What this rule of three calculator does
Enter three values, mark the one you are looking for, and the calculator solves it with the full working: the given pair first, then the single-unit step, then the answer. You can switch between proportional and inverse, and solve for any of the four values. Unit labels are free text, and everything is computed locally in your browser.
How does the rule of three work, step by step?
Always in three lines. Example: 3 kg of apples cost £7.50, and you want 5 kg.
| kg | £ | |
|---|---|---|
| 3 | 7.50 | given |
| 1 | 2.50 | ÷ 3 on both sides |
| 5 | 12.50 | × 5 on both sides |
The middle line is the actual trick: work out what a single unit costs, then scale it up. Written as a proportion the same thing is a : b = c : d, so a · d = b · c — and any missing value follows by cross multiplication.
What is the difference between proportional and inverse?
Proportional means "more gives more": twice the amount, twice the value. Inverse means "more gives less": twice the painters need half the days. The difference sits exactly in the single-unit step.
| proportional | inverse | |
|---|---|---|
| Single-unit step | divide the right column | multiply the right column |
| Final step | multiply the right column | divide the right column |
| What stays constant | the ratio a ÷ b = c ÷ d | the product a · b = c · d |
Four painters need 9 days. Treated as proportional, six painters would come out at 13.5 days — obviously nonsense. The inverse approach is right: 4 · 9 = 36 person-days, divided by 6 painters gives 6 days.
How do I tell which type a problem is?
Double the first quantity in your head and ask what happens to the second one:
- Does it double too? → proportional. Typical: prices and quantities, distance and time at constant speed, recipe ingredients, material requirements.
- Does it halve? → inverse. Typical: workers and working time, speed and travel time, machines and runtime, supplies and number of people.
This is not a technicality, it is the heart of the task: get the type wrong and you can do every remaining step perfectly and still land on a wrong answer.
Where does the rule of three come from?
The method is old. Ratios belong to elementary mathematics and are already treated in Euclid's Elements; the rule appears as regula de tri in the arithmetic books of Adam Ries in the 16th century, which is where the German name "Regeldetri" — later "Dreisatz" — comes from (source: Wikipedia, article "Dreisatz"). The name simply means a rule built from three given terms.
Honest limits
- The rule of three only applies to relationships that are purely proportional or purely inverse. Tiered pricing, bulk discounts, progressive taxation and compound interest are none of those — it will mislead you there.
- The calculator does not check whether your problem really matches the type you picked. That judgement stays with you; the tool only shows what follows from your choice.
- A zero in the single-unit step stops the calculation, because that step divides by it. The tool says so rather than printing an infinity.
- Multi-stage problems with three quantities (compound rule of three) need two passes — the calculator covers one step at a time.
Privacy
Your numbers never leave the device. The calculation runs entirely in the browser, and your last entries are kept in local storage only, so the problem is still there next time.