Pre-Algebra

Ratios and Proportions

Lesson

A ratio compares two quantities. We can write it as $3 : 4$ or as the fraction $\frac{3}{4}$ .

A proportion says that two ratios are equal:

\frac{a}{b} = \frac{c}{d}

To solve a proportion for a missing value, use cross-multiplication: multiply diagonally and set the products equal. So $\frac{a}{b} = \frac{c}{d}$ means $a \cdot d = b \cdot c$ .

Worked example 1

Solve $\frac{x}{6} = \frac{2}{3}$ .

Cross-multiply:

x \cdot 3 = 6 \cdot 2

3x = 12

x = 4

Worked example 2 — real-world setup

A recipe needs 2 cups of flour for every 3 cookies. How many cups for 12 cookies?

Set up the proportion (flour over cookies, on each side):

\frac{2}{3} = \frac{x}{12}

Cross-multiply:

2 \cdot 12 = 3x

x = 8 \text{ cups}

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

\frac{x}{2} = \frac{6}{4}

Problem 2

\frac{3}{x} = \frac{6}{8}

Problem 3

\frac{x}{5} = \frac{4}{10}

Problem 4

\frac{2}{3} = \frac{x}{6}

Practice

Standard problems matching the lesson.

Problem 5

\frac{x}{6} = \frac{2}{3}

Problem 6

\frac{5}{x} = \frac{15}{18}

Problem 7

\frac{x}{8} = \frac{3}{4}

Problem 8

\frac{4}{5} = \frac{x}{20}

Problem 9

\frac{12}{x} = \frac{4}{5}

Problem 10

\frac{x}{14} = \frac{2}{7}

Problem 11

\frac{9}{12} = \frac{x}{4}

Problem 12

\frac{x}{10} = \frac{7}{2}

Problem 13

\frac{6}{x} = \frac{2}{9}

Problem 14

\frac{8}{3} = \frac{x}{12}

Problem 15

\text{2 cups flour : 3 cookies. } x \text{ cups for 12 cookies?}

Problem 16

\text{60 miles : 1 hour. } x \text{ miles in 4 hours?}

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 17

\frac{x}{15} = \frac{8}{12}

Problem 18

\frac{21}{x} = \frac{7}{4}

Problem 19

\frac{x}{18} = \frac{5}{6}

Problem 20

\text{3 in. : 5 mi. on a map. How many miles is 12 in.?}

Problem 21

\text{4 books cost \$18. How much for 10 books?}

Problem 22

\frac{x}{1.5} = \frac{8}{3}