Rule of Three Calculator guide
Direct or Inverse? How to Tell Which You Need
Ask what should happen to your answer if the known quantity increases. If the answer should also increase, the proportion is direct and x = (b × c) ÷ a. If the answer should decrease, it is inverse and x = (a × b) ÷ c. Direct keeps the ratio constant; inverse keeps the product constant.
This is the step that goes wrong
Almost nobody gets the arithmetic of the rule of three wrong. The multiplication and division are straightforward, and a calculator handles them anyway.
What goes wrong is picking the wrong kind of proportion at the start. Once you have chosen direct when you needed inverse, everything afterwards is careful, tidy and wrong — and because the number looks reasonable, it is easy to miss.
So it is worth spending ten seconds on this before you do anything else.
The question to ask
Before calculating, ask yourself: if I increase the known quantity, should my answer get bigger or smaller?
Bigger means direct. Smaller means inverse. That is the whole test, and it works because you already have an intuition about the situation even when you do not have one about the formula.
More petrol should cost more money — direct. More taps filling a bath should take less time — inverse. You knew both of those before reading this sentence; the test just makes you consult that knowledge deliberately instead of hoping the formula sorts it out.
Side by side
| Direct proportion | Inverse proportion | |
|---|---|---|
| As one goes up… | the other goes up | the other goes down |
| What stays constant | the ratio, a ÷ b | the product, a × b |
| Formula | x = (b × c) ÷ a | x = (a × b) ÷ c |
| Typical example | quantity and price | workers and time |
| Graph shape | a straight line through zero | a curve that never touches the axes |
Examples of each
Direct — these all get bigger together:
- Litres of petrol and the amount you pay
- Hours worked and wages earned, at a fixed hourly rate
- Servings of a recipe and grams of each ingredient
- Distance travelled and time taken, at a steady speed
And the inverse cases
Inverse — as one grows, the other shrinks:
- Number of workers and the days a job takes
- Speed and the time to cover a fixed distance
- Number of guests and the size of each slice of one cake
- Taps running and the time to fill a bath
When it is neither
Some relationships are not proportional at all, and no version of the rule of three will help.
If a taxi charges a flat £3 pickup fee plus £2 per mile, that is not a proportion — doubling the distance does not double the fare, because the £3 is there regardless. The giveaway is that at zero miles the cost is not zero.
A quick test: a direct proportion must pass through zero. If none of the quantity means none of the other thing, it is likely proportional. If there is a fixed amount that applies even at zero, you need a different tool.
