National 5MathsSQARevision NotesAlgebraic SkillsAlgebraic FractionsMultiplication & Division with Algebraic Fractions

Multiplication & Division with Algebraic Fractions (SQA National 5 Maths): Revision Note

Exam code: X847 75

Roger B

Written by: Roger B

Reviewed by: Dan Finlay

Updated on

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

Express fraction numerator 5 over denominator x plus 3 end fraction divided by 6 over open parentheses x plus 3 close parentheses squared comma space space x not equal to negative 3 comma as a single fraction in its simplest form.

Answer:

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

fraction numerator 5 over denominator x plus 3 end fraction divided by 6 over open parentheses x plus 3 close parentheses squared equals fraction numerator 5 over denominator x plus 3 end fraction cross times open parentheses x plus 3 close parentheses squared over 6

Check the numerators and denominators to see if any factors cancel out

  • Remember that open parentheses x plus 3 close parentheses squared equals open parentheses x plus 3 close parentheses open parentheses x plus 3 close parentheses

equals space fraction numerator 5 over denominator 1 up diagonal strike x plus 3 end strike end fraction cross times fraction numerator open parentheses x plus 3 close parentheses up diagonal strike open parentheses x plus 3 close parentheses end strike over denominator 6 end fraction equals 5 over 1 cross times fraction numerator x plus 3 over denominator 6 end fraction

Multiply the remaining numerators and denominators together

equals fraction numerator 5 open parentheses x plus 3 close parentheses over denominator 6 end fraction

There are no other factors that can be cancelled, so that answer is in simplest form

fraction numerator 5 open parentheses x plus 3 close parentheses over denominator 6 end fraction space space open parentheses or space space fraction numerator 5 x plus 15 over denominator 6 end fraction space space or space space 5 over 6 open parentheses x plus 3 close parentheses close parentheses

Examiner Tips and Tricks

For the purpose of multiplying and dividing algebraic fractions in questions like the Worked Example, you can disregard conditions like "x not equal to negative 3".

That is just there to make sure that the denominators of the fractions can never be equal to zero.

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    Author
    Reviewer
    Roger B

    Author: Roger B

    Expertise: Maths Content Creator

    Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.

    Dan Finlay

    Reviewer: Dan Finlay

    Expertise: Maths Subject 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.