Solving Proportions

Using cross multiplication to solve proportions

Proportions

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

Example: $\frac{2}{3} = \frac{4}{6}$ is a proportion because both simplify to the same value.

To solve a proportion, use cross multiplication:

If $\frac{a}{b} = \frac{c}{d}$ , then $ad = bc$

Example: Solve $\frac{x}{5} = \frac{3}{15}$

Cross multiply: $15x = 5 \cdot 3$ $15x = 15$ $x = 1$

Two ratios form a proportion if their cross products are equal.

Example: Is $\frac{3}{4} = \frac{9}{12}$ ?

Check: $3 \cdot 12 = 36$ and $4 \cdot 9 = 36$ ✓

Yes, they form a proportion!

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

💡 Show Solution

Use cross multiplication:

$3x = 6 \cdot 2$ $3x = 12$ $x = 4$

Check: $\frac{4}{6} = \frac{2}{3}$ → both equal $\frac{2}{3}$ ✓

Answer: $x = 4$

Solve: $\frac{5}{x+2} = \frac{3}{4}$

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Cross multiply:

$5 \cdot 4 = 3(x + 2)$ $20 = 3x + 6$

Subtract 6: $14 = 3x$

Divide by 3: $x = \frac{14}{3}$

Answer: $x = \frac{14}{3}$

A recipe calls for 2 cups of flour for every 3 cups of sugar. If you use 8 cups of flour, how many cups of sugar do you need?

💡 Show Solution

Set up a proportion: $\frac{\text{flour}}{\text{sugar}} = \frac{2}{3} = \frac{8}{x}$

Cross multiply: $2x = 3 \cdot 8$ $2x = 24$ $x = 12$

Answer: 12 cups of sugar

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