Multiplying/dividing fractions | Year 7

01 / LEARN

Understand the idea.

Taking part of an amount

Multiplication by a positive fraction can mean taking part of an amount. Multiply numerators to count the selected parts and denominators to name their size. A factor between zero and one produces a smaller positive amount.

Simplify before or after multiplying. Cancelling a common factor divides the combined numerator and denominator equally, so the value stays unchanged. Cancel factors in a product, never unrelated terms joined by an addition or subtraction.

Counting fractional groups

Division asks how many groups of the divisor's size fit in the starting amount. Dividing two litres into half-litre groups gives four groups, so a division can have a numerical answer larger than its starting number.

To divide by a non-zero fraction, multiply by its reciprocal. This works because a fraction multiplied by its reciprocal equals one. Convert mixed numbers first, then interpret whether the question needs complete groups or permits a fractional group.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Take part of a mass

Find $\frac{3}{5}$ of $\frac{10}{9}$ kg.

Worked solution

1
Translate “of” into multiplication.
$\frac{3}{5} \times \frac{10}{9} kg$
Three fifths of a mass scales that mass by the factor three fifths.
2
Multiply the fractions.
$\frac{3}{5} \times \frac{10}{9} = \frac{30}{45}$
The numerators multiply together, as do the denominators.
3
Simplify and retain the unit.
$\frac{30}{45} = \frac{2}{3}, mass = \frac{2}{3} kg$
Dividing both parts by 15 preserves the value; the result is smaller than the original positive mass.

Answer

$\frac{2}{3}$ kg

EXAMPLE 02

Fill cups

A jug contains $\frac{3}{4}$ L. Each cup holds $\frac{1}{8}$ L. How many full cups can be filled?

Worked solution

1
Divide the total volume by one cup’s volume.
$\frac{3}{4} \div \frac{1}{8}$
The quotient counts how many eighth-litre groups fit into three quarters of a litre.
2
Multiply by the reciprocal.
$\frac{3}{4} \div \frac{1}{8} = \frac{3}{4} \times 8 = 6$
Multiplying by 8 counts the eighths contained in the total volume.
3
Check the six full cups.
$6 \times \frac{1}{8} = \frac{6}{8} = \frac{3}{4} L$
Six full cups use exactly the jug’s volume, with no remainder.

Answer

6 cups

03 / YOUR TURN

Try it yourself.

Check your understanding

Calculate $\frac{2}{3} \div \frac{4}{5}$ .

Show answer & reasoning

$\frac{5}{6}$

Use the reciprocal of the divisor: $\frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6}$ . Only the divisor is inverted.

Keep practising

6 worksheet questions · 15 marks

Open worksheet Worked solutions

Read the questions online

$\frac{2}{5} \times \frac{3}{4}$
2 marks
$\frac{7}{9} \times \frac{3}{14}$
2 marks
$\frac{5}{6} \div \frac{1}{3}$
2 marks
$1 \frac{1}{2} \times \frac{4}{5}$
2 marks
$2 \frac{1}{4} \div \frac{3}{8}$
3 marks
A $2 \frac{1}{2}$ m ribbon is cut into $\frac{3}{4}$ m pieces. How many complete pieces fit, and what length remains?
4 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

$\frac{2}{5} \times \frac{3}{4}$
2 marks
$\frac{6}{20} = \frac{3}{10}$ .
$\frac{3}{10}$
$\frac{7}{9} \times \frac{3}{14}$
2 marks
Cancel 7 with 14 and 3 with 9 to obtain $\frac{1}{3} \times \frac{1}{2}$ .
$\frac{1}{6}$
$\frac{5}{6} \div \frac{1}{3}$
2 marks
$\frac{5}{6} \times 3 = \frac{15}{6} = \frac{5}{2}$ .
$2 \frac{1}{2}$
$1 \frac{1}{2} \times \frac{4}{5}$
2 marks
$\frac{3}{2} \times \frac{4}{5} = \frac{12}{10} = \frac{6}{5}$ .
$1 \frac{1}{5}$
$2 \frac{1}{4} \div \frac{3}{8}$
3 marks
$\frac{9}{4} \times \frac{8}{3} = \frac{72}{12} = 6$ .
6
A $2 \frac{1}{2}$ m ribbon is cut into $\frac{3}{4}$ m pieces. How many complete pieces fit, and what length remains?
4 marks
$\frac{5}{2} \div \frac{3}{4} = \frac{10}{3} = 3 \frac{1}{3}$ , so only three complete pieces fit.
$\frac{5}{2} - 3 \times \frac{3}{4} = \frac{1}{4}$ m.
3 complete pieces; one quarter of a metre remains.

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Curriculum reference & lesson scope

Australian Curriculum v9 / SA Curriculum Mathematics

Equivalent fractions and mixed numbers; ordering rational numbers on number lines; adding/subtracting fractions; multiplying/dividing fractions; four operations with decimals; converting between fractions and decimals; multi-step problems and checking estimates.

Official curriculum source ↗

In this lesson

Understand the idea.

Taking part of an amount

Counting fractional groups

See the method in action.

Take part of a mass

Translate “of” into multiplication.

Multiply the fractions.

Simplify and retain the unit.

Fill cups

Divide the total volume by one cup’s volume.

Multiply by the reciprocal.

Check the six full cups.

Try it yourself.

Check your understanding

Keep practising