How to Solve Proportions: Cross Multiplication Step by Step

A proportion is an equation that says two ratios are equal: a/b = c/d. When one of the four numbers is unknown, cross multiplication finds it in two short steps. It's one of the most useful techniques in school math, and it shows up in cooking, shopping, maps, science and finance.

Why cross multiplication works

Start with a/b = c/d. Multiply both sides by b, then by d. The denominators cancel and you're left with a × d = b × c. The two products are called cross products because they connect diagonally across the equals sign. Any true proportion has equal cross products, and that single fact lets you solve for the missing value.

The steps

Write the proportion with the unknown as a letter, such as x.
Multiply diagonally to form the equation a × d = b × c.
Divide both sides by the number multiplying x.
Check by substituting x back into the original ratios.

Example 1: unknown in the numerator

3/4 = x/20

Cross multiply: 4 × x = 3 × 20, so 4x = 60.
Divide by 4: x = 15.
Check: 3/4 = 0.75 and 15/20 = 0.75 ✓

Example 2: unknown in the denominator

5/x = 8/12

Cross multiply: 8x = 5 × 12 = 60.
Divide by 8: x = 7.5.

Example 3: decimals

2.4/1.5 = 6/x gives 2.4x = 9, so x = 3.75. Decimals change nothing about the method; a calculator just makes the arithmetic easier. The proportion calculator shows each line of working so you can copy it.

Example 4: an expression for x

(x + 2)/6 = 5/3

Cross multiply: 3(x + 2) = 30.
Distribute: 3x + 6 = 30.
Solve: 3x = 24, so x = 8.

Setting up word problems

The hardest part is writing the proportion correctly. Keep the same kind of quantity in the same position on both sides.

A car travels 150 miles on 5 gallons. How far on 8 gallons?

150 miles / 5 gallons = x miles / 8 gallons

5x = 1,200, so x = 240 miles. Writing units next to the numbers is the easiest way to catch a ratio set up upside down.

When not to use a proportion

Proportions only work when two quantities change at a constant rate. If you add a fixed fee, such as a taxi with a $3 base fare plus $2 per mile, the total cost is not proportional to distance. Likewise, when one quantity goes up as the other goes down, the relationship is inverse; see direct vs. inverse proportion.

Common mistakes

Multiplying straight across (a × c) instead of diagonally.
Flipping one ratio but not the other.
Mixing units, such as minutes on one side and hours on the other.
Dividing by zero: if the number multiplying x is 0, there is no single answer.

Practice

Try these, then check them in the calculator: 7/x = 21/30 (x = 10); 9/12 = x/16 (x = 12); 1.2/x = 3/10 (x = 4). For ratios with more than two terms, use the ratio calculator instead.