Division of Fractions

To divide by a fraction, multiply by its reciprocal: the same fraction turned upside down. This page explains why that works, then shows how to divide with whole numbers, mixed numbers, complex fractions, negative fractions and fractions with letters.

The reciprocal of a fraction

Two numbers are reciprocals of each other when their product is 1. To find the reciprocal of a fraction, swap its numerator and denominator.

The reciprocal of $\frac{3}{5}$ is $\frac{5}{3}$, because $\frac{3}{5}\cdot\frac{5}{3}=1$.
The reciprocal of $\frac{1}{4}$ is $\frac{4}{1}=4$.
The reciprocal of $7=\frac{7}{1}$ is $\frac{1}{7}$.
For a mixed number, change it to an improper fraction first: $2\frac{1}{3}=\frac{7}{3}$, so its reciprocal is $\frac{3}{7}$.
0 has no reciprocal, because no number times 0 gives 1. That's why you can't divide by 0.

The rule

Keep the first fraction, change ÷ to ×, and flip the second fraction. $$\frac{a}{b}\div\frac{c}{d}=\frac{a}{b}\cdot\frac{d}{c}=\frac{ad}{bc}$$

$$\frac{3}{4}\div\frac{2}{5}=\frac{3}{4}\cdot\frac{5}{2}=\frac{15}{8}=1\frac{7}{8}$$

Why it works

Counting pieces. $3\div\frac{1}{4}$ asks how many quarters fit into 3. Each whole holds 4 quarters, so 3 wholes hold $3\cdot 4=12$ of them. Dividing by $\frac{1}{4}$ gives the same answer as multiplying by 4, its reciprocal.

Checking with multiplication. Division undoes multiplication, so the answer times the divisor gives back the number you divided. For the example above: $\frac{15}{8}\cdot\frac{2}{5}=\frac{30}{40}=\frac{3}{4}$, the number we started with. In general $\frac{ad}{bc}\cdot\frac{c}{d}=\frac{a}{b}$, so the rule always passes this check.

Whole numbers

Write the whole number as a fraction over 1.

A fraction divided by a whole number: the whole number ends up in the denominator.

$$\frac{6}{7}\div 3=\frac{6}{7}\cdot\frac{1}{3}=\frac{6}{21}=\frac{2}{7}$$

When the whole number divides the numerator, it's even quicker to divide the numerator: $6\div 3=2$, so the answer is $\frac{2}{7}$.

A whole number divided by a fraction:

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

The answer is bigger than 4. Dividing by a number less than 1 makes the result bigger: six pieces of size $\frac{2}{3}$ fit into 4.

Mixed numbers

Change mixed numbers to improper fractions, then flip the divisor and multiply:

$$\begin{aligned}2\frac{1}{4}\div 1\frac{1}{2}&=\frac{9}{4}\div\frac{3}{2}=\frac{9}{4}\cdot\frac{2}{3}\\&=\frac{18}{12}=\frac{3}{2}=1\frac{1}{2}\end{aligned}$$

Negative fractions

Divide as usual. The sign follows the same rules as in multiplication: different signs give a negative answer, equal signs a positive one.

$$\begin{aligned}-\frac{4}{9}\div\frac{2}{3}&=-\frac{4}{9}\cdot\frac{3}{2}\\&=-\frac{12}{18}=-\frac{2}{3}\end{aligned}$$

Complex fractions

A fraction with fractions in its numerator or denominator is called a complex fraction. Its main fraction bar means division:

$$\frac{\dfrac{a}{b}}{\dfrac{c}{d}}=\frac{a}{b}\div\frac{c}{d}=\frac{ad}{bc}$$

Example 1

$$\frac{\dfrac{3}{4}}{\dfrac{5}{8}}=\frac{3}{4}\cdot\frac{8}{5}=\frac{24}{20}=\frac{6}{5}$$

Example 2

When there are sums in the numerator or denominator, it's easier to multiply the top and the bottom by the least common denominator of all the small fractions, here 6:

$$\begin{aligned}\frac{1+\dfrac{1}{2}}{2-\dfrac{1}{3}}&=\frac{\left(1+\dfrac{1}{2}\right)\cdot 6}{\left(2-\dfrac{1}{3}\right)\cdot 6}\\&=\frac{6+3}{12-2}=\frac{9}{10}\end{aligned}$$

Algebraic fractions

The rule is the same: flip the divisor, then multiply and cancel common factors. Cancel only after you have turned the division into a multiplication.

Example 3: $\frac{m}{2d}\div\frac{3h}{y}$

$$\frac{m}{2d}\div\frac{3h}{y}=\frac{m}{2d}\cdot\frac{y}{3h}=\frac{my}{6dh}$$

Example 4: $\frac{36d}{5}\div\frac{18h}{10y}$

After flipping, 36 cancels with 18 and 10 cancels with 5:

$$\begin{aligned}\frac{36d}{5}\div\frac{18h}{10y}&=\frac{36d}{5}\cdot\frac{10y}{18h}\\&=\frac{2d\cdot 2y}{h}=\frac{4dy}{h}\end{aligned}$$

Example 5: $\frac{x^2-4}{x}\div\frac{x+2}{x^2}$

$$\begin{aligned}&\frac{x^2-4}{x}\div\frac{x+2}{x^2}\\&=\frac{(x-2)(x+2)}{x}\cdot\frac{x^2}{x+2}\\&=x(x-2)\end{aligned}$$

for $x\ne 0$ and $x\ne -2$. At $x=-2$ the divisor would be 0.

Example 6: $\frac{a}{b}\div m$

$$\frac{a}{b}\div m=\frac{a}{b}\cdot\frac{1}{m}=\frac{a}{bm}$$

More on multiplying and cancelling in multiplication of fractions.

Common mistakes

Flipping the wrong fraction: flip the divisor (the second fraction), never the first. $\frac{3}{4}\div\frac{2}{5}\ne\frac{4}{3}\cdot\frac{2}{5}$.
Cancelling across the ÷ sign: in $\frac{3}{4}\div\frac{5}{3}$ the 3s don't cancel. Flip first: $\frac{3}{4}\cdot\frac{3}{5}=\frac{9}{20}$.
Dividing mixed numbers part by part: change them to improper fractions first.
Expecting a smaller answer: dividing by a fraction less than 1 gives a bigger number: $5\div\frac{1}{2}=10$.

Practice

Try each problem, then open it to check your answer.

1. $\frac{2}{3}\div\frac{4}{5}$

$\frac{2}{3}\cdot\frac{5}{4}=\frac{10}{12}=\frac{5}{6}$

2. $\frac{5}{8}\div 5$

$\frac{5}{8}\cdot\frac{1}{5}=\frac{5}{40}=\frac{1}{8}$

3. $3\div\frac{3}{4}$

$3\cdot\frac{4}{3}=\frac{12}{3}=4$

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

$\frac{10}{3}\div\frac{5}{4}=\frac{10}{3}\cdot\frac{4}{5}=\frac{40}{15}$, which reduces to $\frac{8}{3}=2\frac{2}{3}$

5. $-\frac{5}{6}\div\frac{10}{9}$

$-\frac{5}{6}\cdot\frac{9}{10}=-\frac{45}{60}=-\frac{3}{4}$

6. $\dfrac{\frac{1}{2}}{\frac{3}{4}}$

$\frac{1}{2}\cdot\frac{4}{3}=\frac{4}{6}=\frac{2}{3}$

7. $\frac{a^2}{b}\div\frac{a}{b^2}$

$\frac{a^2}{b}\cdot\frac{b^2}{a}=\frac{a^2b^2}{ab}=ab$

8. $\frac{x^2-1}{3x}\div\frac{x+1}{6x^2}$

$\frac{(x-1)(x+1)}{3x}\cdot\frac{6x^2}{x+1}$, which simplifies to $2x(x-1)$

You can check any division of numeric fractions, step by step, with the fraction calculator below.