The rule of three

The supermarket shows the price per 100 grams, but you want to buy 250 grams. The recipe serves 4 people, but there are 6 of you. Questions like these come up all the time, and they all share the same core: two quantities are linked, you know one complete pair of values and, of a second pair, only one of the two values. The rule of three is the method that safely gets you the missing one.

Proportional relationships

The classic rule of three works when two quantities are directly proportional: twice as much of one means twice as much of the other. If 3 apples cost 2 euros, then 6 apples cost 4 euros and 9 apples cost 6 euros. If you plot such value pairs in a coordinate system, all the points lie on a straight line through the origin: zero apples cost zero euros.

Going via one unit

The trick of the rule of three: first scale down to 1, then scale up to the amount you want. If 3 apples cost 2 euros, divide by 3: one apple costs 2/3 of a euro. For 7 apples, multiply that by 7, which is 14/3 euros, so about 4.67 euros. Important: round only at the very end, not at the intermediate step, otherwise the result becomes inaccurate. A rule-of-three table with three rows keeps this tidy: the given amount, the unit, the amount you want. On the left and the right you always do the same operation: first divide by 3, then multiply by 7.

When the rule of three fails

Not everything is directly proportional. If a child is 1.20 metres tall at age 6, it won't be 2.40 metres tall at 12, because growth does not follow a fixed factor. For other quantities the relationship reverses: if 2 painters paint a wall in 6 hours, 4 equally fast painters need less time, not more, namely 3 hours. Here it is not the ratio that stays the same but the product: 2 · 6 = 4 · 3 = 12 working hours. For that there is the inverse rule of three. So before every calculation, check: are the quantities directly proportional, inversely proportional, or neither?