Solving Proportions

A proportion is an equation that says two ratios are equal. Solving a proportion means finding the unknown value that makes the two ratios equal. The main tool is cross-multiplication, which transforms a proportion into a simple equation.

If you need a refresher on what ratios are, start with Ratios and Proportions first.

The Cross-Multiplication Method

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

$a \times d = b \times c$

This works because equivalent fractions have equal cross products.

Example 1: Solve $\frac{3}{5} = \frac{x}{20}$

Step 1: Cross-multiply:

$3 \times 20 = 5 \times x$

$60 = 5x$

Step 2: Divide both sides by 5:

$x = \frac{60}{5} = 12$

Check: $\frac{3}{5} = \frac{12}{20}$. Simplify: $\frac{12}{20} = \frac{3}{5}$ ✓

Example 2: Solve $\frac{7}{x} = \frac{21}{30}$

Step 1: Cross-multiply:

$7 \times 30 = x \times 21$

$210 = 21x$

Step 2: Divide both sides by 21:

$x = \frac{210}{21} = 10$

Check: $\frac{7}{10} = \frac{21}{30}$. Simplify: $\frac{21}{30} = \frac{7}{10}$ ✓

Example 3: Solve $\frac{x}{9} = \frac{8}{12}$

Step 1: Cross-multiply:

$x \times 12 = 9 \times 8$

$12x = 72$

Step 2: Divide:

$x = \frac{72}{12} = 6$

Check: $\frac{6}{9} = \frac{8}{12}$. Both simplify to $\frac{2}{3}$ ✓

Setting Up Proportions from Word Problems

The key to word problems is making sure matching quantities are in the same position on both sides. There are two ways to set it up — both work:

Option A — same units across:

$\frac{\text{miles}}{\text{hours}} = \frac{\text{miles}}{\text{hours}}$

Option B — same scenario down:

$\frac{\text{miles}_1}{\text{miles}_2} = \frac{\text{hours}_1}{\text{hours}_2}$

Example 4: Scaling a Recipe

A recipe serves 4 people and calls for 6 cups of flour. How much flour do you need to serve 10 people?

Set up: Flour and people should be in the same positions:

$\frac{6 \text{ cups}}{4 \text{ people}} = \frac{x \text{ cups}}{10 \text{ people}}$

Cross-multiply: $6 \times 10 = 4 \times x$

$60 = 4x$

$x = 15$

Answer: 15 cups of flour.

Example 5: Map Scale

On a map, 2 inches represents 50 miles. How many miles does 7 inches represent?

$\frac{2 \text{ in}}{50 \text{ mi}} = \frac{7 \text{ in}}{x \text{ mi}}$

Cross-multiply: $2x = 350$

$x = 175$

Answer: 175 miles.

Example 6: Unit Price

If 3 pounds of apples cost $4.50, how much do 8 pounds cost?

$\frac{3 \text{ lb}}{4.50} = \frac{8 \text{ lb}}{x}$

Cross-multiply: $3x = 36$

$x = 12$

Answer: $12.00.

Proportions with Decimals

Cross-multiplication works the same way with decimal values.

Example 7: Solve $\frac{2.5}{4} = \frac{x}{10}$

$2.5 \times 10 = 4 \times x$

$25 = 4x$

$x = 6.25$

Common Mistake: Mixing Up Positions

The most frequent error is placing quantities in mismatched positions:

Wrong: $\frac{6 \text{ cups}}{4 \text{ people}} = \frac{10 \text{ people}}{x \text{ cups}}$ ← cups and people swapped on the right

Correct: $\frac{6 \text{ cups}}{4 \text{ people}} = \frac{x \text{ cups}}{10 \text{ people}}$ ← same units in same positions

Label your ratios to avoid this mistake.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• A proportion states that two ratios are equal
• Cross-multiplication converts a proportion into a solvable equation: $\frac{a}{b} = \frac{c}{d}$ means $ad = bc$
• Always put matching units in matching positions when setting up proportions
• Check your answer by substituting back and verifying the ratios are equal
• Watch out for inverse proportions — more workers means less time, not more

Return to Arithmetic for more foundational math topics.