Dividing Fractions

A recipe uses quarter-cup scoops of flour, and you have three-quarters of a cup in the bag — how many scoops can you fill? That is a division by a fraction: how many quarters fit inside three-quarters. Questions like this — splitting an amount into fraction-sized pieces — are where dividing fractions earns its keep.

Dividing by a fraction has a famous magic trick attached to it — one that looks like cheating the first time you see it. The trick: to divide by a fraction, flip it upside down (take its reciprocal) and multiply instead. That's it.

\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{a\,d}{b\,c}

Before we trust the trick, let's see what division is even asking. A division like \frac{3}{4} \div \frac{1}{2} means "how many halves fit inside three-quarters?" — just like 12 \div 3 means "how many 3s fit inside 12?". Fractions are no different; the pieces are just smaller.

Try the neat one: how many quarters fit in a half?

\frac{1}{2} \div \frac{1}{4} = \frac{1}{2} \times \frac{4}{1} = \frac{4}{2} = 2

Two quarters fit in a half, so the answer is 2 — and you can see it's right just by picturing two quarter-slices sitting inside one half-slice. Notice something odd: dividing by a number smaller than 1 made the answer bigger than what we started with. Splitting into small pieces gives you lots of them.

Keep, change, flip — worked examples

A handy chant for the trick is "keep, change, flip": keep the first fraction, change the \div into a \times, and flip the second fraction over. Then it's just an ordinary multiply.

1. The intro one again, slowly. \frac{1}{2} \div \frac{1}{4}: keep \frac{1}{2}, change to \times, flip \frac{1}{4} to \frac{4}{1}. That gives \frac{1}{2} \times \frac{4}{1} = \frac{4}{2} = 2.

2. Neither one is nice. Work out \frac{3}{4} \div \frac{2}{3}. Keep, change, flip: \frac{3}{4} \times \frac{3}{2}. Multiply straight across:

\frac{3}{4} \div \frac{2}{3} = \frac{3}{4} \times \frac{3}{2} = \frac{9}{8} = 1\tfrac{1}{8}

3. Dividing a fraction by a whole number. A whole number 2 is \frac{2}{1}, and its flip is \frac{1}{2}. So \frac{1}{2} \div 2 = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} — half a pizza shared between two people is a quarter each. Dividing by a whole number makes the pieces smaller, just as you'd expect.

4. A real one from the kitchen. You have 2 cups of flour and a \frac{1}{3}-cup scoop. How many scoops can you fill? 2 \div \frac{1}{3} = 2 \times \frac{3}{1} = 6 scoops. Picture it in the diagram below.

A mixed number, and a common denominators shortcut

Dividing mixed numbers? Turn them into top-heavy (improper) fractions first, then keep-change-flip as usual. For example 1\tfrac{1}{2} \div \frac{3}{4}: the mixed number 1\tfrac{1}{2} is \frac{3}{2}, so

1\tfrac{1}{2} \div \frac{3}{4} = \frac{3}{2} \div \frac{3}{4} = \frac{3}{2} \times \frac{4}{3} = \frac{12}{6} = 2.

A neat shortcut when the bottoms already match. If both fractions share the same denominator, you can just divide the tops! Since \frac{6}{7} \div \frac{2}{7} asks "how many \frac{2}{7}s are in \frac{6}{7}?", and the pieces are the same size, the answer is simply 6 \div 2 = 3. (Flip and multiply gives \frac{6}{7} \times \frac{7}{2} = \frac{42}{14} = 3 too — same answer, the 7s cancel.)

See it: how many scoops fit?

Two cups, and each scoop is \frac{1}{3} of a cup. Every cup holds three scoops, so two cups hold 6 — that's 2 \div \frac{1}{3} = 6. Step through it and count the pieces.

The chant is "keep, change, flip" — and it's the second fraction (the one you are dividing by) that flips. Keep the first one exactly as it was. Flipping the first fraction, or flipping both, gives the wrong answer every time. For \frac{1}{2} \div \frac{1}{4} the correct move is \frac{1}{2} \times \frac{4}{1} = 2, not \frac{2}{1} \times \frac{1}{4}.

And here's the fact that surprises everyone: dividing by a fraction less than 1 makes the answer bigger. We're so used to "dividing shrinks things" that \frac{1}{2} \div \frac{1}{4} = 2 looks impossible. But it's asking "how many little quarters fit in a half?" — and the answer to that is naturally more than one. Small pieces means lots of pieces.

It isn't magic — there's a satisfying reason. Dividing by \frac{1}{2} means asking "how many halves are in this?" — and every whole thing contains two halves, so the answer doubles. Dividing by \frac{1}{2} is exactly the same as multiplying by 2 — and 2 is the flip of \frac{1}{2}! In the same way, dividing by \frac{1}{3} means "how many thirds?", which triples things, i.e. multiplying by 3 — the flip of \frac{1}{3}.

This "small divisor, big answer" idea is everywhere once you spot it. Fitting lots of little scoops into a big tub, working out how many 250-millilitre glasses a 2-litre bottle fills, changing between units, or turning a rate into a count — they all ask "how many of the small thing fit in the big thing?", and the answer is always more than the big thing's own size in whole units.