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How to solve proportions, one step at a time

Turn two equal ratios into one simple equation, then check your answer.

A proportion says that two ratios have the same value. In 3/4 = x/20, the unknown is the top number on the right. The goal is to keep the two ratios equal, not to make their individual numbers equal.

Start with the scale factor

Look at the matching positions. The left denominator is 4 and the right denominator is 20. Multiplying 4 by 5 gives 20, so multiply the left numerator by the same 5: 3 × 5 = 15. Therefore x = 15. This is often the quickest approach when one number is an obvious multiple of another.

Use cross multiplication when scaling is awkward

For a/b = c/d, with b and d nonzero, multiplying both sides by bd gives ad = bc. This is why the diagonal products must match. It is an algebraic consequence of equality, rather than a new rule to memorize without explanation.

1. Write 3/4 = x/20.
2. Multiply across: 3 × 20 = 4 × x.
3. Simplify: 60 = 4x.
4. Divide both sides by 4: x = 15.

Check using the original ratios

Substitute 15 into the original expression: 3/4 = 15/20. Both simplify to 3/4, so the result works. Checking the original expression also catches forbidden zero denominators that a transformed equation might hide.

The unknown can be in any position

If the problem is 3/x = 9/12, cross multiplication gives 36 = 9x, so x = 4. Do not automatically multiply the two visible bottom numbers: identify the diagonal pairs first. For a complete set of arrangements, see the proportion formula reference.

Two habits that prevent mistakes

Keep the order of your quantities consistent. If one side is apples per dollar, the other side must also be apples per dollar. Also, scale by multiplication rather than addition: adding 5 to both 3 and 4 changes 3/4 into 8/9, which is a different ratio.

When an exercise includes fractional entries, use the same method and exact fraction arithmetic. The fraction guide walks through this without rounding. Then try a few questions in practice mode, where an incorrect answer is a chance to inspect the method.