Dividing Fractions — Flip and Multiply Method

To divide fractions, keep the first fraction, flip the divisor, and multiply. This shortcut works as long as the divisor is not zero.

For example,

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

The answer gets larger here because dividing by $\frac{1}{2}$ asks how many halves fit into $\frac{3}{4}$.

In general,

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

as long as $b \ne 0$, $d \ne 0$, and $\frac{c}{d} \ne 0$.

How To Divide Fractions

The flipped fraction is the reciprocal. The reciprocal of $\frac{2}{3}$ is $\frac{3}{2}$ because the numerator and denominator switch places.

Use this process:

1. Keep the first fraction unchanged.
2. Flip the second fraction, which is the divisor.
3. Multiply across.
4. Simplify the result.

Why Flip And Multiply Works

Dividing by a number is the same as multiplying by its multiplicative inverse. For a nonzero fraction $\frac{c}{d}$, that inverse is $\frac{d}{c}$ because

$$\frac{c}{d} \times \frac{d}{c} = 1$$

So dividing by $\frac{c}{d}$ gives the same result as multiplying by $\frac{d}{c}$. That is the reason the rule works, not just a trick to memorize.

Worked Example: $\frac{3}{4} \div \frac{1}{2}$

Start with

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

Keep the first fraction and flip the divisor:

$$\frac{3}{4} \times \frac{2}{1}$$

Multiply:

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

Simplify:

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

So

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

This also makes sense in words: "How many halves fit into three fourths?" The answer is $1\frac{1}{2}$ halves.

Why Dividing By A Fraction Can Make The Answer Bigger

Students often expect division to make numbers smaller. That is true when you divide by a positive number greater than $1$, but not when you divide by a positive fraction less than $1$.

If you divide by $\frac{1}{2}$, you are counting halves. Since halves are smaller pieces than wholes, you can often fit more than one of them into the original amount. That is why $\frac{3}{4} \div \frac{1}{2}$ is bigger than $\frac{3}{4}$.

Common Mistakes In Fraction Division

Flipping the wrong fraction

You only flip the second fraction, the divisor. The first fraction stays the same.

Forgetting the zero condition

You cannot divide by $0$, so the divisor cannot be the zero fraction. For instance, $\frac{5}{6} \div 0$ is undefined.

Dividing top by top and bottom by bottom

That is not the rule for fraction division. After flipping the divisor, you multiply across.

Forgetting to rewrite whole numbers as fractions

If a whole number appears, write it over $1$. For example, $2 \div \frac{2}{3}$ means $\frac{2}{1} \div \frac{2}{3}$.

Missing an easy simplification

You can multiply first and simplify at the end, but sometimes it is easier to cancel common factors before multiplying. Either approach is fine if the algebra is valid.

When You Use Dividing Fractions

Dividing fractions comes up in measurement, recipes, unit rates, and scaling problems. If you know the size of one piece and want to know how many such pieces fit into a total amount, fraction division is often the right model.

For example, if a recipe uses $\frac{2}{3}$ cup of milk per batch and you have $2$ cups of milk, the question "How many batches can I make?" becomes

$$2 \div \frac{2}{3}$$

That is fraction division even though one number is a whole number.

A Quick Check Before You Move On

After solving, ask whether the size of the answer is reasonable.

• If you divide by a positive fraction less than $1$, the result should get larger.
• If you divide by a positive number greater than $1$, the result should get smaller.

This does not replace the calculation, but it is a good way to catch a flipped fraction or sign error.

Try A Similar Problem

Try $\frac{5}{6} \div \frac{2}{3}$ and decide whether the answer should be smaller or larger than $\frac{5}{6}$ before you calculate. If you want another case to check your steps, solve a similar problem with GPAI Solver.