Doubling and halving

Turn an awkward multiplication into an easy one by halving one number and doubling the other, without changing the answer.

Turn an awkward multiplication into an easy one, exactly.

The trick

Halve one number and double the other. The answer doesn’t change. Repeat until one of them is something you can multiply by without thinking.

Halve the even number. Halving an odd one is still correct — 35 × 14 is genuinely 17.5 × 28 — but a half is exactly the thing you were trying to avoid.

Why it works

Halving one factor and doubling the other multiplies the product by ½ and then by 2, and ½ × 2 = 1. Nothing has been approximated:

That is the whole justification, and it explains when the move is worth making. It always holds, so the only question is whether the new pair is easier than the old one. Going from 14 × 35 to 7 × 70 helps because 70 is a round number. Going from 13 × 41 to 6.5 × 82 does not help at all, even though it is just as true.

It goes further

The step can be taken more than once, which is what makes 25 easy: halve twice and double twice, and 25 becomes 100.

It runs the other way too. If a number is awkward because it is large, double the small one instead: 15 × 12 is 30 × 6, which is 180.