Solving Proportions

What is a Proportion?

A proportion states that two ratios are equal.

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

Example: $\frac{2}{3} = \frac{4}{6}$ is a proportion.

Cross Products

Cross Product Property: $\frac{a}{b} = \frac{c}{d} \text{ means } ad = bc$

Visual: $\frac{a}{b} = \frac{c}{d} \rightarrow a \times d = b \times c$

Solving Proportions

Use cross products to find the missing value.

Example: Solve $\frac{3}{x} = \frac{9}{15}$

Step 1: Cross multiply $3 \times 15 = 9 \times x$ $45 = 9x$

Step 2: Solve $x = 5$

Word Problems

Strategy:

1. Set up the proportion (keep units aligned)
2. Cross multiply
3. Solve for the unknown

Example: If 3 apples cost \2$, how much do 12 apples cost?

$\frac{3 \text{ apples}}{\$2} = \frac{12 \text{ apples}}{x}$

Scale Problems

Maps and models use proportions for scale.

Example: If 1 inch represents 50 miles, how many miles does 3.5 inches represent? $\frac{1 \text{ in}}{50 \text{ mi}} = \frac{3.5 \text{ in}}{x \text{ mi}}$