Solving Proportions

Use cross multiplication to solve proportion problems

What is a Proportion?

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

To solve proportions, use cross multiplication:

$\frac{a}{b} = \frac{c}{d} \quad \Rightarrow \quad ad = bc$

Example: $\frac{3}{4} = \frac{x}{12}$ $3 \times 12 = 4 \times x$ $36 = 4x$ $x = 9$

Substitute back to verify: $\frac{3}{4} = \frac{9}{12} \quad \checkmark$

Solve for $x$ : $\frac{2}{5} = \frac{x}{15}$

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Solution:

Cross multiply: $2 \times 15 = 5 \times x$ $30 = 5x$ $x = 6$

Check: $\frac{2}{5} = \frac{6}{15}$ ✓ (both simplify to $\frac{2}{5}$ )

Answer: $x = 6$

If 3 pounds of apples cost $\$ 4.50$, how much do 7 pounds cost?

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Solution:

Set up proportion: $\frac{3 \text{ pounds}}{\$4.50} = \frac{7 \text{ pounds}}{x}$

Cross multiply: $3x = 4.50 \times 7$ $3x = 31.50$ $x = 10.50$

Answer: $\$ 10.50$

On a map, 2 inches represents 50 miles. Two cities are 7 inches apart on the map. What is the actual distance between them?

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Solution:

Set up proportion: $\frac{2 \text{ inches}}{50 \text{ miles}} = \frac{7 \text{ inches}}{x \text{ miles}}$

Cross multiply: $2x = 50 \times 7$ $2x = 350$ $x = 175$

Answer: 175 miles

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