Topic 2 of 12

Dividing fractions

Step 8 · Ratios and negative numbers topics

To divide by a fraction, multiply by its reciprocal. This lesson shows the rule and why it works.

What division asks

$6 \div 2$ asks: how many 2s fit into 6? Dividing by a fraction asks the same kind of question. $3 \div \frac{1}{2}$ asks: how many halves fit into 3? Each whole holds 2 halves, so 3 wholes hold 6 halves.

3 \div \frac{1}{2} = 6

The answer is bigger than the number you started with. That makes sense: lots of small pieces fit into 3.

Reciprocals

The reciprocal of a fraction is the fraction turned upside down. The reciprocal of $\frac{2}{3}$ is $\frac{3}{2}$ . A number times its reciprocal is always 1:

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

A whole number such as 4 can be written $\frac{4}{1}$ , so its reciprocal is $\frac{1}{4}$ .

The rule

To divide by a fraction, multiply by its reciprocal:

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

Here is why. Dividing by $\frac{1}{4}$ counts the quarters, and there are 4 in every whole, so it is the same as multiplying by 4. Dividing by $\frac{3}{4}$ counts groups of three quarters: count the quarters (multiply by 4), then put them in groups of 3 (divide by 3). Multiplying by $\frac{4}{3}$ does exactly those two things.

Example: Fraction divided by fraction

Work out $\frac{3}{5} \div \frac{2}{3}$ .

Keep the first fraction: $\frac{3}{5}$ .
Turn the second fraction upside down: $\frac{2}{3}$ becomes $\frac{3}{2}$ .
Multiply tops and bottoms: $\frac{3}{5} \times \frac{3}{2} = \frac{9}{10}$ .
Check by multiplying back: $\frac{9}{10} \times \frac{2}{3} = \frac{18}{30} = \frac{3}{5}$ .

Answer $\frac{9}{10}$

Example: A word problem

A ribbon is $\frac{3}{4}$ of a yard long. How many pieces $\frac{1}{8}$ of a yard long can be cut from it?

The question is how many eighths fit into three quarters: $\frac{3}{4} \div \frac{1}{8}$ .
Multiply by the reciprocal: $\frac{3}{4} \times \frac{8}{1} = \frac{24}{4}$ .
Simplify: $\frac{24}{4} = 6$ .

Answer6 pieces

Example: Mixed numbers

Work out $2\frac{1}{2} \div \frac{5}{6}$ .

Change the mixed number to an improper fraction: $2\frac{1}{2} = \frac{5}{2}$ .
Multiply by the reciprocal: $\frac{5}{2} \times \frac{6}{5} = \frac{30}{10}$ .
Simplify: $\frac{30}{10} = 3$ .

Answer3

Try it yourself

Work each one on paper first, then open the answer to check.

$\frac{1}{2} \div \frac{1}{4}$
Show answer
2, because $\frac{1}{2} \times \frac{4}{1} = \frac{4}{2} = 2$ . Two quarters fit into a half.
$\frac{5}{6} \div \frac{2}{3}$
Show answer
$\frac{5}{4}$ , or $1\frac{1}{4}$ , because $\frac{5}{6} \times \frac{3}{2} = \frac{15}{12} = \frac{5}{4}$ .
$4 \div \frac{2}{5}$
Show answer
10, because $\frac{4}{1} \times \frac{5}{2} = \frac{20}{2} = 10$ .
$1\frac{1}{2} \div \frac{3}{8}$
Show answer
4, because $\frac{3}{2} \times \frac{8}{3} = \frac{24}{6} = 4$ .
One serving of rice is $\frac{2}{3}$ of a cup. How many servings are in 4 cups?
Show answer
6 servings, because $4 \div \frac{2}{3} = \frac{4}{1} \times \frac{3}{2} = \frac{12}{2} = 6$ .