Dividing Fractions

Here is the good news: multiplying fractions is the easiest fraction operation there is. No common denominators, no rewriting — you multiply straight across, top times top and bottom times bottom. Dividing is one extra move: keep the first fraction, change division to multiplication, flip the second fraction.

Students often mix these rules up with the addition rules and start hunting for common denominators where none are needed. This article keeps the two skills side by side so the difference sticks: multiply straight across, divide by flipping.

Multiplying: straight across

To multiply two fractions, multiply the numerators to get the new numerator and multiply the denominators to get the new denominator. That is the entire rule: $\dfrac{2}{3} \cdot \dfrac{5}{7} = \dfrac{2 \cdot 5}{3 \cdot 7} = \dfrac{10}{21}$ .

It even makes sense in words. Taking $\dfrac{1}{2}$ of $\dfrac{1}{3}$ means cutting a third into two pieces and keeping one — a sixth. The word of in fraction problems almost always means multiply.

Simplify at the end, or better, simplify before you multiply. If a numerator and a denominator share a factor — even across the two fractions — divide it out first. This is called cross-canceling, and it keeps the numbers small.

Dividing: keep, change, flip

To divide fractions: keep the first fraction as it is, change the division sign to multiplication, and flip the second fraction (swap its numerator and denominator). Then multiply straight across like normal. So $\dfrac{2}{5} \div \dfrac{3}{4}$ becomes $\dfrac{2}{5} \cdot \dfrac{4}{3}$ .

The flipped fraction is called the reciprocal. Dividing by a number and multiplying by its reciprocal are the same operation — dividing by $2$ is the same as multiplying by $\dfrac{1}{2}$ , and dividing by $\dfrac{1}{4}$ is the same as multiplying by $4$ . Asking how many quarters fit in $3$ is asking $3 \div \dfrac{1}{4}$ , and the answer is $3 \cdot 4 = 12$ .

Only the second fraction flips — the one you are dividing by. Flipping the first one is the most common mistake on this skill.

Whole numbers and mixed numbers

A whole number is a fraction in disguise: write it over $1$ . So $8 \cdot \dfrac{3}{4}$ becomes $\dfrac{8}{1} \cdot \dfrac{3}{4} = \dfrac{24}{4} = 6$ .

A mixed number must be converted to an improper fraction before you multiply or divide. Rewrite $1\dfrac{1}{2}$ as $\dfrac{3}{2}$ first — multiplying the whole-number parts and fraction parts separately does not work.

Worked examples

Example 1: multiply straight across

Multiply $\dfrac{2}{3} \cdot \dfrac{5}{7}$ .

Multiply the numerators

2 \cdot 5 = 10

Multiply the denominators

3 \cdot 7 = 21

10

and

21

share no common factor, so it is already simplest form

\dfrac{10}{21}

Answer: $\dfrac{10}{21}$

Example 2: cross-cancel before multiplying

Multiply $\dfrac{3}{8} \cdot \dfrac{4}{9}$ .

Divide the

3

and the

9

3

\dfrac{1}{8} \cdot \dfrac{4}{3}

Divide the

4

and the

8

4

\dfrac{1}{2} \cdot \dfrac{1}{3}

Multiply straight across

\dfrac{1}{6}

Answer: $\dfrac{1}{6}$

Example 3: divide two fractions

Divide $\dfrac{2}{5} \div \dfrac{3}{4}$ .

Keep, change, flip

\dfrac{2}{5} \cdot \dfrac{4}{3}

Multiply the numerators

2 \cdot 4 = 8

Multiply the denominators

5 \cdot 3 = 15

Answer: $\dfrac{8}{15}$

Example 4: a mixed number

Divide $1\dfrac{1}{2} \div \dfrac{3}{8}$ .

Convert the mixed number to an improper fraction

1\dfrac{1}{2} = \dfrac{3}{2}

Keep, change, flip

\dfrac{3}{2} \cdot \dfrac{8}{3}

Multiply straight across

\dfrac{24}{6}

Simplify

4

Answer: $4$

Try one yourself

Common questions

Do I need a common denominator to multiply or divide fractions?

No — that is only for adding and subtracting. Multiplying goes straight across, and dividing flips the second fraction and then goes straight across. Hunting for a common denominator here just adds extra work.

Why does flipping the second fraction work?

Division asks how many of the second number fit inside the first. Smaller pieces fit more times, so dividing by $\dfrac{1}{4}$ multiplies the count by $4$ . In general, dividing by a fraction and multiplying by its reciprocal give the same answer every time.

Which fraction do I flip?

Only the second one — the fraction you are dividing by. The first fraction stays exactly as it is. If you flip the first one instead, you get the reciprocal of the right answer.

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