Rule of Three
Find the fourth number of a proportion — direct or inverse, because the two give different answers.
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gives this
10
- The working
- 6 × 5 ÷ 3
- What stays the same (the ratio, or the product)
- 2
If three apples cost six, five cost ten. If three workers take six hours, five take three hours and thirty-six minutes — not ten. Both are called the rule of three, and telling them apart is the whole skill.
How it works
In a direct proportion the ratio stays the same: whatever one unit is worth, it is worth that for every unit. Multiply across and divide by the first number — the cross multiplication taught at school.
In an inverse proportion the product stays the same. Three workers times six hours is eighteen worker-hours, and eighteen is what the job costs however many people turn up. Five workers therefore take eighteen divided by five.
The page shows the constant — the value of one unit, or the fixed product — because it is the single number the whole answer turns on, and seeing it lets you check the result rather than trust it.
Examples
| Case | Input | Result |
|---|---|---|
| Direct: more apples, more money | 3 apples cost 6, so 5 cost… | 10 — one apple is worth 2 |
| Inverse: more workers, less time | 3 workers take 6 hours, so 5 take… | 3.6 hours — the job is 18 worker-hours |
| The same numbers, the wrong rule | 3 workers, 6 hours, 5 workers, treated as direct | 10 hours — more people, somehow slower |
Frequently asked questions
How do I know which one to pick?
Ask what happens as the first quantity grows. If the second grows with it — more litres, more money — it is direct. If the second shrinks — more workers, less time; faster speed, shorter journey — it is inverse. Picking wrong gives an answer that looks reasonable and is not, which is why both are offered here rather than just the common one.
Is every relationship one of the two?
No, and that is worth knowing. Doubling the side of a square quadruples its area, which is neither. The rule of three applies when the ratio or the product genuinely holds — and whether it does is something about the situation, not about the numbers.
Can I use decimals or negative numbers?
Decimals, yes. Negative numbers are accepted and the arithmetic is correct, but a proportion between quantities that can go negative rarely means what it appears to — read the result carefully.
Good to know
- The fourth proportional is a number, not a measurement. Whatever unit the second field carried, the answer carries too — keep them consistent, because the page cannot check that for you.