Rule of Three Calculator guide
The Inverse Rule of Three, Explained with Examples
In an inverse proportion, one quantity goes up as the other goes down, so the products stay constant instead of the ratios. If a corresponds to b and c corresponds to x, then a × b = c × x, which rearranges to x = (a × b) ÷ c. Use it for workers and time, speed and journey time, or any pairing where doubling one halves the other.
The situation this covers
Two workers take 16 days to build a fence. How long would four workers take?
If you reach for the ordinary rule of three here, you will get 32 days — which is worse than useless, because it says that adding people makes the job take longer. Something about the setup is different, and it is worth understanding what.
The difference is direction. With flour and people, more of one meant more of the other. Here, more workers means fewer days. The quantities move in opposite directions, and that changes the arithmetic.
What stays constant instead
In a direct proportion, the ratio between the two quantities stays the same. In an inverse proportion, it is the product that stays the same.
Think about what is really fixed in the fence problem: the total amount of work. Two workers for 16 days is 32 worker-days of labour. Four workers still have to supply those same 32 worker-days, so they need 32 ÷ 4 = 8 days.
That is the whole idea. Multiply the pair you know to find the constant, then divide by the new value.
The formula
Laid out the same way as before:
- 2 workers → 16 days
- 4 workers → x days
Working it through
Because this is inverse, you multiply across the row you know, then divide by the value in the other row: x = (2 × 16) ÷ 4 = 32 ÷ 4 = 8 days.
Compare that to the direct formula, x = (b × c) ÷ a. The numbers are the same three values; what changes is which one ends up as the divisor. In a direct proportion you divide by a. In an inverse proportion you divide by c.
And the sanity check works as before, just in reverse: you doubled the workers, so the time should halve. 16 became 8. That is the right direction and the right size.
Everyday inverse proportions
- Workers and time — more people, less time (up to a point).
- Speed and journey time — drive twice as fast, arrive in half the time.
- Number of servings and portion size — the same cake split between more people gives smaller slices.
- Pressure and volume of a gas at constant temperature — squeeze it into half the space and the pressure doubles.
- Machines and production time — twice the machines, half as long to fill the order.
A caution about the worker example
Textbooks lean on the workers-and-days example because the maths is clean, but it is worth saying plainly that real projects do not behave this way indefinitely.
Doubling from two workers to four may genuinely halve the time. Going to a hundred workers will not make it a hundred times faster — they will run out of fence to work on, and get in each other's way.
The maths is correct; the assumption that the relationship stays inverse forever is what breaks. That is worth remembering any time an inverse proportion is applied to people rather than to physics.
