Dividing Fractions: Why You Multiply by the Reciprocal

Q: What is the reciprocal of a whole number?

Write the whole number over 1 and flip it. The reciprocal of 7 is 1/7.

What multiplying fractions means, why "keep, change, flip" works for division, how to cancel before multiplying, and what a reciprocal is.

3 min read Supports 3 lessons

Multiplying first

Taking a fraction of something means multiplying. Half of three quarters is $\frac{1}{2} \times \frac{3}{4}$ . Picture three quarters of a bar, then take half of that: you get three eighths.

\tfrac{3}{4}

\tfrac{3}{8}

Half of $\frac{3}{4}$ is $\frac{3}{8}$ .

Multiplying fractions

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

No common denominator needed.

Tip

Cancel before you multiply. In $\frac{4}{9} \times \frac{3}{8}$ , the $4$ cancels into the $8$ and the $3$ into the $9$ , leaving $\frac{1}{3} \times \frac{1}{2} = \frac{1}{6}$ .

What division asks

$12 \div 3$ asks: how many $3$ s fit into $12$ ? Division by a fraction asks the same question. $3 \div \frac{1}{2}$ asks how many halves fit into $3$ . Each whole holds two halves, so three wholes hold six.

\tfrac{6}{2}

Six halves make three wholes, so $3 \div \frac{1}{2} = 6$ .

Notice that dividing by $\frac{1}{2}$ gave the same answer as multiplying by $2$ . That is no accident.

The reciprocal

The reciprocal of a number is what you multiply it by to get $1$ . For a fraction, just turn it upside down: the reciprocal of $\frac{2}{3}$ is $\frac{3}{2}$ , because $\frac{2}{3} \times \frac{3}{2} = 1$ . The reciprocal of $5$ is $\frac{1}{5}$ .

Keep, change, flip

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

Keep the first fraction, change $\div$ to $\times$ , flip the second.

Why flipping works

Write the division as one big fraction, then multiply its top and bottom by the reciprocal of the bottom. The bottom becomes $1$ and disappears.

$\frac{2}{3} \div \frac{4}{5} = \dfrac{\;\frac{2}{3}\;}{\;\frac{4}{5}\;}$
A division is a fraction.

1/4

Worked example

Compute $2\frac{1}{2} \div \frac{3}{4}$ .

1
Write the mixed number as an improper fraction.
$\frac{5}{2} \div \frac{3}{4}$

Watch out

Flip the second fraction, the one you are dividing by, never the first.

Key idea

Dividing by a number less than $1$ makes the answer bigger. That feels odd until you remember the question: lots of small pieces fit into a whole.

Reciprocals reappear with negative exponents: $a^{-1}$ means $\frac{1}{a}$ , the reciprocal of $a$ .

Common questions

Why do you flip the second fraction when dividing?: Dividing by a number is the same as multiplying by its reciprocal. Multiplying top and bottom of the division by that reciprocal turns the bottom into $1$ , leaving a multiplication.
What is the reciprocal of a whole number?: Write the whole number over $1$ and flip it. The reciprocal of $7$ is $\frac{1}{7}$ .

Practise multiplying and dividing fractions

The Dividing Fractions lesson has worked examples and unlimited practice that adapts to you.

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Common questions

Why do you flip the second fraction when dividing?

Dividing by a number is the same as multiplying by its reciprocal. Multiplying top and bottom of the division by that reciprocal turns the bottom into $1$ , leaving a multiplication.

What is the reciprocal of a whole number?

Write the whole number over $1$ and flip it. The reciprocal of $7$ is $\frac{1}{7}$ .