Equivalent fractions | Year 6

01 / LEARN

Understand the idea.

Same amount, more pieces

The denominator counts equal parts in one whole; the numerator counts the selected parts. Equivalent fractions describe the same amount of the same whole. Their different numbers show that the whole has been divided into different-sized pieces.

If every fifth is split into four pieces, the whole has twenty pieces and the selected three fifths have twelve pieces. Multiply both numerator and denominator by the same positive whole number to keep the selected amount unchanged.

Simplify by joining pieces

Simplifying reverses the process: combine equal-sized pieces into larger groups by dividing numerator and denominator by the same common factor. A fraction is in simplest form when those two numbers share no factor greater than one.

The whole must stay fixed in a fraction diagram. Half a large cake and half a small cake are both one half, but the amounts of cake differ. Equivalent fractions compare shares of the same whole, not differently sized objects.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Worked example 1

Write $\frac{3}{5}$ with denominator 20.

Worked solution

1
Find the denominator’s scale factor.
$20 \div 5 = 4$
Each fifth must be split into four equal pieces to make twentieths.
2
Scale the numerator by the same factor.
$3 \times 4 = 12$
All three selected fifths are split, giving twelve selected pieces.
3
Write the equivalent fraction.
$\frac{3}{5} = \frac{3 \times 4}{5 \times 4} = \frac{12}{20}$
Both counts have been multiplied by four, so the amount is unchanged.

Answer

$\frac{12}{20}$ .

EXAMPLE 02

Worked example 2

Simplify $\frac{18}{24}$ .

Worked solution

1
Choose a shared factor.
$18 = 3 \times 6, 24 = 4 \times 6$
Six divides both the numerator and denominator exactly.
2
Divide both counts by six.
$\frac{18}{24} = \frac{18 \div 6}{24 \div 6} = \frac{3}{4}$
Grouping six small pieces into one larger piece preserves the fraction.
3
Check that the result is simplest.
$Factors of 3 : 1, 3; factors of 4 : 1, 2, 4$
There is no shared factor greater than one, so three quarters cannot be simplified further.

Answer

$\frac{3}{4}$ .

03 / YOUR TURN

Try it yourself.

Check your understanding

Is $\frac{4}{6}$ equivalent to $\frac{2}{3}$ ?

Show answer & reasoning

Yes.

$\frac{4}{6} = \frac{4 \div 2}{6 \div 2} = \frac{2}{3}$ . Dividing both counts by two changes the size of the pieces while keeping the amount the same.

Keep practising

6 worksheet questions · 8 marks

Open worksheet Worked solutions

Read the questions online

Complete $\frac{2}{3} = \frac{?}{12}$ .
1 mark
Write $\frac{5}{8}$ with denominator 24.
1 mark
Simplify $\frac{12}{20}$ .
1 mark
Simplify $\frac{21}{28}$ .
1 mark
Write the shaded fraction, then simplify it.
2 marks
Noah says $\frac{2}{5} = \frac{4}{7}$ because he added 2 to both numbers. Explain the error and give an equivalent fraction.
2 marks

Read the worked solutions

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

Complete $\frac{2}{3} = \frac{?}{12}$ .
1 mark
Multiply numerator and denominator by 4.
8.
Write $\frac{5}{8}$ with denominator 24.
1 mark
Multiply both counts by 3.
$\frac{15}{24}$ .
Simplify $\frac{12}{20}$ .
1 mark
Divide both counts by 4.
$\frac{3}{5}$ .
Simplify $\frac{21}{28}$ .
1 mark
Divide both counts by 7.
$\frac{3}{4}$ .
Write the shaded fraction, then simplify it.
2 marks
Six of eight parts are shaded; divide both counts by 2.
$\frac{6}{8} = \frac{3}{4}$ .
Noah says $\frac{2}{5} = \frac{4}{7}$ because he added 2 to both numbers. Explain the error and give an equivalent fraction.
2 marks
Multiply both numerator and denominator by 2 instead.
The amounts differ; one correct equivalent is $\frac{4}{10}$ .

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Curriculum reference & lesson scope

Australian Curriculum v9 and SA Curriculum Mathematics

Equivalence; comparing/ordering; addition/subtraction; practical quantities.

Official curriculum source ↗

In this lesson

Understand the idea.

Same amount, more pieces

Simplify by joining pieces

See the method in action.

Worked example 1

Find the denominator’s scale factor.

Scale the numerator by the same factor.

Write the equivalent fraction.

Worked example 2

Choose a shared factor.

Divide both counts by six.

Check that the result is simplest.

Try it yourself.

Check your understanding

Keep practising