3 Determining the Best Buy

The best buy is the container with the lowest cost per unit. All the jars are measured in ounces. Find the cost per ounce for each one by dividing the price of the jar by the number of ounces in it. Round to the nearest thousandth if necessary.

18 oz. 24 oz. 28 oz.

$2.39$3.09 $3.69

ParallelExample 3continued

Determining the Best Buy

The lowest cost per ounce is $0.129, so the 24-ounce jar is the best buy.

Size Cost per Unit (rounded)

18 ounces

24 ounces

28 ounces

$2.39$0.133 per ounce

18 ounces

$3.09$0.129 per ounce

24 ounces

$3.69$0.132 per ounce

28 ounces

highest

lowest

Slide 5.2- 14

Juice is sold as a concentrated can as well as in a ready to serve carton. Which of the choices below is the best buy?

ParallelExample 4 Solving Best Buy Applications

12 oz can makes 48 ounces of juice for $1.69 60 oz carton for $2.59

To determine the best buy, divide the cost by the number of ounces.

Slide 5.2- 15

ParallelExample 4continued

Solving Best Buy Applications

Concentrate

Slide 5.2- 16

$1.69

48 ounces

Carton $2.59

60 ounces

Although, you must mix it yourself, the concentrated can of juice is the better buy.

12 oz can makes 48 ounces of juice for $1.69 60 oz carton for $2.59

$0.0352 per ounce

$0.0432 per ounce

Slide 5.4- 17

Four numbers are used in a proportion. If any three of these numbers are known, the fourth can be found.

Find the unknown number in each proportion. Round answers to the nearest hundredth when necessary.

ParallelExample 1

Solving Proportions for Unknown Numbers

Slide 5.4- 18

a.

Ratios can be written in lowest terms. You can do that before finding the cross products.

30 48

40x

30 6.

5x

can be written in lowest terms as , which gives the proportion

6

548

40

ParallelExample 1continued

Solving Proportions for Unknown Numbers

Slide 5.4- 19

30 6

5x

Show that the cross products are equivalent.

Step 1 6x

30 5Find the cross products

Step 2 6 30 5x

6 150x

Step 3 6 150

6 6

x

1

1

25x

ParallelExample 1continued

Solving Proportions for Unknown Numbers

Slide 5.4- 20

3 20

7 x

Show that the cross products are equivalent.

Step 1

3 x

7 20

Find the cross products

Step 2 3 140x

Step 3 3 140

3 3

x

1

46.67x

3 20

7 xb.

1

Rounded to the nearest hundredth.

Find the unknown number in each proportion.

ParallelExample 2

Solving Proportions with Mixed Numbers and Decimals

Slide 5.4- 21

a.

Find

236

8 36

x

26 36

3

8 x

Find the cross products

236

8 36

x

26 36.

3

20 36

3 1

26 36

3

240

1

1

12

240

Show the cross products are equivalent. 8 240x

Divide both sides by 8. 8 240

8 8

x 30x

ParallelExample 2continued

Solving Proportions with Mixed Numbers and Decimals

Slide 5.4- 22

26 36

3

8 30

Equal

236 30

8 36

The cross products are equal, so 30 is the correct solution.

240

240

Find the unknown number in each proportion.

ParallelExample 2continued

Solving Proportions with Mixed Numbers and Decimals

Slide 5.4- 23

b.

Show that cross products are equivalent.

10.4 6.76

12.4 x

10.4 ( ) (12.4)(6.76)x

Divide both sides by 10.4.

8.06x

10.4 83.824

10.4 10.4

x

10.4 ( ) 83.824x

ParallelExample 2continued

Solving Proportions with Mixed Numbers and Decimals

Slide 5.4- 24

10.4 6.76

12.4 8.06 Equal

10.4 ∙ 8.06 = 83.824

12.4 ∙ 6.76 = 83.824

The cross products are equal, so 8.06 is the correct solution.

Similar Triangles

• Similar Triangles whose angles have the same measure, but their sides have different lengths.

• The triangles will look identical, but one will be smaller than the other.

Slide 1- 25

Slide 1- 26

x

y

Slide 1- 27

Slide 1- 28

Solution

𝑌𝑜𝑢𝑟 h𝐻𝑒𝑖𝑔 𝑡𝑇𝑟𝑒𝑒 h𝐻𝑒𝑖𝑔 𝑡

=𝑌𝑜𝑢𝑟 h𝑆 𝑎𝑑𝑜𝑤 h𝐿𝑒𝑛𝑔𝑡𝑇𝑟𝑒𝑒 h𝑠 𝑎𝑑𝑜𝑤 h𝑙𝑒𝑛𝑔𝑡

Slide 1- 29

Hw section 2.4

1-20

Try 21

Slide 1- 30