Dividing fractions

Keep, change, flip

Keep the first fraction, change the sign to multiply, flip the second. The flipped fraction is its reciprocal.

$$\frac{3}{4} \div \frac{1}{2} = \frac{3}{4} \times \frac{2}{1} = \frac{6}{4} = \frac{3}{2}$$

Only the second fraction flips. Flipping both, or flipping the first, is the usual error.

Why flipping works

Division asks a counting question: how many of these fit into that?

$$3 \div \tfrac{1}{2} \quad\text{asks}\quad \text{how many halves fit into } 3?$$

Each whole holds two halves and there are three wholes, so the answer is $6$ — which is $3$ multiplied by $2$. Dividing by $\tfrac{1}{2}$ became multiplying by $2$ because each whole contains two halves.

For any divisor $\tfrac{c}{d}$, each whole contains $\tfrac{d}{c}$ of them, so you multiply by $\tfrac{d}{c}$. That is the reciprocal, and that is the whole reason for the flip. Full reasoning in dividing fractions.

The answer often grows

Dividing by less than $1$ makes the answer bigger, because small pieces means many pieces.

$$6 \div \frac{1}{2} = 12$$

Twelve halves really do make six. This is the mirror image of what happens when multiplying.

Worked examples

Work out $\tfrac{2}{3} \div \tfrac{1}{6}$.

$$\frac{2}{3} \times \frac{6}{1} = \frac{12}{3} = 4$$

Work out $\tfrac{5}{6} \div 5$.

Write the whole number over $1$:

$$\frac{5}{6} \times \frac{1}{5} = \frac{5}{30} = \frac{1}{6}$$

Zero has no reciprocal

There is no number you can multiply $0$ by to get $1$, so $\tfrac{c}{d}$ can never be zero here. That is the same fact as “you cannot divide by zero”, seen from another angle.