Unit Rates

Unit Rates

Vocabulary

• A rate is a ratio that compares two quantities measured in different units.

• The unit rate is the rate for one unit of a given quantity. Unit rates have a denominator of 1.

ExamplesRate: 150 heartbeats

2 minutes

Unit Rate (Divide to get it):

150 ÷ 2 = 75 heartbeats per minute.

Find the Unit RateAmy can read 88 pages in 4 hours. What is the unit rate? (How many pages can she read per hour?)

88 pages4 hours

22 pages / hour

Using Unit Rates

• You can find the missing terms of equal ratios.

• Use the unit rate, and set it equal to another ratio.

• Solve for what is missing by dividing or multiplying.

ExampleJoe’s car goes 25 miles per gallon of gasoline. How far can it go on 8 gallons of gasoline?

25 miles1 gallonUnit

Rate

=8 gallons

x 8

x 8

25 x 8 = 200. Joe’s car can go 200 miles on 8 gallons of gas.

Comparing Unit Prices

• Use division to find the unit prices of the two products in question.

• The unit rate that is smaller (costs less) is the better value.

Example

Juice is sold in two different sizes. A 48-fluid ounce bottle costs $2.07. A 32-fluid ounce bottle costs $1.64. Which is the better buy?

$2.0748 fl.oz. $0.04 per fl.oz.

$1.6432 fl.oz.

$0.05 per fl.oz.

The 48 fl.oz. bottle is the better value.

Marge can clean the house in 3 hrs. Lisa can clean it in 5 hrs.

How long will it take them to clean the house if they both work together?

Marge can clean the house in 3 hrs.,

so she does 1/3 of the house per hour.

Lisa can clean the house in 5 hrs.,

so she does 1/5 of the house per hour.

Let T T be the time in hours.

13 TT + 1

5 TT = 1

1/3T + 1/5T = 1

15(1/3T + 1/5T) = 15(1)

5T + 3T = 15

8T = 15

T = 15/8

17/8 hrs.

Bart can wash the car in 20 min.Homer can wash it in 30 min.How long will it take them to wash the car if they both work together?

Bart can wash the car in 20 min.,

so he does 1/20 of the car per min.

Homer can wash the car in 30 min.,

so he does 1/30 of the car per min.

Let M M be the time in minutes.

120 MM + 1

30 MM = 1

1/20M + 1/30M = 1

60(1/20M + 1/30M) = 60(1)

3M + 2M = 60

5M = 60

M = 12 min.

Mr. Skinner can enroll 15 students per hr.

Mr. Chalmer can enroll 20 students per hr.

How long will it take them to enroll 140 students working together?

Minutes work better than hours.

Mr. Skinner enrolls 15 students per 60 min.,

so he enrolls 15/60 or 1/4 students per min.

Minutes work better than hours.

Mr. Chalmer enrolls 20 students per 60 min.,

so he enrolls 20/60 or 1/3 students per min.

1/4M + 1/3M = 140

12(1/4M + 1/3M) = 12(140)

3M + 4M = 1680

7M = 1680M = 240 min.

= 4 hrs.

Mrs. Krabappel can grade a set of papers in 11/2 hrs.

Miss Hoover can grade them in 80 min.

How long will it take them to grade one set if they both work together?

Minutes work better than hours.

Mrs. Krabappel grades 1/90 of a set per min.

Miss Hoover grades 1/80 of a set per min.

1/90M + 1/80M = 1

720(1/90M + 1/80M) = 720(1)

8M + 9M = 720

17M = 720

M = 426/17 min.

Approximately how long would it take them to grade 3 sets if they both work together?

1/90M + 1/80M = 3

720(1/90M + 1/80M) = 720(3)

8M + 9M = 2160

17M = 2160

M = 1271/17 min. A little more than 2 hrs.