Rule of Three Calculator
Find the missing fourth value in any proportion — for both direct and inverse relationships.
Direct or inverse
Pick direct when both quantities rise together, inverse when one rises as the other falls.
Divisor of zero
The result is undefined when the divisor is zero — A for a direct proportion, C for an inverse one.
What is the rule of three?
Find the missing value in a proportion
The rule of three — known in German as the Dreisatz — finds an unknown fourth value when you already know three values that share a fixed ratio. Written as A : B = C : x, it answers questions like "if 3 apples cost 12, what do 5 apples cost?" It is the everyday name for cross-multiplication and underpins recipe scaling, currency conversion, unit pricing, and map scales.
For a direct proportion, multiply the diagonal pair and divide by the value opposite the unknown.
x = (B × C) ÷ AAn inverse proportion keeps the product constant rather than the ratio, so the formula becomes x = (A × B) ÷ C. The calculator switches between the two automatically based on the type you select, and divides by the value that anchors the relationship.
Suppose 3 notebooks cost 12, and you want the cost of 5 notebooks.
Set up the proportion
Write it as 3 : 12 = 5 : x, with x the unknown cost.Cross-multiply
Multiply the known diagonal: 12 × 5 = 60.Divide by the opposite value
60 ÷ 3 = 20, so 5 notebooks cost 20.
The result x completes the proportion you set up, so it only makes sense if the relationship is genuinely fixed. Choose direct (proportional) when the two quantities move together — twice the ingredients for twice the servings. Choose inverse (inverse proportional) when more of one means less of the other: more workers finish a job in less time, and a higher speed covers a route in fewer minutes. Picking the wrong type is the most common mistake, so check the direction of the relationship before reading the number.
The arithmetic is exact, but the rule of three only models a constant relationship.
A constant ratio is an assumption
The rule of three assumes the relationship stays perfectly proportional across the whole range. Real-world quantities often break this — bulk discounts, fixed setup costs, or diminishing returns mean doubling the input does not always double the output. It also cannot divide by zero, so the divisor (A for direct, C for inverse) must not be zero.