GCSEMathsWJECMaths & Numeracy (Double Award)FoundationRevision NotesAlgebraAlgebraic FractionsAdding & Subtracting Algebraic Fractions

Adding & Subtracting 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

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Adding & subtracting algebraic fractions

How do I add (or subtract) two algebraic fractions?

  • The rules for adding and subtracting algebraic fractions are the same as they are for fractions with numbers

  • STEP 1
    Find the lowest common denominator (LCD)

    • Sometimes the LCD can be found by multiplying the denominators together

      • E.g. The LCD for the fractions fraction numerator 3 x over denominator 4 end fraction and fraction numerator 5 x over denominator 7 end fraction is 4 cross times 7 equals 28

    • Although multiplying the denominators will always give you a common denominator, it is not necessarily the lowest

      • E.g. The LCD for the fractions fraction numerator 2 x over denominator 9 end fraction and fraction numerator 5 x over denominator 18 end fraction is 9 (not 9×18)

  • STEP 2

    Write each fraction over the lowest common denominator

    Multiply the numerator of each fraction by the same amount as the denominator

    • E.g. table row cell fraction numerator 3 x over denominator 4 end fraction plus fraction numerator 5 x over denominator 7 end fraction end cell equals cell fraction numerator 3 x open parentheses 7 close parentheses over denominator 4 open parentheses 7 close parentheses end fraction plus fraction numerator 5 x open parentheses 4 close parentheses over denominator 7 open parentheses 4 close parentheses end fraction equals fraction numerator 21 x over denominator 28 end fraction plus fraction numerator 20 x over denominator 28 end fraction end cell end table

  • STEP 3

    Write as a single fraction over the lowest common denominator and simplify the numerator

    • Do this by adding or subtracting the numerators

      • Take particular care if subtracting

    • E.g. fraction numerator 21 x plus 20 x over denominator 28 end fraction equals fraction numerator 41 x over denominator 28 end fraction

  • STEP 4

    Check at the end to see if the top factorises and the fraction can be simplified

    • E.g. An answer of fraction numerator 6 x over denominator 10 end fraction could be simplified to fraction numerator 2 open parentheses 3 x close parentheses over denominator 2 open parentheses 5 close parentheses end fraction equals fraction numerator 3 x over denominator 5 end fraction

Examiner Tips and Tricks

In foundation tier, the denominators when adding or subtracting algebraic fractions are always numbers, not letters.

Worked Example

(a) Express fraction numerator 5 y over denominator 8 end fraction minus fraction numerator 3 y over denominator 10 end fraction as a single fraction.

Answer:

Write the fractions over the lowest common denominator

The lowest common denominator of 8 and 10 is 40

You could also write them with a common denominator of 80, but this makes the numbers slightly harder to work with

fraction numerator 5 y open parentheses 5 close parentheses over denominator 8 open parentheses 5 close parentheses end fraction minus fraction numerator 3 y open parentheses 4 close parentheses over denominator 10 open parentheses 4 close parentheses end fraction fraction numerator 25 y over denominator 40 end fraction minus fraction numerator 12 y over denominator 40 end fraction

Write as a single fraction

fraction numerator 25 y minus 12 y over denominator 40 end fraction

Simplify

fraction numerator 13 y over denominator 40 end fraction

Check if the answer can be simplified

In this case the answer does not simplify any further (as 13 is prime)

fraction numerator 13 y over denominator 40 end fraction

(b) Express fraction numerator 2 x plus 7 over denominator 4 end fraction minus fraction numerator x minus 3 over denominator 6 end fraction as a single fraction.

Answer:

Write the fractions over the lowest common denominator

The lowest common denominator of 4 and 6 is 12

You could also write them with a common denominator of 24, but this makes the numbers slightly harder to work with

fraction numerator open parentheses 2 x plus 7 close parentheses open parentheses 3 close parentheses over denominator 4 open parentheses 3 close parentheses end fraction minus fraction numerator open parentheses x minus 3 close parentheses open parentheses 2 close parentheses over denominator 6 open parentheses 2 close parentheses end fraction

Simplify, being careful to expand each term in the numerators

fraction numerator 6 x plus 21 over denominator 12 end fraction minus fraction numerator 2 x minus 6 over denominator 12 end fraction

Write as one fraction

To help avoid mistakes when dealing with the subtraction, use brackets around the numerator of each fraction

fraction numerator open parentheses 6 x plus 21 close parentheses minus open parentheses 2 x minus 6 close parentheses over denominator 12 end fraction

Simplify the numerator, being careful with the subtraction

fraction numerator 6 x plus 21 minus 2 x plus 6 over denominator 12 end fraction

fraction numerator 4 x plus 27 over denominator 12 end fraction

Check if the answer can be simplified

4 and 12 share a factor of 4, but 27 does not have a factor of 4

So this cannot be further simplified

fraction numerator 4 x plus 27 over denominator 12 end fraction

The question asks for a single fraction, so you cannot write fraction numerator 4 x over denominator 12 end fraction plus 27 over 12 or x over 3 plus 9 over 4, even though they are equivalent

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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.