Simplifying algebraic fractions

GCSE Tutoring Programme

"Our chosen students improved 1.19 of a grade on average - 0.45 more than those who didn't have the tutoring."

In order to access this I need to be confident with:

Fractions Algebraic fractions Simplifying fractions Rearranging equations Factorising quadratics Difference of two squares Expanding brackets

This topic is relevant for:

GCSE Maths Algebra Algebraic Expressions Simplifying Expressions Algebraic Fractions

Simplifying Algebraic Fractions

Here we will learn about simplifying algebraic fractions, including different powers of x, quadratics, and the difference of two squares.

There are also simplifying algebraic fractions worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What is simplifying algebraic fractions?

Simplifying algebraic fractions is simplifying a fraction that contains algebra so that the numerator and the denominator do not contain any common factors.

To do this, we need to be able to find common factors between the numerator and the denominator which can be cancelled down.

In order to simplify algebraic fractions, you must be confident with calculating with fractions.

If we want to express $\frac{a}{b}+\frac{c}{d}$ or $\frac{a}{b} - \frac{c}{d}$ as a single fraction, we need to find a common denominator and then add or subtract the resultant fractions.

The general form when adding fractions is:

$\frac{a}{b}+\frac{c}{d}=\frac{a\times{d}}{b\times{d}}+\frac{b\times{c}}{b\times{d}}=\frac{ad+bc}{bd}$

The general form for subtracting fractions is:

$\frac{a}{b}-\frac{c}{d}=\frac{a\times{d}}{b\times{d}}-\frac{b\times{c}}{b\times{d}}=\frac{ad-bc}{bd}$

E.g.
Calculate $\frac{2}{5}+\frac{1}{4}$

So if $a = 2, b = 5, c = 1$ and $d = 4$ , we can therefore say that

\frac{2}{5}+\frac{1}{4}=\frac{8}{20}+\frac{5}{20}=\frac{8+5}{20}=\frac{13}{20}

Below is a visual representation of this problem,

By converting the fractions so that they have a common denominator (in this case $20$ ), we divide the fractions into smaller parts, then total those parts, all by using knowledge of equivalent fractions.

We can use this to simplify algebraic expressions:

E.g.
Let’s look at $\frac{5x}{15}$

Here, the terms $5x$ and $15$ share a common factor of $5$ .

If we wrote this out as two products, we would get $\frac{5\times{x}}{5\times{3}}$ .

We can simplify this by cancelling the common factor of $5$ from the numerator and the denominator of the fraction $\frac{5\times{x}}{5\times{3}}=\frac{x}{3}$ .

So by finding the highest common factor of the numerator and the denominator, we can show that the fraction $\frac{5x}{15}$ can be expressed as as the more simple algebraic fraction $\frac{x}{3}$ .

We can use knowledge of simplifying algebraic fractions to solve equations that include algebraic fractions. For more information on this, see the lesson on algebraic fractions.

Step-by-step guide: Algebraic fractions

What is simplifying algebraic fractions?

$What is simplifying algebraic fractions?$

Simplify algebraic products and quotients

Sometimes algebraic fractions can be simplified by using the index laws. Here is an example where a product and a quotient can be simplified.

E.g.

Simplify

\frac{4x^5 \times 3x^2}{2x^3}

This can be simplified by first considering the numerator can be simplified using the product law of indices.

\frac{4x^5 \times 3x^2}{2x^3}=\frac{12x^7}{2x^3}

Then the fraction can be further simplified by using the quotient law of indices.

\frac{12x^7}{2x^3}=6x^4

How to simplify algebraic fractions

In order to simplify algebraic fractions:

Complete any calculations required in the question $(+ − \times\div)$ .
Factorise the numerator and the denominator.
Cancel the common factor and simplify further if required.

Explain how to simplify algebraic fractions

$Explain how to simplify algebraic fractions$

$Simplifying algebraic fractions worksheet$

Simplifying algebraic fractions worksheet

$Simplifying algebraic fractions worksheet$

Get your free simplifying algebraic fractions worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREE

$Simplifying algebraic fractions worksheet$

Simplifying algebraic fractions worksheet

$Simplifying algebraic fractions worksheet$

Get your free simplifying algebraic fractions worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREE

Simplifying algebraic fractions examples

Example 1: Multiplying algebraic fractions

Simplify the algebraic expression

\frac{5}{x+2}\times\frac{x+2}{10}

Complete any calculations required in the question $(+−\times\div)$ .

\frac{5}{x+2}\times\frac{x+2}{10}=\frac{5x+10}{10x+20}

2Factorise the numerator and the denominator.

\frac{5x+10}{10x+20}=\frac{5(x+2)}{10(x+2)}

3Cancel the common factor and simplify further if required.

\frac{5(x+2)}{10(x+2)}=\frac{5}{10}=\frac{1}{2}

Example 2: Adding algebraic fractions

Simplify the algebraic expression

\frac{10}{6+2x}+\frac{3}{x+3}

Complete any calculations required in the question $(+−\times\div)$ .

\begin{aligned} &\frac{10}{6+2x}+\frac{3}{x+3}=\frac{10(x+3)}{(6+2x)(x+3)}+\frac{3(6+2x)}{(x+3)(6+2x)}\\\\ &=\frac{10(x+3)+3(6+2x)}{(6+2x)(x+3)} \\\\ &=\frac{10x+30+18+6x}{(6+2x)(x+3)} \\\\ &=\frac{16x+48}{(6+2x)(x+3)} \end{aligned}

Top tip: don’t expand the double brackets on the denominator straight away. One of the brackets may be a common factor with the numerator and will eventually cancel out.

Factorise the numerator and the denominator.

\frac{16x+48}{(6+2x)(x+3)}=\frac{16(x+3)}{2(3+x)(x+3)}

Cancel the common factor and simplify further if required.

\frac{16(x+3)}{2(3+x)(x+3)}=\frac{16}{2(3+x)}=\frac{8}{3+x}

Example 3: Linear ÷ linear

Simplify the algebraic fraction

\frac{6x+12}{10x+20}

Complete any calculations required in the question $(+−\times\div)$ .

Here there is no calculation to complete as we have a single fraction.

Factorise the numerator and the denominator.

\frac{6x+12}{10x+20}=\frac{6(x+2)}{10(x+2)}

Cancel the common factor and simplify further if required.

\frac{6(x+2)}{10(x+2)}=\frac{6}{10}=\frac{3}{5}

Example 4: Linear ÷ quadratic

Simplify the algebraic fraction

\frac{2x+6}{x^{2}+11x+24}

Complete any calculations required in the question $(+−\times\div)$ .

Here there is no calculation to complete as we have a single fraction.

Factorise the numerator and the denominator.

\frac{2x+6}{x^{2}+11x+24}=\frac{2(x+3)}{(x+3)(x+8)}

Cancel the common factor and simplify further if required.

\frac{2(x+3)}{(x+3)(x+8)}=\frac{2}{x+8}

Example 5: Quadratic ÷ quadratic

Simplify the algebraic fraction

\frac{x^{2}-3x-10}{x^{2}+2x-35}

Complete any calculations required in the question $(+−\times\div)$ .

Here there is no calculation to complete as we have a single fraction.

Factorise the numerator and the denominator.

\frac{x^{2}-3x-10}{x^{2}+2x-35}=\frac{(x+2)(x-5)}{(x-5)(x+7)}

Cancel the common factor and simplify further if required.

\frac{(x+2)(x-5)}{(x-5)(x+7)}=\frac{x+2}{x+7}

Example 6: Quadratic ÷ quadratic (difference of two squares)

Simplify the algebraic fraction

\frac{x^{2}+15x+56}{x^{2}-64}

Complete any calculations required in the question $(+−\times\div)$ .

Here there is no calculation to complete as we have a single fraction.

Factorise the numerator and the denominator.

\frac{x^{2}+15x+56}{x^{2}-64}=\frac{(x+7)(x+8)}{(x+8)(x-8)}

Cancel the common factor and simplify further if required.

\frac{(x+7)(x+8)}{(x+8)(x-8)}=\frac{x+7}{x-8}

Common misconceptions

Ignoring the denominators

When given an equation including an algebraic fraction, the denominators are ignored so the question is answered incorrectly.

Simplifying Algebraic Fractions Common Misconceptions 1

Adding denominators

When adding two fractions, the denominator must be the same. A common misconception for adding two fractions is to add the numerators and the denominators together because this method is emphasised when looking at multiplying fractions.

Simplifying Algebraic Fractions Common Misconceptions 2

Cancelling terms incorrectly

Seeing the same term on the numerator and denominator can result in the misconception that they can both be cancelled.

Simplifying Algebraic Fractions Common Misconceptions 3

Expanding double brackets incorrectly

When expanding double brackets, each term in the bracket needs to be multiplied by each term in the other bracket. By not treating the numerator as a single expression, mistakes are made when multiplying brackets together.

Simplifying Algebraic Fractions Common Misconceptions 4

Collecting un-like terms

Simple algebraic expressions such as $5x + 4$ can be incorrectly simplified to get $9x$ as the $x$ is ignored; the assumption is that $5x + 4$ is the same as $(5 + 4)x$ which is not correct.

Factorising into a single bracket incorrectly

When factorising into a single bracket, the incorrect method occurs when the bracket is simply placed between the number and the $x$ of the first term, then around the last term in the expression.

E.g.

Factorising $3x + 12$ incorrectly, we would get $3(x + 12)$ .

Here the bracket is placed between the $3$ and the $x$ , and the $12$ has not been factorised at all. When factorising, the common factor is placed on the outside of the bracket, and the factor pair that generates the term is placed on the inside, so for the example given $3x + 12 = 3(x + 4)$ .

Practice simplifying algebraic fractions questions

\frac{2}{\frac{x+1}{x+5}}

\frac{8x+5}{2x+3}

\frac{2(x+5)}{x+1}

\frac{2}{3}

\begin{aligned} &\frac{2x+6}{x+1}\div{\frac{x+3}{x+5}}\\\\ &=\frac{2x+6}{x+1}\times{\frac{x+5}{x+3}} \\\\ &=\frac{2(x+3)}{x+1}\times{\frac{x+5}{x+3}}\\\\ &=\frac{2(x+3)(x+5)}{(x+1)(x+3)}\\\\ &=\frac{2(x+5)}{x+1} \end{aligned}

\frac{3-2x}{4x^{2}}

\frac{2}{2x}

\frac{2}{4x^{2}-2x}

\frac{3-2x}{4x^{2}}

\begin{aligned} &\frac{3}{4x^{2}}-\frac{1}{2x}=\frac{3}{4x^{2}}-\frac{2x}{4x^{2}}\\\\ &=\frac{3-2x}{4x^{2}} \end{aligned}

1

\frac{22x}{33x}

\frac{2}{3}

\frac{2(x+12)}{3(x+18)}

\frac{10x+12}{15x+18}=\frac{2(5x+6)}{3(5x+6)}=\frac{2}{3}

\frac{9}{(x-7)}

\frac{3(3x-9)}{(x+7)(x-3)}

\frac{9}{x+7}

\frac{-18}{x-17}

\begin{aligned} &\frac{9x-27}{x^{2}+4x-21}=\frac{9(x-3)}{(x+7)(x-3)}\\\\ &=\frac{9}{x+7} \end{aligned}

\frac{x-4}{x-5}

\frac{x-22}{x-31}

\frac{x+4}{x+5}

\frac{x+8}{x+10}

\begin{aligned} &\frac{x^{2}+2x-24}{x^{2}+x-30}=\frac{(x+6)(x-4)}{(x+6)(x-5)}\\\\ &=\frac{x-4}{x-5} \end{aligned}

\frac{x+1}{x+7}

\frac{x-1}{7-x}

\frac{x-1}{48}

\frac{x-1}{x-7}

\begin{aligned} &\frac{x^{2}+6x-7}{49-x^{2}}=\frac{(x+7)(x-1)}{(7+x)(7-x)}\\\\ &=\frac{(x+7)(x-1)}{(x+7)(7-x)}\\\\ &=\frac{x-1}{7-x} \end{aligned}

Simplifying algebraic fractions GCSE questions

1. Simplify the algebraic fraction

\frac{x^{2}-9x+20}{5x-20}

Write your solution in the form $\frac{ax+b}{c}$ where $a, b,$ and $c$ are integers.

(3 marks)

Show answer

Numerator is factorised: $(x-5)(x-4)$

(1)

Denominator is factorised: $5(x-4)$

(1)

Fraction is simplified: $\frac{x-5}{5}$

(1)

2. (a) Justin is simplifying the algebraic fraction $\frac{x^{2}-4}{x+2}$

His work is written below.

Simplifying Algebraic Fractions Exam Question 2a

Is Justin correct?

Explain your answer.

(b) Show that the fraction $\frac{9x^{2}-16}{3x-4}$ can be simplified to $3x+4$ .

(5 marks)

Show answer

(a)

Justin has the correct answer but he has not shown the correct understanding

(1)

He should have factorised the numerator to get $(x+2)(x-2)$

(1)

He can then cancel the common factor of $(x+2)$ , leaving the solution $x-2$

(1)

(b)

9x^2-16=(3x+4)(3x-4)

(1)

\frac{(3x+4)(3x-4)}{(3x-4)}=3x+4

(1)

3. Simplify fully

\frac{2x+1}{2x+10}\div{\frac{3+6x}{x+5}}

(4 marks)

Show answer

\frac{2x+1}{2x+10}\div{\frac{3+6x}{x+5}}=\frac{2x+1}{2x+10}\times{\frac{x+5}{3+6x}}

(1)

=\frac{2x+1}{2(x+5)}\times{\frac{x+5}{3(1+2x)}}

(1)

=\frac{(2x+1)(x+5)}{6(x+5)(2x+1)}

(1)

=\frac{1}{6}

(1)

Learning checklist

You have now learned how to:

Simplify and manipulate algebraic expressions (including those involving surds {and algebraic fractions}) by:

Factorising quadratic expressions of the form x² + bx + c, including the difference of 2 squares; {factorising quadratic expressions of the form ax² + bx + c}
Simplifying expressions involving sums, products and powers, including the laws of indices

The next lessons are

Still stuck?

Prepare your KS4 students for maths GCSEs success with Third Space Learning. Weekly online one to one GCSE maths revision lessons delivered by expert maths tutors.

Find out more about our GCSE maths tuition programme.

Introduction

What is simplifying algebraic fractions?

Simplify algebraic products and quotients

How to simplify algebraic fractions

Simplifying algebraic fractions worksheet

Simplifying algebraic fractions examples

↓

Example 1: Multiplying algebraic fractions Example 2: Adding algebraic fractions Example 3: Linear ÷ linear Example 4: Linear ÷ quadratic Example 5: Quadratic ÷ quadratic Example 6: Quadratic ÷ quadratic (difference of two squares)

Common misconceptions

Practice simplifying algebraic fractions questions

Simplifying algebraic fractions GCSE questions

Learning checklist

The next lessons are

Still stuck?