Rule of Three Calculator
Solve a proportion from three known values. Pick a direct or inverse rule of three, enter the two known pairs, and the unknown is filled in with the working shown.
- Proportion
- Direct
- x (the unknown)
- 10
Calculator
Worked solution
Direct: a ∶ b = c ∶ x ⇒ x = (b × c) ÷ a = (6 × 5) ÷ 3 = 10
About this calculator
The rule of three is the classic way to solve a proportion: you know three numbers and want the fourth. Two quantities are linked, you have a complete pair for one situation, and a single value for a second situation — the calculator fills in the missing partner. It handles both kinds of link. In a direct proportion the two quantities rise and fall together (more items, more cost), while in an inverse proportion one rises as the other falls so their product stays constant (more workers, fewer days). Pick the type, type the three values you know, and the unknown appears with the arithmetic written out.
How to read your results
The large figure is x, the unknown you were solving for, shown in the same units as the third value you entered. Beneath it the three known numbers — A, B and C — are echoed so you can check nothing was mistyped, and the worked-solution panel shows exactly which formula was used and the numbers slotted into it. If you switch between direct and inverse the answer recomputes instantly, because the same three inputs are combined differently: direct multiplies the cross pair and divides by A, inverse multiplies the first pair and divides by C.
How it's calculated
A proportion equates two ratios. For a DIRECT proportion the ratios match, a ∶ b = c ∶ x, and cross-multiplication (Wikipedia: Cross-multiplication) gives a·x = b·c, so x = b·c ÷ a. For an INVERSE proportion the product is the invariant, a·b = c·x, so x = a·b ÷ c — this is the y = k/x relationship from the mathematics of proportionality. The calculator validates the inputs with Zod and refuses the case that would divide by zero (A = 0 in direct mode, C = 0 in inverse mode); negative and fractional values are allowed because the arithmetic is the same.
Worked example
A direct proportion: 3 items cost 6, so what do 5 items cost? (A = 3, B = 6, C = 5.)
Set up 3 ∶ 6 = 5 ∶ x and cross-multiply: x = (6 × 5) ÷ 3 = 30 ÷ 3 = 10. Five items cost 10. Switch the same numbers to inverse and you would instead get x = (3 × 6) ÷ 5 = 3.6 — which is why choosing the right relationship matters.
Frequently asked questions
What is the rule of three?
It is a method for finding an unknown fourth value when you already know three values of a proportion. You write the known pair as a ratio, set it equal to the ratio that contains the unknown, and cross-multiply to isolate the unknown. It is the everyday tool behind unit pricing, recipe scaling, map distances and currency conversion.
What is the difference between a direct and an inverse rule of three?
In a direct proportion the two quantities move together: double one and the other doubles, so x = B × C ÷ A. In an inverse proportion they move oppositely and their product is fixed: double one and the other halves, so x = A × B ÷ C. Speed-and-time and workers-and-duration problems are inverse; price-and-quantity problems are direct.
How do I know which values are A, B and C?
A and B are the complete pair you already know (A in the first column, B the value that goes with it). C is the new value in the same column as A, and x — the answer — is the value in the same column as B. For "3 items cost 6, what do 5 cost?" that is A = 3, B = 6, C = 5, and x is the cost of 5.
Can I use decimals or negative numbers?
Yes. Any finite number works for A, B and C, including decimals and negatives, and the result follows the same formula. The only forbidden case is dividing by zero — A cannot be zero in a direct proportion and C cannot be zero in an inverse one, because that would leave the unknown undefined.
Sources
Reviewed by the YouCalc Team · Last reviewed
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