Equivalent fractions and mixed numbers | Year 7

01 / LEARN

Understand the idea.

Equal amounts, different parts

A fraction counts equal parts of one whole. Its denominator names the number of parts in that whole; its numerator counts the chosen parts. The two equal bars show why two thirds and four sixths cover the same amount.

Equivalent fractions rename an amount without changing it. Multiply or divide numerator and denominator by the same non-zero factor. To simplify fully, remove their highest common factor; the final numerator and denominator then share no factor greater than 1.

Counting complete wholes

An improper fraction contains at least one whole when its positive numerator is at least its denominator. Divide the numerator by the denominator: the quotient counts complete wholes and the remainder counts leftover parts of the same size.

To reverse the conversion, count all the fractional parts in the whole-number portion before adding the leftover numerator. For $2 \frac{3}{5}$ , two wholes contain ten fifths, and the extra three fifths produce thirteen fifths altogether.

\begin{aligned} 2 \frac{3}{5} & = \frac{2 \times 5 + 3}{5} \\ = \frac{13}{5} \end{aligned}

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Simplify a fraction

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

Worked solution

1
Find the highest common factor.
$\begin{aligned} 18 & = 6 \times 3 \\ 24 & = 6 \times 4 \end{aligned}$
Six divides both numbers, and the remaining factors 3 and 4 are coprime.
2
Divide numerator and denominator by 6.
$\frac{18}{24} = \frac{18 \div 6}{24 \div 6} = \frac{3}{4}$
Dividing both by the same non-zero factor preserves the fraction’s value.
3
Check that simplification is complete.
$HCF (3, 4) = 1$
No common factor greater than 1 remains, so three quarters is in simplest form.

Answer

$\frac{3}{4}$

EXAMPLE 02

Find complete wholes

Write $\frac{17}{5}$ as a mixed number.

Worked solution

1
Divide the numerator into groups of five.
$17 = 3 \times 5 + 2$
Three complete groups of five fit, with two parts left over.
2
Write the whole and fractional parts.
$\frac{17}{5} = 3 \frac{2}{5}$
Each complete group is one whole; the remaining two parts are still fifths.
3
Convert back to check.
$3 \frac{2}{5} = \frac{3 \times 5 + 2}{5} = \frac{17}{5}$
Counting every fifth recovers the original numerator.

Answer

$3 \frac{2}{5}$

03 / YOUR TURN

Try it yourself.

Check your understanding

Which equals $\frac{3}{5}$ : $\frac{6}{10}$ or $\frac{6}{8}$ ?

Show answer & reasoning

$\frac{6}{10}$

Multiplying the numerator and denominator of $\frac{3}{5}$ by 2 gives $\frac{6}{10}$ . The denominator must also double, so $\frac{6}{8}$ is not equivalent.

Keep practising

6 worksheet questions · 13 marks

Open worksheet Worked solutions

Read the questions online

Simplify $\frac{15}{20}$ .
2 marks
Complete $\frac{4}{7} = \frac{□}{28}$ .
2 marks
Write $2 \frac{3}{5}$ as an improper fraction.
2 marks
Write $\frac{23}{6}$ as a mixed number.
2 marks
Explain why $\frac{6}{8}$ is not in simplest form.
2 marks
One board is $1 \frac{1}{2}$ m long. Another is $\frac{6}{4}$ m. Are their lengths equal?
3 marks

Read the worked solutions

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

Simplify $\frac{15}{20}$ .
2 marks
Divide numerator and denominator by 5.
$\frac{3}{4}$
Complete $\frac{4}{7} = \frac{□}{28}$ .
2 marks
28 is $7 \times 4$ , so the numerator must also be multiplied by 4.
16
Write $2 \frac{3}{5}$ as an improper fraction.
2 marks
$2 \times 5 + 3 = 13$ ; keep denominator 5.
$\frac{13}{5}$
Write $\frac{23}{6}$ as a mixed number.
2 marks
$23 \div 6$ is 3 remainder 5.
$3 \frac{5}{6}$
Explain why $\frac{6}{8}$ is not in simplest form.
2 marks
6 and 8 share the factor 2.
$\frac{6}{8} = \frac{3}{4}$
One board is $1 \frac{1}{2}$ m long. Another is $\frac{6}{4}$ m. Are their lengths equal?
3 marks
$1 \frac{1}{2} = \frac{3}{2} = \frac{6}{4}$ .
Both amounts use metres.
Yes.

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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.

Equal amounts, different parts

Counting complete wholes

See the method in action.

Simplify a fraction

Find the highest common factor.

Divide numerator and denominator by 6.

Check that simplification is complete.

Find complete wholes

Divide the numerator into groups of five.

Write the whole and fractional parts.

Convert back to check.

Try it yourself.

Check your understanding

Keep practising