Proportion and Scaling Without a Calculator

Almost every proportion question opens up once you know the value of one. Twelve dollars buys three model cars, so one car is twelve divided by three, which is four dollars. Now forty-eight dollars buys forty-eight divided by four, which is twelve cars. You could also notice that forty-eight is four times twelve and so buys four times three cars, and that is quicker when the numbers happen to be friendly — but per-one works every time, whatever the numbers. Write the unit value down rather than holding it in your head.

Halving one number while doubling the other leaves the product unchanged, and it can turn an awkward multiplication into an obvious one. Eighteen times five becomes nine times ten, which is ninety. The same trick works with any matching pair: twenty-four times twenty-five becomes six times a hundred by dividing one side by four and multiplying the other by four, and six hundred is immediate. Aim to produce a ten, a hundred, or a small single digit, and the multiplication does itself.

Multiplying and dividing can be done in any order, so cancel before you multiply. Take thirty-six times fifteen divided by twelve. Done slowly that is five hundred and forty, then divided by twelve, which is forty-five — two hard steps. Done properly, notice that thirty-six divided by twelve is three, so the whole thing becomes three times fifteen, which is forty-five in one easy step. The answer is identical; only the size of the numbers you have to handle changes. With no calculator, keeping the numbers small is not tidiness, it is the method — and if you find yourself writing three-digit workings, you have missed a cancellation.

A ratio counts parts, so always add the parts first. Sharing sixty dollars in the ratio two to three means five parts altogether, and one part is sixty divided by five, which is twelve dollars — that is the value-of-one step again in a different costume. So the first share is two twelves, twenty-four dollars, and the second is three twelves, thirty-six. Check by adding: twenty-four and thirty-six make sixty. The commonest error by far is dividing by two or by three rather than by five, because a ratio of two to three does not mean halves or thirds — it means fifths.

Scaling a recipe is the same machinery once more. Three hundred grams of flour serves four, so one serving needs seventy-five grams, and ten servings need seven hundred and fifty. The useful part is that every ingredient scales by the same factor — here two and a half times — so you work out the factor once and then only the multiplication changes from ingredient to ingredient. That is the whole lesson: find the value of one, halve and double to make multiplications easy, cancel before multiplying, and remember that a ratio counts parts. Quiz and exercises now.