What the rule of three is and how to calculate it
The rule of three is the simplest method for solving problems of proportionality: given
three values of a relationship, it lets you find the fourth. The classic set-up is "A is to B as C is to X",
where X is the unknown. It is probably the most widely used piece of maths in everyday life: scaling the
quantities in a recipe, comparing prices per kilo, splitting a bill, converting currencies or estimating how
long a task will take if you change the pace.
The only tricky part is picking the right type of proportion. There are two variants and each has its own formula:
the direct rule of three, when the two quantities grow or shrink together, and the
inverse rule of three, when one grows while the other shrinks. This calculator solves both
live: enter A, B and C, choose the mode and you get X straight away, with a breakdown of the formula applied.
Direct rule of three: more A, more B
In direct proportion, the ratio between the quantities stays constant: if one doubles, so does the other.
The formula is X = (B × C) / A. This is the case with cooking recipes: if a recipe for
4 people uses 300 grams of rice, for 6 people you will need (300 × 6) / 4 = 450 grams. And with
prices per unit: if 3 kilos of oranges cost £4.50, 5 kilos will cost (4.50 × 5) / 3 = £7.50.
In both examples, as the first quantity increases the second one increases too, which is why direct is the correct choice.
Inverse rule of three: more A, less B
In inverse proportion, what stays constant is the product of the quantities, not the ratio.
The formula changes to X = (A × B) / C. The classic example is workers and days on a job:
if 4 workers build a wall in 6 days, 8 workers (twice the hands) will do it in half the time:
(4 × 6) / 8 = 3 days. Another everyday case is speed and journey time: if at 60 mph you take
2 hours to arrive, at 80 mph you will take (60 × 2) / 80 = 1.5 hours. The faster you go, the less time you need:
opposite directions, inverse proportion.
Table of worked examples
| Problem |
Type |
Formula |
Result |
| If 3 tickets cost £12, how much do 5 cost? |
Direct |
(12 × 5) / 3 |
£20 |
| If 2.5 m of fabric cost £10, how much do 7 m cost? |
Direct |
(10 × 7) / 2.5 |
£28 |
| If 4 workers take 6 days, how long do 8 workers take? |
Inverse |
(4 × 6) / 8 |
3 days |
| If at 60 mph you take 2 h, how long at 80 mph? |
Inverse |
(60 × 2) / 80 |
1.5 hours |
The most common mistake: using direct when the relationship is inverse
The slip that produces the most absurd results is using the direct formula on an inverse problem.
Take the workers example: if you apply the direct rule to "4 workers take 6 days, how long for 8?", you get
(6 × 8) / 4 = 12 days — that is, twice as many workers take twice as long. Obviously it is the other way round.
The defence is always the same: before you calculate, put the relationship into words ("if I add more workers, do the
days go up or down?") and check that the final result moves in the direction you expect. Another frequent error is mixing
units (grams with kilos, minutes with hours): convert everything to the same unit before you set up the proportion.
The rule of three and percentages
Percentages are a special case of the direct rule of three in which one of the quantities
is always 100. "15% of 80" is set up as "100 is to 15 as 80 is to X" → X = (15 × 80) / 100 = 12. And the other
way round: "what percentage is 30 of 150?" is set up as "150 is to 100 as 30 is to X" → X = (100 × 30) / 150 = 20%.
Mastering the rule of three lets you solve any percentage, discount, mark-up or proportional split without
memorising different formulas: they are all the same proportion under different names.
Note: This rule of three calculator is a general-purpose tool. For specific calculations of
tax, discounts or dates, we recommend our dedicated calculators for percentages, discounts, days between
dates and VAT, linked below.