Math Core

Multiplying and dividing fractions is actually simpler than adding them: no common denominator is needed. In this lesson you'll review both operations, learn to cancel before you multiply, and handle negative signs with the same rules you use for integers.

Multiplying fractions

Multiply the numerators, and multiply the denominators.

\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}

For example, $\dfrac{2}{3} \cdot \dfrac{4}{5} = \dfrac{8}{15}$ . You can think of it as " $\dfrac{2}{3}$ of $\dfrac{4}{5}$ ": take $\dfrac{4}{5}$ of something, then take two thirds of that.

Cancel first

Multiplying big numbers and then simplifying works, but it's slow and error-prone. Instead, cancel any factor that appears in a numerator and a denominator before multiplying. It can be the same fraction or across fractions.

Worked example: Cancelling across fractions

Find $\dfrac{9}{14} \cdot \dfrac{7}{12}$ .

9 and 12 share a factor of 3: $9 \to 3$ and $12 \to 4$ .
7 and 14 share a factor of 7: $7 \to 1$ and $14 \to 2$ .

\frac{9}{14} \cdot \frac{7}{12} = \frac{3}{2} \cdot \frac{1}{4} = \frac{3}{8}

Without cancelling you'd get $\dfrac{63}{168}$ and still have to simplify it.

Signs

Fractions follow the integer sign rules:

factors	product or quotient
same signs ( $+ \cdot +$ or $- \cdot -$ )	positive
different signs ( $+ \cdot -$ or $- \cdot +$ )	negative

The easiest method is to decide the sign first, then work with the numbers as if they were positive. With several factors, count the negatives: an even number of negatives gives a positive result and an odd number gives a negative result.

Worked example: Multiplying with negatives

$-\dfrac{3}{4} \cdot \dfrac{8}{15}$
$\left(-\dfrac{5}{6}\right)\left(-\dfrac{9}{10}\right)$

Solutions.

One negative, so the answer is negative. Cancel 8 with 4 and 3 with 15: $\dfrac{1}{1} \cdot \dfrac{2}{5}$ . The answer is $-\dfrac{2}{5}$ .
Two negatives, so the answer is positive. Cancel 5 with 10 and 9 with 6: $\dfrac{1}{2} \cdot \dfrac{3}{2} = \dfrac{3}{4}$ .

Dividing fractions

Two numbers are reciprocals if their product is 1. The reciprocal of $\dfrac{a}{b}$ is $\dfrac{b}{a}$ . The reciprocal of $-\dfrac{2}{7}$ is $-\dfrac{7}{2}$ : a reciprocal keeps the same sign, because $\left(-\dfrac{2}{7}\right)\left(-\dfrac{7}{2}\right) = 1$ . Zero has no reciprocal, which is why you can't divide by zero.

Dividing fractions

To divide by a fraction, multiply by its reciprocal.

\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}

This works because division asks "what times $\dfrac{c}{d}$ gives $\dfrac{a}{b}$ ?" If you check $\dfrac{a}{b} \cdot \dfrac{d}{c}$ , multiplying it by $\dfrac{c}{d}$ gives $\dfrac{a}{b} \cdot 1 = \dfrac{a}{b}$ .

Worked example: Dividing with negatives

Find $\dfrac{4}{9} \div \left(-\dfrac{2}{3}\right)$ .

Different signs, so the answer is negative. Flip the divisor and multiply:

\frac{4}{9} \cdot \frac{3}{2} = \frac{2}{3} \cdot \frac{1}{1} = \frac{2}{3}

(4 and 2 share 2; 3 and 9 share 3.) The answer is $-\dfrac{2}{3}$ .

Check: $\left(-\dfrac{2}{3}\right)\left(-\dfrac{2}{3}\right) = \dfrac{4}{9}$ . Correct.

Common mistake

Don't cancel before you flip. In $\dfrac{3}{5} \div \dfrac{3}{10}$ , the two 3s are not a numerator-over-denominator pair yet. Rewrite first: $\dfrac{3}{5} \cdot \dfrac{10}{3}$ . Now cancel: the 3s cancel and 10 with 5 gives 2, so the answer is 2.

Mixed numbers and complex fractions

Always convert mixed numbers to improper fractions before multiplying or dividing. Multiplying the whole parts and the fraction parts separately gives the wrong answer.

Worked example: Mixed numbers

Find $-2\dfrac{1}{4} \div 1\dfrac{1}{2}$ .

Convert: $-2\dfrac{1}{4} = -\dfrac{9}{4}$ and $1\dfrac{1}{2} = \dfrac{3}{2}$ .

-\frac{9}{4} \div \frac{3}{2} = -\frac{9}{4} \cdot \frac{2}{3} = -\frac{3}{2} \cdot \frac{1}{1} = -\frac{3}{2} = -1\frac{1}{2}

A fraction bar is a division sign, so a fraction whose numerator or denominator is itself a fraction (a complex fraction) is just a division problem:

\frac{\;\frac{5}{8}\;}{\;-\frac{1}{4}\;} = \frac{5}{8} \div \left(-\frac{1}{4}\right) = \frac{5}{8} \cdot (-4) = -\frac{5}{2}

You'll see these often in algebra, for example when you compute a slope.

Tip

Estimate to catch mistakes. Dividing by a number between 0 and 1 makes a positive number bigger, and multiplying by one makes it smaller. Since $\dfrac{3}{5} \div \dfrac{3}{10}$ divides by a small number, an answer of 2 (bigger than $\dfrac{3}{5}$ ) makes sense.