Dividing Fractions

To divide by a fraction, multiply by its reciprocal — flip the second fraction and multiply.

The Reciprocal

The reciprocal of a fraction is obtained by swapping numerator and denominator.

Fraction	Reciprocal
3/4	4/3
5/1	1/5
2/7	7/2

Dividing Fractions: Keep–Change–Flip

1
Keep the first fraction as is
$\frac{3}{5}$
2
Change ÷ to ×
$\frac{3}{5} \times$
3
Flip the second fraction (write its reciprocal)
$\frac{3}{5} \times \frac{4}{2}$
4
Multiply and simplify
$\frac{3 \times 4}{5 \times 2} = \frac{12}{10} = \frac{6}{5} = 1\frac{1}{5}$

Division of Fractions

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

Multiply by the reciprocal of the divisor.

Note The phrase "Keep–Change–Flip" (KCF) is a handy memory aid: keep the first fraction, change the operation, flip the second fraction.

Dividing with Mixed Numbers

Convert to improper fractions first, then apply KCF.

$2\frac{1}{3} \div 1\frac{3}{4} = \frac{7}{3} \div \frac{7}{4} = \frac{7}{3} \times \frac{4}{7} = \frac{28}{21} = \frac{4}{3} = 1\frac{1}{3}$

How many 1/4 pieces fit in 3/2?

$\frac{3}{2} \div \frac{1}{4} = \frac{3}{2} \times \frac{4}{1} = \frac{12}{2} = 6$

Six pieces fit.

Real-World Applications

Real world: cutting lengths of ribbon

You have $\dfrac{3}{4}$ m of ribbon and need to cut pieces $\dfrac{1}{8}$ m long. How many pieces can you cut?

$\dfrac{3}{4} \div \dfrac{1}{8} = \dfrac{3}{4} \times \dfrac{8}{1} = \dfrac{24}{4} = 6$ pieces.

Real world: portioning flour for loaves

A cook has $2\dfrac{1}{2}$ kg of flour. Each loaf needs $\dfrac{5}{8}$ kg. How many loaves can be made?

$\dfrac{5}{2} \div \dfrac{5}{8} = \dfrac{5}{2} \times \dfrac{8}{5} = \dfrac{40}{10} = 4$ loaves.

Real world: sharing a road trip

Three friends share equal thirds of a $\dfrac{9}{10}$ -motorway stretch. How long is each person's section?

$\dfrac{9}{10} \div 3 = \dfrac{9}{10} \times \dfrac{1}{3} = \dfrac{9}{30} = \dfrac{3}{10}$ of the motorway each.

Tip

Where do you see this? Dividing food into equal portions, calculating how many tiles fit in a space, measuring fabric or wood into equal lengths, and splitting shifts between workers all rely on dividing fractions.

Practice Problems

Calculate 5/6 ÷ 5/12.

Show answer

5/6 × 12/5 = 60/30 = 2.

Calculate 1 1/2 ÷ 3/4.

Show answer

3/2 ÷ 3/4 = 3/2 × 4/3 = 12/6 = 2.