Rule of Three Calculator: Cross Multiplication and Ratios

The rule of three finds an unknown value when two quantities are proportional: if 4 notebooks cost $6, how much do 10 notebooks cost? This tool does the cross multiplication and shows every step. It also handles inverse proportion (more workers, fewer days) and the simplification of a ratio such as 1920:1080 to 16:9, with scaling.

Two quantities are proportional when you go from one to the other by always multiplying by the same number, called the constant of proportionality. In a four-cell table where A corresponds to B and C to X, we have A/B = C/X, and cross multiplication gives A × X = B × C, so X = (B × C) ÷ A. X is called the fourth proportional.

The name “rule of three” comes from the three known values used to find the fourth. The method sometimes goes through the unit value: if 4 notebooks cost $6, one notebook costs 6 ÷ 4 = $1.50, so 10 notebooks cost $15. Cross multiplication gives the same result in a single operation: (6 × 10) ÷ 4 = 15.

Two quantities are inversely proportional when their product stays constant: if one doubles, the other is halved. This is the case for the number of workers and the duration of a job, or for speed and travel time over a fixed distance. Then A × B = C × X, so X = (A × B) ÷ C.

A ratio A:B compares two quantities. It is simplified like a fraction: decimal terms are first multiplied to obtain whole numbers, then all the terms are divided by their GCD. Scaling a ratio means multiplying all its terms by the same factor, which preserves the proportions.

Calculations are done with exact values: the fractions and decimals you enter undergo no intermediate rounding. When the result has no finite decimal form, the tool shows a value rounded to six decimal places together with the exact fraction.