Multiplying & dividing fractions

You want half of a third of a pizza, or you pour 1.5 litres of juice into quarter-litre glasses: as soon as parts meet parts, you need to multiply and divide fractions. The good news: both follow two simple rules, and here you'll also see why they work.

Fraction times fraction: a part of a part

One half times one third means: half of a third. Picture a chocolate bar: you split it into 3 columns and take one, then split that column into 2 rows and take one. In your hand you hold 1 of the 6 equal pieces of the whole bar, so 1/6. That's exactly what the rule says: numerator times numerator, denominator times denominator. So 1/2 · 1/3 = 1/6, and in the same way 2/3 · 4/5 = 8/15. The little word of means times, whole numbers included: 3/4 of 12 is 3/4 · 12/1 = 36/4 = 9. In your head the other order is often quicker, first 12 : 4 = 3, then 3 · 3 = 9.

Simplifying first saves work

For 2/3 · 3/4 you may cancel before you multiply: the 3 in the numerator and the 3 in the denominator cancel out, and 2 and 4 become 1 and 2. What's left is 1/2, with no big numbers at all. If you multiply first, you get 6/12 and still have to simplify at the end. Cancelling first is not a trick, it's the same path with less work.

Dividing by a fraction means multiplying by the reciprocal

What does 2 : 1/4 mean? It asks: how many quarters fit into 2 wholes? Each whole contains 4 quarters, so 2 wholes contain 8. That's why 2 : 1/4 = 2 · 4 = 8. In general: you divide by a fraction by multiplying by its reciprocal, which means swapping numerator and denominator. So 3/4 : 3/8 becomes 3/4 · 8/3, and after cancelling you're left with 2.