Solve Proportions

A proportion is a statement that two ratios are equal. We write it with an equals sign between two fractions: 2/3 = 8/12. When one of the four numbers is missing, we can solve for it using a powerful shortcut called cross-multiplication.

What Is a Proportion?

• A ratio compares two quantities, like 2 cups of flour to 3 cups of milk (about 480 mL to 720 mL): 2/3.
• A proportion says two ratios name the same relationship: 2/3 = 4/6 = 8/12.
• Equivalent ratios are produced by multiplying (or dividing) the top and bottom by the same factor.
• If the cross products of two ratios are equal, the ratios form a proportion.

Cross-Multiplication — The Solving Trick

1. Write the proportion as two fractions joined by =, with x for the unknown.
2. Multiply the top of one fraction by the bottom of the other (the cross products).
3. Set the two cross products equal to each other to make a one-step equation.
4. Divide both sides by the number multiplied by x to get x by itself.

Worked Example: 2/3 = x/12

1. Cross products: 2 times 12 and 3 times x.
2. Set equal: 2 times 12 = 3 times x, so 24 = 3x.
3. Divide both sides by 3: x = 8.
4. Check: 2/3 and 8/12 — both reduce to 2/3, so the proportion is true.

Real-World Proportions

• Recipes: if 2 cups of flour serve 3 people, how much flour serves 12 people? Set up 2/3 = x/12.
• Maps: if 1 inch (about 2.5 cm) on a map is 50 miles (about 80 km), what does 4 inches stand for? Set up 1/50 = 4/x.
• Speed: a car going 60 miles in 1 hour (about 96 km/h) will go how far in 3 hours? Set up 60/1 = x/3.
• Always pair the same kind of quantity in matching positions (top to top, bottom to bottom).