Training Ratios and Proportions Ratios and Rates
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Ratios and Rates

20 min Ratios and Proportions

Ratios

A ratio is a comparison of two quantities, and a rate is a ratio that compares quantities measured in different units. When you say a car travels 60 miles per hour or a recipe calls for 2 cups of flour per 3 cups of sugar, you are using rates and ratios.

Ratios can be expressed in three ways — with the word "to," with a colon, or as a fraction — and understanding this flexibility is the first step toward solving proportion problems.

This lesson introduces ratios, rates, and unit rates, showing you how to simplify ratios and convert rates to their most useful per-unit form.

Definition

A ratio compares two quantities using division. It can be written:

  • $a$ to $b$
  • $a : b$
  • $\dfrac{a}{b}$

Ratios should be simplified like fractions.

Example 1

A class has 15 boys and 20 girls. Write the ratio of boys to girls in simplest form.

$$15 : 20 = 3 : 4$$

Example 2

A recipe uses 2 cups sugar and 5 cups flour. Ratio of sugar to total?

Total $= 2 + 5 = 7$. Ratio $= 2 : 7$.

Rates and Unit Rates

Definition

A rate compares two quantities with different units. A unit rate has a denominator of 1.

Example 3

240 miles in 4 hours. Find the unit rate.

$$\frac{240 \text{ mi}}{4 \text{ hr}} = 60 \text{ mi/hr}$$

Unit Price (Comparison Shopping)

Example 4

Brand A: 12 oz for $\$3.60$. Brand B: 16 oz for $\$4.48$. Which is the better deal?

A: $\dfrac{3.60}{12} = \$0.30$/oz. B: $\dfrac{4.48}{16} = \$0.28$/oz. Brand B is cheaper.

Example 5

A printer produces 45 pages in 3 minutes. Pages per minute?

$\dfrac{45}{3} = 15$ pages per minute.

Practice Problems

1. Simplify $24 : 36$.
2. A bag has 8 red, 12 blue, and 5 green marbles. Ratio of red to total?
3. 350 km in 5 hours. Unit rate?
4. Pack A: 6 rolls for $\$5.10$. Pack B: 8 rolls for $\$6.40$. Better deal?
5. A factory makes 720 items in an 8-hour shift. Rate per hour?
6. Ratio of 45 minutes to 2 hours (in same units).
7. A car uses 12 gal for 360 miles. Miles per gallon?
8. Simplify $1.5 : 2.5$.
9. Mix paint 3 parts blue to 2 parts white. How much blue for 10 parts total?
10. Earnings of $\$525$ for 35 hours. Hourly rate?
11. Write $\dfrac{2}{3}$ to $\dfrac{5}{6}$ as a ratio of whole numbers.
12. A runner covers 5 km in 22 minutes. Pace in min/km?
Show Answer Key

1. $2 : 3$

2. $8 : 25$

3. $70$ km/hr

4. Pack A: $\$0.85$/roll; Pack B: $\$0.80$/roll. Pack B

5. $90$ items/hr

6. $45 : 120 = 3 : 8$

7. $30$ mpg

8. $3 : 5$

9. $6$ parts blue

10. $\$15$/hr

11. $4 : 5$ (multiply both by 6)

12. $4.4$ min/km

Setting Up and Solving Proportions