How to Use the Rule of Three (Direct Proportion)

What the rule of three actually does

The rule of three is the piece of arithmetic you use dozens of times a week without naming it. If four people need 300g of flour, how much do six people need? If three shirts cost £36, what do seven cost? Every one of those is the same shape of problem: you know three numbers, and you want the fourth.

The name comes from exactly that — three known values, one unknown. Some textbooks call it "solving a proportion" or "finding the fourth proportional", but it is all the same procedure.

Lay it out before you calculate

The single most useful habit here is to write the problem down in two rows before touching any arithmetic, with matching quantities lined up underneath each other.

For the flour question:

• 4 people → 300 g
• 6 people → x g

Why the layout matters more than the formula

Almost every wrong answer in a rule of three problem comes from mixing up which number belongs where, not from bad multiplication. Writing the two rows out with units attached makes that mistake very hard to make, because a mismatched row looks obviously wrong on the page.

The rule to hold onto: the two numbers in the left column are the same kind of thing as each other, and so are the two in the right column. People with people, grams with grams. If your rows do not read like that, rearrange them before going further.

Now do the arithmetic

Once it is laid out, the calculation is always the same. Multiply diagonally — the two numbers that are not next to the x — then divide by the remaining one.

For our example: x = (300 × 6) ÷ 4 = 1800 ÷ 4 = 450 grams.

Written as a formula that is x = (b × c) ÷ a, but you may find it easier to remember as "multiply the two that face each other, divide by the one left over".

Check your answer in two seconds

Before you write the answer down, ask whether it points the right way. Six people is more than four, so they should need more flour than 300g. 450 is more than 300, so the answer is at least pointing in a sensible direction.

This catches the most common serious error, which is dividing when you meant to multiply. If your answer had come out as 200g, that "less flour for more people" oddity would have told you something was wrong before you handed it in.

A second check: how many times bigger is 6 than 4? One and a half. And 450 is one and a half times 300. Both sides scale by the same factor, which is precisely what "in proportion" means.

When the rule of three does not apply

It is worth knowing the limits, because the method is so easy to use that it gets applied where it should not be.

The rule of three assumes the two quantities scale together evenly. That is true for ingredients, prices per unit, distances at a steady speed and currency conversions. It is not true for things like the area of a shape as its side grows, or the cost of postage as weight increases in bands, or how long a task takes as you add people past a certain point.

If doubling one quantity does not exactly double the other, the rule of three will give you a confident, tidy, wrong answer. That is worth pausing over for a second before you use it.