Ratios and Proportions

Ratios

A ratio compares two quantities: $\frac{a}{b}$ or $a : b$

Proportions

A proportion is an equation stating two ratios are equal: $\frac{a}{b} = \frac{c}{d}$

Cross Multiplication

To solve a proportion, cross multiply: $\frac{a}{b} = \frac{c}{d} \implies ad = bc$

Example: $\frac{3}{4} = \frac{x}{20}$ $3 \times 20 = 4 \times x$ $60 = 4x$ $x = 15$

Word Problems with Proportions

If 3 apples cost \2.25$, how much do 8 apples cost?

$\frac{3}{2.25} = \frac{8}{x}$ $3x = 2.25 \times 8 = 18$ $x = \$6.00$

Scale Factor

The ratio of corresponding lengths in similar figures:

If a model car is $\frac{1}{24}$ scale and the model is 7.5 inches: $\text{Real length} = 7.5 \times 24 = 180 \text{ inches} = 15 \text{ feet}$

Similar Figures

Figures are similar if:

• Corresponding angles are equal
• Corresponding sides are proportional

Tip: Set up proportions so the same type of quantity is in each fraction's numerator (or denominator).