Multiplication and Division of Fractions

This lesson covers how to multiply and divide fractions quickly, without ever taking an LCM, and how to deal with signs, mixed numbers, and cancelling before you multiply.

Multiplying fractions

To multiply fractions, multiply the numerators together and the denominators together. No LCM is required.

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

For example:

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

You can also cancel common factors before multiplying:

$\frac{5}{6} \times \frac{2}{10} = \frac{5 \times 2}{6 \times 10} = \frac{10}{60} = \frac{1}{6}$

Mixed numbers

Change any mixed number to an improper fraction first, then multiply.

$1\tfrac{1}{5} \times 3\tfrac{1}{2} = \frac{6}{5} \times \frac{7}{2}$

Cancel the $6$ and the $2$ by $2$ to get $3$ and $1$:

$\frac{3}{5} \times \frac{7}{1} = \frac{21}{5}$

Multiplying without cancelling first gives the same answer: $\frac{6}{5} \times \frac{7}{2} = \frac{42}{10} = \frac{21}{5}$.

Signs

Decide the sign first, then multiply the numbers.

$\frac{5}{8} \times \left(-\frac{2}{15}\right)$

Cancel $5$ with $15$ (leaving $3$) and $2$ with $8$ (leaving $4$). The result is negative:

$-\frac{1}{4} \times \frac{1}{3} = -\frac{1}{12}$

With three factors, multiply the signs first. For example:

$\left(-\frac{5}{12}\right) \times \left(-\frac{1}{5}\right) \times \frac{3}{7} = \frac{1}{7 \times 4} = \frac{1}{28}$

Two negatives make a positive. If any factor is $0$, the whole product is $0$, so there is nothing to cancel.

Dividing fractions

Division needs no LCM either. Keep the first fraction, change the division to multiplication, and use the reciprocal of the second fraction (swap its numerator and denominator).

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

For example, $\frac{1}{2} \div 2$:

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

And $-\frac{1}{5} \div \frac{3}{2}$:

$-\frac{1}{5} \times \frac{2}{3} = -\frac{2}{15}$

Mixed numbers in division

Convert to improper fractions first, then flip and multiply.

$1\tfrac{1}{2} \div 3\tfrac{1}{4} = \frac{3}{2} \div \frac{13}{4} = \frac{3}{2} \times \frac{4}{13} = \frac{6}{13}$

Brackets combined with division

Work out the brackets first, then divide.

$\left(\frac{3}{5} + \frac{1}{5}\right) \div \left(\frac{8}{7} - \frac{3}{7}\right) = \frac{4}{5} \div \frac{5}{7} = \frac{4}{5} \times \frac{7}{5} = \frac{28}{25}$

A full mixed-operation example

Do the bracket first, taking the LCM of $8$ and $2$, which is $8$:

$\left(\frac{1}{8} - \frac{1}{2}\right) \times \frac{8}{5} \div \frac{2}{5}$

Bracket: $\frac{1}{8} - \frac{4}{8} = -\frac{3}{8}$.

Change the division to multiplication by the reciprocal of $\frac{2}{5}$:

$-\frac{3}{8} \times \frac{8}{5} \times \frac{5}{2}$

Cancel $5$ with $5$ and $8$ with $8$:

$-\frac{3}{2}$

Key takeaways

• To multiply, take the product of numerators over the product of denominators; no LCM is needed.
• To divide, keep the first fraction and multiply by the reciprocal of the second.
• Fix the sign first, change mixed numbers to improper fractions, and cancel common factors before multiplying to keep the numbers small.