Ratios & Proportions

Ratios compare two quantities. Proportions say two ratios are equal. Both show up constantly in everyday life and on tests.

What Is a Ratio?

A ratio compares two values. You can write it three ways:

  3 to 5
  3:5
  3/5

A class has 12 boys and 16 girls. The ratio of boys to girls is 12:16, which simplifies to 3:4.

Order Matters "Boys to girls" (12:16) is different from "girls to boys" (16:12). Always check what the ratio is comparing and in which order.

Simplifying Ratios

Divide both numbers by their greatest common factor (GCF).

  Simplify 18:24
  GCF of 18 and 24 = 6
  18 ÷ 6 = 3
  24 ÷ 6 = 4
  Simplified ratio: 3:4

Unit Rate

A unit rate is a ratio with a denominator of 1. It answers "how much per one?"

Examples

  60 miles in 2 hours → 60 ÷ 2 = 30 miles per hour
  $4.50 for 3 apples → $4.50 ÷ 3 = $1.50 per apple
  180 words in 3 minutes → 60 words per minute

What Is a Proportion?

A proportion states that two ratios are equal: a/b = c/d. If one ratio holds true, the other must as well.

Is 3/4 = 9/12 a true proportion?

  Cross multiply: 3 × 12 = 36
                  4 × 9  = 36
  Both sides equal 36 → YES, it is a proportion.

Solving Proportions

Use cross multiplication to solve for a missing value.

Solve: 5/8 = x/24

  Cross multiply: 5 × 24 = 8 × x
                      120 = 8x
                        x = 120 ÷ 8
                        x = 15

Solve: x/6 = 10/15

  Cross multiply: x × 15 = 6 × 10
                     15x = 60
                       x = 60 ÷ 15
                       x = 4

Scale and Scaling Up

Proportions let you scale recipes, maps, blueprints, and models.

Recipe example

A recipe for 4 people needs 2 cups of flour. How much flour for 10 people?

  Set up proportion:  2/4 = x/10
  Cross multiply:     2 × 10 = 4 × x
                          20 = 4x
                           x = 5

  You need 5 cups of flour for 10 people.

Quick Quiz

Check your understanding. Click an answer to see if you got it right.

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