How do you simplify a fraction?

Divide the numerator and denominator by their highest common factor (HCF). For 18/24 the HCF is 6, so 18/24 = 3/4.

How do I know when a fraction is fully simplified?

When the only number that divides into both the top and the bottom is 1. 3/4 is fully simplified; 6/8 isn't, because 2 divides into both.

How do I simplify an improper fraction?

Simplify it the same way, then change it to a mixed number if the question asks. 45/12 simplifies to 15/4, which is 3¾ as a mixed number.

Can I simplify by dividing by a smaller number first?

Yes. Dividing by any common factor works, as long as you keep going until nothing else divides into both. Using the HCF just gets there in one step.

What year level learns simplifying fractions?

Equivalent fractions start in primary school. In the Victorian Curriculum 2.0, Year 7 uses simplifying in fraction operations and ratios, and finds HCFs using prime factors. Simplifying algebraic expressions builds through Years 8 and 9 with factorising, and algebraic fractions are a Year 10 topic.

How to Simplify Fractions: HCF Method, Examples and Fraction Simplifier (Years 7 to 10)

What simplest form means

$\dfrac{6}{8}$ and $\dfrac{3}{4}$ are equivalent fractions: they're the same amount. $\dfrac{3}{4}$ is in simplest form (lowest terms) because no number except 1 divides into both 3 and 4.

Dividing the top and bottom by the same number never changes a fraction's value. It just writes it with smaller numbers.

The HCF method (one step)

Find the highest common factor (HCF) of the numerator and the denominator.
Divide both by it.

Worked example

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

Factors of 18: 1, 2, 3, 6, 9, 18. Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24. The HCF is 6.

\frac{18}{24} = \frac{18 \div 6}{24 \div 6} = \frac{3}{4}

Worked example: bigger numbers

Simplify $\dfrac{84}{126}$ .

Use prime factors: $84 = 2^2 \times 3 \times 7$ and $126 = 2 \times 3^2 \times 7$ . The HCF is the product of the primes they share: $2 \times 3 \times 7 = 42$ .

\frac{84}{126} = \frac{84 \div 42}{126 \div 42} = \frac{2}{3}

Dividing step by step

If you can't spot the HCF, divide by any number that goes into both, and repeat until you can't go any further.

Quick checks: both even, divide by 2. Digits of both add to a multiple of 3, divide by 3. Both end in 0 or 5, divide by 5.

Worked example

Simplify $\dfrac{48}{72}$ .

\frac{48}{72} = \frac{24}{36} \quad (\div 2)

\frac{24}{36} = \frac{12}{18} \quad (\div 2)

\frac{12}{18} = \frac{2}{3} \quad (\div 6)

You get the same answer as dividing by the HCF, 24, in one go.

Fraction simplifier (with working)

Enter the numerator and denominator. You’ll get the HCF, the simplified fraction, and the mixed number if it’s improper, with every step.

Improper fractions and mixed numbers

An improper fraction has a numerator bigger than its denominator. Simplify it first, then divide to turn it into a mixed number: the whole number is how many times the bottom goes in, and the remainder goes over the bottom.

Worked example

Simplify $\dfrac{45}{12}$ and write it as a mixed number.

\frac{45}{12} = \frac{45 \div 3}{12 \div 3} = \frac{15}{4}

$15 \div 4 = 3$ remainder $3$ , so $\dfrac{15}{4} = 3\dfrac{3}{4}$ .

To simplify a mixed number, leave the whole number alone and simplify the fraction part: $2\dfrac{6}{8} = 2\dfrac{3}{4}$ .

Simplifying ratios

Ratios simplify exactly like fractions: divide every part by the HCF.

$12 : 18$ . The HCF is 6, so $12 : 18 = 2 : 3$ .
$1.5 : 2.5$ . Multiply by 10 to clear the decimals: $15 : 25$ , then divide by 5: $3 : 5$ .
$50$ cm $: 2$ m. Make the units the same first: $50 : 200$ , then divide by 50: $1 : 4$ .
$8 : 12 : 20$ . The HCF of all three is 4: $2 : 3 : 5$ .

Simplifying algebraic fractions (Years 8 to 10)

The rule is the same: cancel factors that appear in both the top and the bottom. Numbers cancel by their HCF, and letters cancel using the index laws ( $\dfrac{a^2}{a} = a$ ).

Worked examples

\frac{6x}{9} = \frac{2x}{3}

\frac{4a^2b}{10ab} = \frac{2a}{5}

When there's a bracket or a sum, factorise first, then cancel the common factor:

\frac{3x + 6}{x + 2} = \frac{3(x + 2)}{x + 2} = 3

\frac{x^2 - 9}{x + 3} = \frac{(x + 3)(x - 3)}{x + 3} = x - 3

You can only cancel something that multiplies the whole top and the whole bottom. $\dfrac{x + 3}{3}$ does not simplify to $x$ , because the 3 on top is added, not multiplied. (These cancellations also assume the bottom isn't zero.)

Common mistakes

Stopping too early. $\dfrac{12}{18} = \dfrac{6}{9}$ is right, but not finished: it simplifies again to $\dfrac{2}{3}$ .
Dividing only the top or only the bottom. Whatever you do to one, do to the other.
Subtracting instead of dividing. Taking 2 off the top and bottom of $\dfrac{5}{7}$ gives $\dfrac{3}{5}$ , which is a different amount. Only multiplying or dividing keeps a fraction equivalent.
Cancelling terms instead of factors in algebra, like turning $\dfrac{x + 3}{3}$ into $x$ .

Practice questions with answers

Try each one before you open the answer.

Simplify $\dfrac{15}{25}$ .
Show the answer
The HCF is $5$ : $\dfrac{15 \div 5}{25 \div 5} = \dfrac{3}{5}$
Simplify $\dfrac{36}{48}$ .
Show the answer
The HCF is $12$ : $\dfrac{36 \div 12}{48 \div 12} = \dfrac{3}{4}$
Simplify $\dfrac{56}{84}$ .
Show the answer
The HCF is $28$ : $\dfrac{56 \div 28}{84 \div 28} = \dfrac{2}{3}$
Simplify $\dfrac{42}{12}$ and write it as a mixed number.
Show the answer
The HCF is $6$ : $\dfrac{42}{12} = \dfrac{7}{2} = 3\dfrac{1}{2}$
Simplify the ratio $24 : 40$ .
Show the answer
The HCF is $8$ : $24 : 40 = 3 : 5$
Simplify the ratio $0.6 : 1.5$ .
Show the answer
Multiply both by $10$ : $6 : 15$ . The HCF is $3$ : $2 : 5$
Simplify $\dfrac{12xy}{18x}$ .
Show the answer
The HCF of $12$ and $18$ is $6$ , and $x$ cancels: $\dfrac{12xy}{18x} = \dfrac{2y}{3}$
Simplify $\dfrac{5x - 10}{x - 2}$ .
Show the answer
Factorise the top: $\dfrac{5(x - 2)}{x - 2} = 5$ (for $x \neq 2$ )

Simplifying fractions by year level

Following the Victorian Curriculum 2.0. Schools sequence topics differently, so check your course.

Year 7

Powers and prime factors

Prime factors and factor trees, which make finding the HCF quick.

Worksheets and topic test · Free PDF with answers

Year 7

Fraction operations

Simplifying answers when adding, subtracting, multiplying and dividing fractions, and mixed numbers.

Worksheets and topic test · Free PDF with answers

Year 7

Ratios

Writing and simplifying ratios, and equivalent ratios.

Worksheets and topic test · Free PDF with answers

Year 8

Algebra

Factorising by taking out a common factor, the first step in simplifying algebraic fractions.

Worksheets and topic test · Free PDF with answers

Year 10

Algebraic fractions

Simplifying and using the 4 operations with algebraic fractions.

Worksheets and topic test · Free PDF with answers

How to simplify fractions

What simplest form means

The HCF method (one step)

Dividing step by step

Fraction simplifier (with working)

Improper fractions and mixed numbers

Simplifying ratios

Simplifying algebraic fractions (Years 8 to 10)

Common mistakes

Practice questions with answers

Simplifying fractions by year level

Simplifying fractions: common questions

More maths help, Years 7 to 10

Got a fractions question wrong in homework?