What is the rule of three?

It is a way to solve a proportion when you know three of four values. If 3 apples cost £6, then 5 apples cost (6 × 5) ÷ 3 = £10. You multiply crosswise and divide by the remaining value — which is why it is also called cross-multiplication.

What is the difference between direct and inverse proportion?

In direct proportion, more of one means more of the other: more apples, more money. In inverse proportion, more of one means LESS of the other: more workers, fewer hours. The formulas differ — direct is (B × C) ÷ A, inverse is (A × B) ÷ C.

How do I know which one to use?

Ask yourself: if I double the first quantity, does the answer double or halve? Doubling workers halves the time, so that is inverse. Doubling apples doubles the price, so that is direct. Getting this backwards is the single most common mistake.

Are my numbers sent anywhere?

No. The whole calculation runs in your browser on your own device. Nothing is uploaded, and the page keeps working offline once loaded.

Rule of Three Calculator

If A gives B, what does C give? Solves both direct and inverse proportion and shows the working. Everything runs in your browser.

If gives

then gives —

—

How the rule of three works

The rule of three solves a proportion when three of four values are known. For direct proportion the formula is x = (B × C) ÷ A: if 3 apples cost £6, then 5 apples cost (6 × 5) ÷ 3 = £10.

For inverse proportion it flips: x = (A × B) ÷ C. If 2 workers need 6 hours, then 4 workers need (2 × 6) ÷ 4 = 3 hours. More workers, less time.

How to tell which one you need

Ask one question: if I double the first quantity, does the answer double or halve? If it doubles, the relationship is direct. If it halves, it is inverse. Mixing these up is by far the most common mistake, and it usually only shows up when the answer looks implausible.

Typically direct: quantity and price, distance and time at a fixed speed, ingredients and servings. Typically inverse: workers and hours, speed and journey time, machines and production time.

A worked example, step by step

250 g of flour serves 4 people. How much for 6? That is direct — more people, more flour. So (250 × 6) ÷ 4 = 375 g. The intermediate step taught in school is to go via one unit: 250 ÷ 4 = 62.5 g per person, then 62.5 × 6 = 375 g. Both routes give the same answer.

Why this turns up everywhere

Scaling a recipe, converting fuel use to litres per 100 km, reading a map scale, currency conversion, mixing ratios for paint or concrete, dosing by body weight — all of it is the same proportion. Outside of plain arithmetic it is probably the most-used calculation there is.

Frequently asked questions

What is the rule of three?: It is a way to solve a proportion when you know three of four values. If 3 apples cost £6, then 5 apples cost (6 × 5) ÷ 3 = £10. You multiply crosswise and divide by the remaining value — which is why it is also called cross-multiplication.
What is the difference between direct and inverse proportion?: In direct proportion, more of one means more of the other: more apples, more money. In inverse proportion, more of one means LESS of the other: more workers, fewer hours. The formulas differ — direct is (B × C) ÷ A, inverse is (A × B) ÷ C.
How do I know which one to use?: Ask yourself: if I double the first quantity, does the answer double or halve? Doubling workers halves the time, so that is inverse. Doubling apples doubles the price, so that is direct. Getting this backwards is the single most common mistake.
Are my numbers sent anywhere?: No. The whole calculation runs in your browser on your own device. Nothing is uploaded, and the page keeps working offline once loaded.