Rule of Three Calculator

When to use it

Three numbers and one missing fourth: that is the shape of many everyday questions — a recipe for four people, a price per kilogram, a map scale, a job that takes four workers six days. All are proportions, solved the same way once you know whether more means more or more means less.

Direct proportion covers what scales together: double the weight, double the price. Inverse covers what trades off: more workers, fewer days. That choice matters more than the arithmetic.

How the maths works

In a direct proportion A is to B as C is to X, so X = B × C ÷ A: B ÷ A is the amount per unit of A, multiplied by C. In an inverse proportion the product stays constant, so X = A × B ÷ C. The test: if C doubles, should X double or halve?

A zero in the wrong place breaks the sum: A cannot be zero in a direct proportion, nor C in an inverse one.

A worked example

Three metres of cable cost 12; what do five metres cost?

1. Price per metre: 12 ÷ 3 = 4.
2. Five metres: 4 × 5 = 20.
3. In the tool: A = 3, B = 12, C = 5, direct proportion.
4. Check: 3 is to 12 as 5 is to 20, and 20 ÷ 5 equals 12 ÷ 3.

The inverse case: four workers finish in six days, so three take 4 × 6 ÷ 3 = 8 days — fewer workers, more days, the same 24 worker-days.

Common proportions

The last row keeps its product constant at 24 worker-days, which is what makes it inverse rather than direct.

Common mistakes

• Choosing the wrong type. Ask whether doubling C should double the answer or halve it.
• Changing units. 1.5 kg and 2,500 g are not the same field until one is converted.
• Assuming everything is linear. Cooking time, paint drying and fuel use per kilometre are not: doubling the quantity rarely doubles the time.
• Forcing a zero through. A zero in A or C leaves no proportion to solve.

Keep exploring

• Ratio Simplifier — reduce a two-number ratio to its simplest form first.
• Unit Price Comparison — the price-per-unit comparison behind most of these questions.
• Fraction Simplifier — the same relationship written as a fraction.
• Guide: Grades, Fractions and Proportions.