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Multiplying & Dividing Algebraic Fractions (Cambridge (CIE) IGCSE International Maths)

Revision Note

Naomi C

Written by: Naomi C

Reviewed by: Dan Finlay

Cambridge (CIE) International Maths: ExtendedEdexcel Maths A (Modular): Higher Unit 1Edexcel Maths A: HigherCambridge (CIE) Maths: Extended

Multiplying & Dividing Algebraic Fractions

How do I multiply algebraic fractions?

  • STEP 1

    Simplify both fractions first by fully factorising

    • E.g. fraction numerator x over denominator 3 x plus 6 end fraction cross times fraction numerator 2 x plus 4 over denominator x plus 7 end fraction equals fraction numerator x over denominator 3 open parentheses x plus 2 close parentheses end fraction cross times fraction numerator 2 open parentheses x plus 2 close parentheses over denominator x plus 7 end fraction

  • STEP 2

    Cancel any common factors on top and bottom (from either fraction)

    • E.g. fraction numerator x over denominator 3 up diagonal strike open parentheses x plus 2 close parentheses end strike end fraction cross times fraction numerator 2 up diagonal strike open parentheses x plus 2 close parentheses end strike over denominator x plus 7 end fraction equals x over 3 cross times fraction numerator 2 over denominator x plus 7 end fraction

  • STEP 3
    Multiply the tops together
    Multiply the bottoms together

    • E.g. fraction numerator 2 x over denominator 3 open parentheses x plus 7 close parentheses end fraction

  • STEP 4

    Check for any further factorising and cancelling

    • E.g. fraction numerator 2 x over denominator 3 open parentheses x plus 7 close parentheses end fraction has no common factors so is in its simplest form

How do I divide algebraic fractions?

  • Flip (find the reciprocal of) the second fraction and replace ÷ with ×

    • So divided by a over b becomes cross times b over a

    • E.g. fraction numerator 3 x minus 12 over denominator x end fraction divided by fraction numerator 2 x plus 8 over denominator x plus 3 end fraction equals fraction numerator 3 x minus 12 over denominator x end fraction cross times fraction numerator x plus 3 over denominator 2 x plus 8 end fraction

  • Then follow the same rules for multiplying two fractions

Worked Example

Divide fraction numerator x plus 3 over denominator x minus 4 end fraction by fraction numerator 2 x plus 6 over denominator x squared minus 16 end fraction, giving your answer as a simplified fraction.

Division is the same as multiplying by the reciprocal (the fraction flipped)

fraction numerator x plus 3 over denominator x minus 4 end fraction divided by fraction numerator 2 x plus 6 over denominator x squared minus 16 end fraction equals fraction numerator x plus 3 over denominator x minus 4 end fraction cross times fraction numerator x squared minus 16 over denominator 2 x plus 6 end fraction

Factorise all numerators and denominators to see which factors cancel out
You need to use the difference of two squares, x squared minus 4 squared equals open parentheses x minus 4 close parentheses open parentheses x plus 4 close parentheses

fraction numerator x plus 3 over denominator x minus 4 end fraction cross times fraction numerator x squared minus 16 over denominator 2 x plus 6 end fraction equals fraction numerator up diagonal strike x plus 3 end strike over denominator up diagonal strike x minus 4 end strike end fraction cross times fraction numerator up diagonal strike open parentheses x minus 4 close parentheses end strike open parentheses x plus 4 close parentheses over denominator 2 up diagonal strike open parentheses x plus 3 close parentheses end strike end fraction

Multiply the remaining numerators and denominators together

fraction numerator 1 cross times open parentheses x plus 4 close parentheses over denominator 1 cross times 2 end fraction equals fraction numerator x plus 4 over denominator 2 end fraction

Check to see if you missed any terms that are the same on the top and bottom that could be cancelled

fraction numerator x plus 4 over denominator 2 end fraction is already in its simplest form

fraction numerator bold italic x bold plus bold 4 over denominator bold 2 end fraction

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    Author
    Reviewer
    Naomi C

    Author: Naomi C

    Expertise: Maths

    Naomi graduated from Durham University in 2007 with a Masters degree in Civil Engineering. She has taught Mathematics in the UK, Malaysia and Switzerland covering GCSE, IGCSE, A-Level and IB. She particularly enjoys applying Mathematics to real life and endeavours to bring creativity to the content she creates.

    Dan Finlay

    Author: Dan Finlay

    Expertise: Maths Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.