How to use the rule of three

The rule of three solves the most common problem in everyday maths: I know three values and I want the fourth. It is simple, it works whenever the relationship is proportional — and most mistakes with it come from applying it where the relationship is not proportional at all.

Direct proportion

Two quantities are directly proportional when one grows in the same ratio as the other. More flour, more loaves. More hours, more pay.

Set up the proportion keeping each quantity in its own column, then cross-multiply:

a / b = c / x → x = (b × c) ÷ a

Example: 3 kg of flour makes 12 loaves. How many loaves from 5 kg?

3 / 12 = 5 / x → x = (12 × 5) ÷ 3 = 20 loaves.

Inverse proportion

Two quantities are inversely proportional when one grows and the other shrinks in the same ratio. More taps, less time. More speed, fewer hours of driving.

Here you invert one of the ratios before cross-multiplying.

Example: 4 taps fill a tank in 6 hours. How long do 6 taps take?

Because it is inverse: 4 × 6 = 6 × x → x = 24 ÷ 6 = 4 hours.

The test for deciding which of the two applies is always the same and takes three seconds: ask whether increasing the first quantity increases or decreases the second. Increases, direct. Decreases, inverse. Getting that question wrong is the number one error in the topic.

Compound proportion

When more than two quantities are involved, the safest route is to convert everything into a single unit of work.

Example: 5 bricklayers, working 8 hours a day for 6 days, build 12 walls. How many days do 8 bricklayers, working 6 hours a day, need to build 18 walls?

1. Work in the first situation: 5 × 8 × 6 = 240 worker-hours for 12 walls.
2. Cost per wall: 240 ÷ 12 = 20 worker-hours per wall.
3. Work needed for 18 walls: 18 × 20 = 360 worker-hours.
4. Daily capacity of the new crew: 8 × 6 = 48 worker-hours per day.
5. Days: 360 ÷ 48 = 7.5 days.

This unit-of-work approach removes the confusion of deciding which ratios to invert — and it works for any number of quantities.

The error the rule of three will not warn you about

The method never checks whether the relationship is actually proportional. It returns a number even when the number is nonsense, and that is where it does damage.

• Work that does not parallelise. If a pregnancy takes nine months, nine women do not produce a baby in one. The example is deliberately absurd, but the realistic version shows up daily: doubling a project team rarely halves the schedule, because communication and coordination grow with headcount.
• Diminishing returns. Four painters in one small room get in each other's way. Proportionality holds only while there is space and equipment for everyone.
• Relationships that are not linear. A roast twice as heavy does not take twice as long — heat enters through the surface, and surface area does not grow in step with volume. The same applies to cooling, to structural strength and to fuel consumption as a function of speed.
• Valid ranges. A per-kilogram medication dose works inside a range; extrapolating beyond it can be dangerous, which is why labels state limits and not just a ratio.

The question that catches all of these: "if I double the input, does the output really double?" If the answer is "roughly", the rule of three is an estimate, not an answer.

Frequently asked questions

How do I know if it is direct or inverse? Mentally increase the first quantity and watch the second. Rises with it, direct; falls, inverse. In compound problems, run that test separately for each quantity, always against the unknown.

Do I need to convert units first? Yes, and before setting up the proportion. Mixing hours with minutes, or pounds with ounces, produces a result that looks correct and is off by an order of magnitude — the kind of error that slips through unnoticed.

Does it work for percentages? It does, and it is the most intuitive way to teach them. "15 is 20% of what?" becomes 15 / 20 = x / 100, so x = 75.

What if the proportion is not exact? Use the rule of three for the order of magnitude and treat the result as an estimate. In construction take-offs, for instance, material is calculated by proportion and then padded for waste and cuts — which the proportion never anticipates.