What is the rule of three?

Given a known proportion A:B = C:D, the rule of three finds the fourth value when three of the four are known. Formula: D = (B * C) / A (or any rearrangement). It's also called cross-multiplication because the working step is to multiply diagonally: A * D should equal B * C.

When does the rule of three fail?

When the relationship between the two quantities isn't a direct proportion. Doubling the bread flour roughly doubles the yield, so the rule applies; doubling the bake time at the same temperature doesn't double the result, so it doesn't. Speed for short distances is direct; income tax is not (it's bracketed). Try a small mental sanity check on whether 'twice the input should give twice the output' before trusting the result.

What's the difference vs the percentage calculator?

Percent is a special case of proportion - the second pair is always 'something to 100'. 'What is 25% of 80?' is the rule of three with A=100, B=80, C=25, find D = 20. The percentage calculator is faster for percent-shaped questions; the rule of three is more general (any pair of units, including kg price, km minutes).

Can I check if four numbers form a proportion?

Fill all four boxes. Underneath the result, we show the cross-products A*D and B*C; if they're equal, the four values are in proportion. Useful for spotting transcription errors when copying figures from a spreadsheet.

Rule of Three Calculator

Solve a proportion: if A is to B, then C is to ? - leave any box empty and we'll fill it in.

Fill any three of the four boxes.

corresponds to

Then

corresponds to

Result (D)

D = (B * C) / A = (9 * 5) / 3 = 15

Type any three of the four boxes; we'll compute the missing fourth using the rule of three (cross-multiplication). The proportion is A is to B as C is to D - if you know A, B and C, then D = (B * C) / A. Useful any time two quantities scale together at a constant ratio: ingredients in a recipe, distance and time at constant speed, area covered per litre of paint, price per kilogram, currency at a fixed rate.

How to use it

Type the three known values
Each box holds one number. The proportion reads A:B = C:D. If you know A, B, and C, leave D empty; if you know B, C, and D, leave A empty.
Read the computed value
The missing box auto-fills below the inputs, with the formula trace showing the cross-product math. Negative or fractional inputs work the same way.
Sanity check the relationship
The rule assumes direct proportionality. Before trusting the result, ask: if the inputs doubled, would the outputs really double? If not (cooking times, tiered taxes), the rule isn't the right model.

What is it?

The rule of three is the oldest, simplest, and most useful proportionality calculation in school maths. Given two quantities that vary together at a constant ratio - cost per kilo, time per page, paint per square metre - and three of the four numbers that describe a pair, the rule recovers the fourth. The mechanism is cross-multiplication: in A/B = C/D, the cross-products A*D and B*C must match, so the unknown drops out with one division. The arithmetic is one keystroke per number; the value is in not muddling which side multiplies and which divides.

When to use it

Recipe scaling when the recipe-scaler isn't enough (e.g. 'I have 320 g of flour, how much sugar does that need given a recipe that uses 250 g flour and 200 g sugar?'). Price-per-unit comparisons at the supermarket. Map distances when you know the scale. Mix ratios for paint, fertilizer, or concrete. Translating per-1000 / per-100k metrics to a specific population. Working back from a rate to find a quantity: '6 hours at 80 km/h, how far' is rule of three with A=1, B=80, C=6, find D.

Common mistakes

Applying it to non-linear relationships - bake time, exposure time, drug dosage by body weight beyond a certain range. Swapping the cross terms: A goes with B and C goes with D, never A with D in the numerator. Treating percent as if it were a quantity - it's a ratio, so 'add 25% then subtract 25%' is not zero net (the bases differ). And rounding intermediate steps - keep full precision until the final figure, then round.

FAQ

What is the rule of three?: Given a known proportion A:B = C:D, the rule of three finds the fourth value when three of the four are known. Formula: D = (B * C) / A (or any rearrangement). It's also called cross-multiplication because the working step is to multiply diagonally: A * D should equal B * C.
When does the rule of three fail?: When the relationship between the two quantities isn't a direct proportion. Doubling the bread flour roughly doubles the yield, so the rule applies; doubling the bake time at the same temperature doesn't double the result, so it doesn't. Speed for short distances is direct; income tax is not (it's bracketed). Try a small mental sanity check on whether 'twice the input should give twice the output' before trusting the result.
What's the difference vs the percentage calculator?: Percent is a special case of proportion - the second pair is always 'something to 100'. 'What is 25% of 80?' is the rule of three with A=100, B=80, C=25, find D = 20. The percentage calculator is faster for percent-shaped questions; the rule of three is more general (any pair of units, including kg <-> price, km <-> minutes).
Can I check if four numbers form a proportion?: Fill all four boxes. Underneath the result, we show the cross-products A*D and B*C; if they're equal, the four values are in proportion. Useful for spotting transcription errors when copying figures from a spreadsheet.