Reciprocal and dividing fractions

To divide by a fraction, simply multiply by its reciprocal. That single trick takes care of all fraction division.

Idea

The reciprocal (or multiplicative inverse) of a non-zero fraction $\tfrac{a}{b}$ is $\tfrac{b}{a}$. Notice the flip: numerator and denominator swap. Examples:

• Reciprocal of $\tfrac{2}{3}$ is $\tfrac{3}{2}$.
• Reciprocal of $5 = \tfrac{5}{1}$ is $\tfrac{1}{5}$.
• Reciprocal of $\tfrac{7}{4}$ is $\tfrac{4}{7}$.

A fraction multiplied by its reciprocal always gives $1$: $\tfrac{a}{b} \times \tfrac{b}{a} = 1$.

Division rule. To compute $\tfrac{a}{b} \div \tfrac{c}{d}$, multiply by the reciprocal of the divisor: $\dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{a}{b} \times \dfrac{d}{c} = \dfrac{a d}{b c}.$

A simple example: $\tfrac{1}{2} \div \tfrac{1}{4}$. Reciprocal of $\tfrac{1}{4}$ is $4$. So $\tfrac{1}{2} \times 4 = 2$. Check: how many quarters fit in a half? Two! ✓

Why does the rule work? Division asks "how many times does the divisor fit in the dividend?" If you divide both by the same quantity, the ratio stays the same. By multiplying numerator and denominator (of the big fraction) by the reciprocal, the denominator becomes $1$ and the numerator becomes the answer.

Dividing by a whole number. Just multiply by $\tfrac{1}{\text{that number}}$. Example: $\tfrac{3}{5} \div 4 = \tfrac{3}{5} \times \tfrac{1}{4} = \tfrac{3}{20}$.

Worked examples

Example 1. Compute $\tfrac{3}{4} \div \tfrac{2}{5}$.

Flip $\tfrac{2}{5}$ to $\tfrac{5}{2}$. So $\tfrac{3}{4} \times \tfrac{5}{2} = \tfrac{15}{8} = 1\tfrac{7}{8}$.

Example 2. Compute $5 \div \tfrac{2}{3}$.

Flip $\tfrac{2}{3}$ to $\tfrac{3}{2}$. So $5 \times \tfrac{3}{2} = \tfrac{15}{2} = 7\tfrac{1}{2}$.

Example 3. Compute $\tfrac{7}{8} \div 14$.

Treat $14 = \tfrac{14}{1}$; reciprocal $\tfrac{1}{14}$. So $\tfrac{7}{8} \times \tfrac{1}{14} = \tfrac{7}{112} = \tfrac{1}{16}$.

Example 4. $2\tfrac{1}{2} \div 1\tfrac{1}{4}$.

Improper: $\tfrac{5}{2} \div \tfrac{5}{4}$. Flip: $\tfrac{5}{2} \times \tfrac{4}{5} = \tfrac{20}{10} = 2$.

Try it yourself

1. Find the reciprocal of $\tfrac{4}{9}$.
2. Find the reciprocal of $7$.
3. Compute $\tfrac{2}{3} \div \tfrac{4}{9}$.
4. Compute $6 \div \tfrac{2}{3}$.
5. Compute $\tfrac{5}{6} \div \tfrac{10}{3}$.
6. Compute $1\tfrac{1}{2} \div \tfrac{3}{4}$.
7. Compute $\tfrac{8}{15} \div 4$.
8. If $\tfrac{2}{3}$ of a number is $10$, what is the number?

Activity

Visualise $\tfrac{1}{2} \div \tfrac{1}{4}$ with paper strips. Take a strip representing $\tfrac{1}{2}$. How many strips of length $\tfrac{1}{4}$ fit into it? Two , so the answer is $2$.

Make a "reciprocal" memory card: write a fraction on one side and its reciprocal on the other. Shuffle and quiz yourself.