Dividing Fractions — Flip and Multiply Method

To divide fractions, keep the first fraction, flip the divisor, and multiply. In general,

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

When you use this method

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

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

It helps to know why the method is valid before applying it. 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

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

The procedure, step by step

1. Keep the first fraction. Copy the first fraction as it is.
2. Flip the divisor. Replace the second fraction with its reciprocal, which means swap its numerator and denominator.
3. Multiply straight across. Multiply numerators together and denominators together.
4. Simplify the result. Reduce the final fraction, or convert it to a mixed number if that form is preferred.
5. Check the divisor. The fraction you divide by must not be zero.

A full example from start to finish

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

Keep the first fraction and flip the divisor:

Multiply:

Simplify:

So

This also makes sense in words: "How many halves fit into three fourths?" The answer is $1\frac{1}{2}$ halves. Notice that the answer got larger. Division makes numbers smaller only when you divide by a positive number greater than $1$. Dividing by $\frac{1}{2}$ counts halves, and since halves are smaller pieces than wholes, more than one often fits, which is why $\frac{3}{4} \div \frac{1}{2}$ is bigger than $\frac{3}{4}$.

Where students get stuck, and how to check each step

A few steps cause most of the trouble:

• 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. 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.

After solving, check 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 $1$, it should get smaller. This does not replace the calculation, but it is a good way to catch a flipped fraction or sign error.

Practice this procedure

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