Rule of Three Calculator
Free rule of three calculator, simple and compound, with direct and inverse proportion. Find the missing value and see the calculation written out step by step.
What it does
The rule of three finds a missing value from three known ones, when two quantities change in proportion. If 3 notebooks cost 12, then 7 cost 28: the cost grows with the number of notebooks, so the proportion is direct. If 6 builders finish a wall in 10 days, 15 builders finish it in 4: more builders, fewer days, so the proportion is inverse. The compound rule of three does the same with more than two quantities at once, such as machines, days and parts produced. The calculator shows the calculation written out, so you can follow it or copy it into your work.
How to use it
- Choose simple or compound.
- Type the known values and say, for each quantity, whether the proportion is direct or inverse.
- Read the result and the calculation that leads to it.
The arithmetic runs in your browser. Nothing you type is sent anywhere and nothing is stored, so you can use these with real numbers.
Frequently asked questions
What is the rule of three?
A way of finding the fourth value of a proportion when three are known. Written as "a is to b as c is to x", the direct rule gives x = b × c ÷ a and the inverse rule gives x = a × b ÷ c.
How do I know whether it is direct or inverse?
Ask what happens to the value you are looking for if the other quantity doubles. If it also doubles, the proportion is direct (more notebooks, higher price). If it halves, it is inverse (more speed, less time).
How does the compound rule of three work?
Compare each quantity with the one you are looking for, one at a time, as if the others stayed the same, and decide whether that pair is direct or inverse. The result is the known value multiplied by each ratio: the new value over the old one for direct quantities, the old over the new for inverse ones.
When does the rule of three give a wrong answer?
When the quantities are not really proportional. A child who is 1 m tall at age 5 will not be 2 m tall at 10, and a recipe for 4 does not always scale exactly to 40. The calculation is only as good as the assumption that one quantity changes in proportion to the other.
Wikipedia — Cross-multiplication
Wikipedia — Proportionality (mathematics)