Three Painters Take Twelve Days; Five Do Not Take Twenty
Published 10/5/2026 · 3 min read · Everyday calculators
Marco Bianchi — Home, DIY & motoring writer at OneKitly
Renovation · Materials
Checked against 2 sources
The rule of three is a single cross-multiplication: if 3 gives 12, then 5 gives 5 × 12 ÷ 3 = 20. It is correct whenever the two quantities rise together — three metres of cable cost 12, so five metres cost 20 — and it is the wrong tool the moment they move in opposite directions. Three painters taking twelve days is thirty-six painter-days of work; five painters need 36 ÷ 5 = 7.2 days, not twenty. The calculation cannot know which situation you are in, because both look identical as three numbers. Decide the direction first, in words: if more of the first thing means more of the second, multiply as above; if more of the first means less of the second, multiply the two known quantities together and divide by the third instead.
The rule of three turns 3 → 12 into 5 → 20, and that is right for anything that scales up together. For workers and days it is exactly backwards: the answer is 7.2.
Some relationships are neither
The rule of three needs the two quantities to be linked by a single multiplication, and anything that scales in more than one dimension is not. A 20 cm cake serving eight does not mean a 30 cm cake serves twelve: the surface grows with the square of the diameter, so the ratio is (30 ÷ 20)² = 2.25 and the answer is eighteen. The same trap catches paint on a larger wall of different proportions, wire for a bigger coil, and the fabric for a scaled-up pattern. Before cross-multiplying, ask whether doubling the input really doubles the output — if it quadruples it, the rule of three will be wrong by a factor of two.
Even the direct version has a ceiling
Proportionality is an assumption about a range, not a law. Cable at 4 a metre is 20 for five metres and probably not 20,000 for five thousand, because at that quantity the price is negotiated; a job that five painters finish in 7.2 days is not finished in half a day by seventy-two painters, because they cannot all reach the wall. Every proportional answer carries an unstated range in which it holds, and the failures happen at the ends — at very small quantities, where a fixed cost dominates, and at very large ones, where something physical or commercial stops scaling.
| Question | Relationship | Answer |
|---|---|---|
| 3 m of cable cost 12; 5 m? | direct | 20 |
| 3 painters take 12 days; 5? | inverse | 7.2 |
Worked with our own calculator
Rule of three
Given
- A
- 2
- gives B
- 6
- C
- 3
Result
- C gives X
- 9
These figures are produced by the calculator below, not typed in by hand — they are recomputed whenever the tool changes.
Run it on your own figures →Frequently asked questions
- How do I tell direct from inverse quickly?
- Ask what stays constant. In a direct relationship it is the ratio: 4 per metre stays 4 per metre however much cable you buy. In an inverse one it is the product: painters × days stays at 36 however many painters turn up. Naming the constant takes a second and settles the question every time, and it also tells you what to write down — thirty-six painter-days is a fact about the job that survives any change of crew, while "twelve days" is only a fact about one particular crew.
- Why does the answer come out as 7.2 days?
- Because the arithmetic is continuous and the calendar is not. Seven point two days means seven full days plus a fifth of another, which in practice is eight working days — and rounding down to seven would leave the last fifth of the job undone. Any quantity that has to be bought, booked or staffed in whole units should be rounded up after the proportional step, and the fraction that gets rounded away is worth keeping in view: on a long job, a fifth of a day lost every week is a day a month.
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