Dividing with Fractions

Division changes its meaning when the divisor changes. Sharing $\frac{3}{4}$ of one chocolate bar among two whole groups makes smaller shares. Counting $\frac{1}{8}$ portions inside $\frac{3}{4}$ asks how many pieces fit. In both cases, the reciprocal belongs to the divisor.

Parents: ask your child whether each example is sharing an amount or counting portions before revealing the reciprocal working. The P6 source specifies without calculator, so every calculation here is written by hand.

By the end you'll be able to divide a proper fraction by a whole number, divide a whole number by a proper fraction, and divide a proper fraction by a proper fraction.

Proper fraction ÷ whole number: share the fraction

Start with one named whole chocolate bar. $\frac{3}{4}$ of that bar is shared equally between two children. Each child receives $\frac{1}{2}$ of the existing $\frac{3}{4}$ amount.

Show the worked solution

Each share is $\frac{3}{8}$ of the original bar.

Use the reciprocal of the whole divisor

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

$$\frac{3}{4} \times \frac{1}{2} = \frac{3}{8}$$

The reciprocal of the whole number two is $\frac{1}{2}$. Writing it keeps the sharing action visible instead of asking you to memorise a shortcut.

Whole number ÷ proper fraction: count portions

Now the question changes. Five whole litres divided by $\frac{2}{3}$ of a litre asks how many $\frac{2}{3}$-litre portions fit into five litres. Because $\frac{2}{3}$ is less than one whole, the positive count can be greater than the dividend.

Whole ÷ non-unit proper fraction

$$5 \div \frac{2}{3} = 5 \times \frac{3}{2}$$

$$5 \times \frac{3}{2} = \frac{15}{2} = 7\frac{1}{2}$$

The unit-fraction bridge has the same reciprocal idea: three wholes contain six portions of size $\frac{1}{2}$ whole.

Whole ÷ unit fraction

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

Proper fraction ÷ proper fraction: count the portions

For a proper fraction divided by a proper fraction, name the same whole and name the portion size. $\frac{3}{4}$ of one metre divided into $\frac{2}{5}$-metre portions gives 1 $\frac{7}{8}$ portions as a numerical quotient.

Proper ÷ non-unit proper fraction

$$\frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \times \frac{5}{2}$$

$$\frac{3}{4} \times \frac{5}{2} = \frac{15}{8} = 1\frac{7}{8}$$

When the portion is $\frac{1}{8}$, the visual count is especially clear: six $\frac{1}{8}$ portions fit inside $\frac{3}{4}$.

Count the portions

$$\frac{3}{4} \div \frac{1}{8} = \frac{3}{4} \times 8$$

$$\frac{3}{4} \times 8 = \frac{24}{4} = 6$$

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Fraction division: share or count portions

🤔 Predict first: For $\frac{3}{4}$ ÷ 2, what does each whole group get?

$\frac{3}{8}$ of the whole$\frac{3}{2}$ wholes$\frac{2}{3}$ of the whole

Use reciprocal for the divisor only. Keep the dividend in place, take the divisor's reciprocal, then multiply.

Practise with worked steps

Start here — share $\frac{3}{4}$ among two groups

Find $\frac{3}{4}$ divided by two.

Share the existing fraction

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

$$\frac{3}{4} \times \frac{1}{2} = \frac{3}{8}$$

The answer is $\frac{3}{8}$.

Build up — count $\frac{2}{3}$ portions in five wholes

Find five divided by $\frac{2}{3}$.

Use the divisor reciprocal

$$5 \div \frac{2}{3} = 5 \times \frac{3}{2}$$

$$5 \times \frac{3}{2} = \frac{15}{2} = 7\frac{1}{2}$$

The answer is 7 $\frac{1}{2}$ portions.

Challenge — count $\frac{2}{5}$ portions in $\frac{3}{4}$

Find $\frac{3}{4}$ divided by $\frac{2}{5}$.

Keep the dividend, reciprocate the divisor

$$\frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \times \frac{5}{2}$$

$$\frac{3}{4} \times \frac{5}{2} = \frac{15}{8} = 1\frac{7}{8}$$

The answer is 1 $\frac{7}{8}$.

Watch out — easily mixed up

Follow the fraction path

Review Fractions for P5 multiplication families and Fraction and Division for whole-number division as a fraction and fraction-to-decimal division. This P6 note does not move the reciprocal algorithm into the P5 page.

Quick recap

🎯 Mastery check

Answer all 6 — your progress is saved on this device.

1. What is $\frac{5}{6}$ ÷ 2?

   $\frac{5}{8}$$\frac{5}{12}$$\frac{5}{3}$

2. What is 4 ÷ $\frac{2}{3}$?

   3$\frac{8}{3}$6

3. What is $\frac{3}{4}$ ÷ $\frac{2}{5}$?

   1 $\frac{7}{8}$$\frac{3}{4}$$\frac{3}{5}$

4. A $\frac{3}{4}$-litre bottle of juice is poured into $\frac{1}{8}$-litre servings. How many servings?

   $\frac{3}{32}$6$\frac{8}{3}$

5. What is $\frac{5}{8}$ ÷ $\frac{3}{4}$?

   $\frac{15}{32}$$\frac{2}{3}$$\frac{5}{6}$

6. A $\frac{2}{3}$-metre ribbon is cut into equal pieces of $\frac{1}{6}$ metre. How many full pieces are made?

   4$\frac{1}{9}$3

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