RATIOS AND PROPORTIONS

INSTRUCTION SHEET

Ratios

A ratio is a comparison of two quantities that have the same units. You can express a ratio in any one of the following ways:

18 18:5 18 to 5

5

Example #1: If one store has 360 items and another store has 100 of the same items, express the ratio of the items.

360 or 360:100 or 360 to 100

100

Ratios are usually written in lowest terms; therefore, the above example would reduce in this way: 360 ÷ 20 (What is the largest number you

100 ÷ 20 can divide both values by?)

18

5

Example #2: John earns $350 a week. His take-home pay, however, is $295.

What is the ratio of his gross pay to his take-home pay.

350 = 70

295 59

Rates

A rate is a comparison of two quantities that have different units. Rates are usually expressed in the fractional form.

Example: Francine paid $16 for her 12-month subscription to Better Homes and Gardens magazine. Express as a rate.

$16.00 = $4.00

12 magazines 3 magazines

If Francine wants to know how much she pays for each (1) magazine, she can divide $4 by 3 magazines. This will give her the price per magazine (also called the unit rate).

$4.00 = $1.33/magazine

3

Proportions

A proportion is a statement that two ratios or rates are equal. It can be given as a sentence in words, but most often a proportion is an algebraic equation.

The arithmetic equation 3 = 21 is a proportion because its cross

5 35

products are equal.

3 × 35 = 105 and 5 × 21 = 105

Proportions are solved by using this cross-product rule.

Example #1: 4 = X Example #2: 72 = 12

9 36 1.5 x

4 × 36 = 9x 72x = 1.5 × 12

144 = 9x 72x = 18

144 = x x = 18

9 72

16 = x x = .25 or ¼

Applied Proportion Problems

Many problems can be solved by setting up a direct proportion (an increase in one quantity leads to a proportional increase in the other quantity) or by setting up equivalent rates.

Example: In one day you earn $75 for 8 hours of work. If you work 37.5 hours for the week, what will your weekly pay be?

8 hours = $75 8 hours = 37.5 hours

37.5 hours x $75 x

8x = 75 × 37.5 8x = 75 × 37.5

8x = 2812.5 8x = 2812.5

or

x = 2812.5 x = 2812.5

8 8

x = $351.56 x = $351.56

RATIOS AND PROPORTIONS

PRACTICE SHEET

A. Write each ratio as a fraction in lowest terms.

1. 2 to 4 6. 3 to 12 11. 35:7

2. 15 7. 7: 4 12. 8

20 28

8. 18

3. 6:18 12 13. 24 to 96

4. 21:15 9. 20:16 14. 9:27

5. 12 10. 15 to 36 15. 11

18 88

B. Write each of the following rates as a unit rate.

1. 3 Tbsp 2. 135 pitches

2 tsp 45 strikes

3. 128 miles 4. 2250 pencils

4 hours 18 boxes

5. $450 6. 2500 meters

18 shares 15 seconds

7. $5,082 8. 750 gallons

475 sq.yds. 14 minutes

C. Solve each proportion and give the answer in simplest form.

1. 6 : 8 = n : 12 2. 2 = 8

7 n

3. n = 11 4. 4 : n = 6 : 9

6 3

5. 3 = 2 6. 0.4 = 12

n 5 1.5 n

7. 2 ½ : 3 ½ = n : 2 8. 1: 2 = n : 9

9. 4 to 8 = 15 to n 10. 18 : n = 3 :11

11. 5 = n 12. 12 = n

6 30 40 25

13. 8 :19 = 14:n 14. 10 = 2

n 1.7

15. 24 : ¼ = n : ⅓ 16. 44 to 121 = n to 11

D. Solve by using a proportion. Round answers to the nearest hundredth if necessary.

1. You jog 3.6 miles in 30 minutes. At that rate, how long will it take you to jog 4.8 miles?

2. You earn $33 in 8 hours. At that rate, how much would you earn in 5 hours?

3. An airplane flies 105 miles in ½ hour. How far can it fly in 1 ¼ hours at the same rate of speed?

4. What is the cost of six filters if eight filters cost $39.92?

5. If one gallon of paint covers 825 sq. ft., how much paint is needed to cover 2640 sq. ft.?

6. A map scale designates 1” = 50 miles. If the distance between two towns on the map is 2.75 inches, how many miles must you drive to go from the first town to the second?

7. Bob is taking his son to look at colleges. The first college they plan to visit is 150 miles from their home. In the first hour they drive at a rate of 60 mph. If they want to reach their destination in 2 ½ hours, what speed must they average for the remainder of their trip?

8. Four employees can wash 20 service vehicles in 5 hours. How long would it take 5 employees to wash the same number of vehicles?

9. These two figures are similar. Use a proportion to find the length of side n.

20 m 12 m 30 m n