How to Solve Proportions: The Cross-Multiplication Method Explained

Published September 6, 2026 · 7 min read

If you have ever needed to scale a recipe, read a map, or resize a photo, you have solved a proportion. The good news is that almost every proportion problem can be cracked with one simple technique: cross-multiplication. This guide walks you through it step by step, with worked examples and the mistakes that trip most people up.

What is a proportion?

A ratio compares two quantities, such as 3:4 ("three to four"). A proportion is a statement that two ratios are equal:

You can write the same thing as two fractions, which is the form that makes the maths easy:

For example, 1:2 = 4:8 is a true proportion because both ratios simplify to one half. In most real problems, you know three of the four values and need to find the missing one.

The cross-multiplication rule

In any true proportion, the product of the outer values equals the product of the inner values. In fraction form, you multiply diagonally across the equals sign:

Once you have that equation, you can isolate whichever value is missing by dividing:

These four formulas are exactly what our ratio calculator uses when you leave one field blank.

Why does it work?

Start with A / B = C / D and multiply both sides by B × D. On the left, the B cancels and you get A × D. On the right, the D cancels and you get C × B. Nothing magical, just multiplying both sides of an equation by the same number.

Solving a proportion in four steps

1. Write it as two fractions. Put matching quantities in matching positions, such as both widths on top and both heights on the bottom.
2. Cross-multiply. Multiply the numbers diagonally to get an equation like 3 × 20 = 4 × x.
3. Divide to isolate the unknown. Divide the known product by the number attached to x.
4. Check your answer. Plug it back in and confirm both cross-products are equal.

Worked examples

Common mistakes to avoid

1. Mixing up the order

The most common error is putting values in inconsistent positions. If A is a width and B is a height, then C must also be a width and D a height. Writing width : height = height : width gives a wrong answer that still looks plausible.

2. Mixing units

If one ratio uses minutes and the other uses hours, convert first. 30 minutes : 2 pages = 1.5 hours : x pages will give nonsense unless you turn 1.5 hours into 90 minutes (the answer is 6 pages).

3. Using a direct proportion for an inverse relationship

Cross-multiplication assumes that when one quantity goes up, the other goes up by the same factor. Some relationships work the other way. If 4 workers take 6 days to finish a job, 8 workers do not take 12 days. They take fewer days. For inverse relationships, the product stays constant: 4 × 6 = 8 × x, so x = 3 days.

A quick sense check: ask yourself whether the answer should be bigger or smaller than the value you started with.

4. Dividing by zero

A ratio with a zero in it, such as 0:5, cannot be scaled to find a missing value on the other side. If you see a "cannot divide by zero" message, check that you have not entered 0 in one of the known fields.

How to check your answer in five seconds

Multiply the outer pair and the inner pair. If the two results are equal, your answer is correct. You can also divide each side: 3 ÷ 4 = 0.75 and 15 ÷ 20 = 0.75, so 3:4 = 15:20 holds.

Frequently asked questions