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Simplifying Algebraic Fractions (WJEC GCSE Maths & Numeracy (Double Award)): Revision Note

Exam code: 3320

Jamie Wood

Written by: Jamie Wood

Reviewed by: Mark Curtis

Updated on

WJEC Maths & Numeracy (Double Award): FoundationWJEC Maths & Numeracy (Double Award): HigherWJEC Eduqas Maths: HigherOCR Maths: HigherAQA Maths: HigherEdexcel Maths: Higher

Simplifying algebraic fractions

What is an algebraic fraction?

  • An algebraic fraction is a fraction with an algebraic expression on the top (numerator) and/or the bottom (denominator)

How do you simplify an algebraic fraction?

  • If possible, factorise fully the top and bottom

    • E.g. fraction numerator 2 x minus 4 over denominator 4 x minus 10 end fraction equals fraction numerator 2 open parentheses x minus 2 close parentheses over denominator 2 open parentheses 2 x minus 5 close parentheses end fraction

  • Cancel common factors

    • This factor may be a single number or letter

      • E.g. fraction numerator up diagonal strike 2 open parentheses x minus 2 close parentheses over denominator up diagonal strike 2 open parentheses 2 x minus 5 close parentheses end fraction equals fraction numerator open parentheses x minus 2 close parentheses over denominator open parentheses 2 x minus 5 close parentheses end fraction

      • The final answer is fraction numerator x minus 2 over denominator 2 x minus 5 end fraction

  • A common mistake is to cancel a factor that is not common to all terms in either the top or the bottom of a fraction

    • E.g. The fraction fraction numerator 6 x over denominator x plus 1 end fraction cannot be simplified

      • x is not common to all terms in the bottom of the fraction

      • i.e. you cannot factorise an x out of the bottom

  • You may need to use laws of indices to help simplify algebraic fractions

    • E.g. fraction numerator 6 z to the power of 7 over denominator 2 z cubed end fraction can be simplified using the fact that z to the power of 7 over z cubed equals z to the power of 7 minus 3 end exponent equals z to the power of 4 and 6 over 2 equals 3

      • fraction numerator 6 z to the power of 7 over denominator 2 z cubed end fraction equals 3 z to the power of 4

Examiner Tips and Tricks

When asked to simplify an algebraic fraction, first factorise top and bottom.

It is very likely that one of the factors will be the same on the top and the bottom.

You can use this fact to help you if one of the expressions is difficult to factorise!

Worked Example

(a) Simplify fraction numerator 3 y plus 12 over denominator 6 end fraction

Answer:

Factorise the top

fraction numerator 3 open parentheses y plus 4 close parentheses over denominator 6 end fraction

This suggests factorising a 3 out of the bottom

fraction numerator 3 open parentheses y plus 4 close parentheses over denominator 3 open parentheses 2 close parentheses end fraction

Cancel out the common factor of 3

fraction numerator up diagonal strike 3 open parentheses y plus 4 close parentheses over denominator up diagonal strike 3 open parentheses 2 close parentheses end fraction

fraction numerator y plus 4 over denominator 2 end fraction

This could also be written as y over 2 plus 2

(b) Simplify fraction numerator 2 h cubed cross times 3 h to the power of 4 over denominator 8 h squared end fraction

Answer:

Simplify the numerator using the laws of indices

h cubed cross times h to the power of 4 equals h to the power of 7

fraction numerator 6 h to the power of 7 over denominator 8 h squared end fraction

Factorise the top and bottom

They both have a common factor of 2

fraction numerator 2 open parentheses 3 h to the power of 7 close parentheses over denominator 2 open parentheses 4 h squared close parentheses end fraction

Cancel out the common factor of 2

fraction numerator 3 h to the power of 7 over denominator 4 h squared end fraction

Use the laws of indices to simplify

h to the power of 7 over h squared equals h to the power of 7 minus 2 end exponent equals h to the power of 5

fraction numerator 3 h to the power of 5 over denominator 4 end fraction or 3 over 4 h to the power of 5

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    Author
    Reviewer
    Jamie Wood

    Author: Jamie Wood

    Expertise: Maths Content Creator

    Jamie graduated in 2014 from the University of Bristol with a degree in Electronic and Communications Engineering. He has worked as a teacher for 8 years, in secondary schools and in further education; teaching GCSE and A Level. He is passionate about helping students fulfil their potential through easy-to-use resources and high-quality questions and solutions.

    Mark Curtis

    Reviewer: Mark Curtis

    Expertise: Maths Content Creator

    Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.