Algebraic Fractions (Edexcel IGCSE Maths A)

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  • What is an algebraic fraction?

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

    E.g. fraction numerator 3 x over denominator 2 end fraction, fraction numerator 7 over denominator x minus 4 end fraction and fraction numerator space 2 x over denominator x plus 5 end fraction are all examples of algebraic fractions.

  • What is a common factor?

    A common factor is a factor that is shared between two or more expressions.

    E.g. x is a common factor for the expressions 3 x plus 8 and x y.

  • What is cancelling in algebraic fractions?

    Cancelling in algebraic fractions means removing common factors that appear in both the numerator and denominator.

    E.g. The algebraic fraction fraction numerator 2 open parentheses x plus 3 close parentheses over denominator 6 end fraction can have the common factor of 2 removed from both the numerator and denominator, leaving fraction numerator x plus 3 over denominator 3 end fraction.

  • Why is it useful to leave an algebraic fraction in factorised form?

    Factorised form means leaving the top and bottom of the fraction each expressed as the product of its factors.

    This is useful because it makes it easier to see if anything cancels at the end.

  • What are the common steps for simplifying an algebraic fraction such as fraction numerator 6 x minus 24 over denominator 5 x minus 20 end fraction?

    To simplify an algebraic fraction such as fraction numerator 6 x minus 24 over denominator 5 x minus 20 end fraction, you should:

    1. Factorise fully top and bottom, fraction numerator 6 open parentheses x minus 4 close parentheses over denominator 5 open parentheses x minus 4 close parentheses end fraction.

    2. Then cancel common factors (including common brackets), 6 over 5.

  • True or False?

    If asked to simplify an algebraic fraction in an exam question, one factor will likely be the same on the top and bottom.

    True.

    If asked to simplify an algebraic fraction in an exam question, one factor will likely be the same on the top and bottom.

    Factorise the easier expression, then use this fact to help you to factorise the more difficult quadratic.

  • How can you find the lowest common denominator (LCD) for a pair of algebraic fractions?

    With algebraic fractions, the lowest common denominator (LCD) is found by multiplying the denominators together if they do not share any factors.

    E.g. The LCD of fraction numerator 3 over denominator x plus 2 end fraction and fraction numerator 5 x over denominator x minus 3 end fraction is open parentheses x plus 2 close parentheses open parentheses x minus 3 close parentheses.

    If they do share factors, find the lowest common denominator by taking the denominator that already includes the other(s).

    E.g. The LCD of 4 over x and fraction numerator 9 over denominator x open parentheses x plus 1 close parentheses end fraction is x open parentheses x plus 1 close parentheses.

  • True or False?

    If x and 2 x are the denominators of two algebraic fractions, then the lowest common denominator is found by multiplying xand 2 x together.

    False.

    2 x already includes the factor x.

    If x and 2 x are the denominators of two algebraic fractions, then the lowest common denominator is 2 x.

  • What is the process for adding two algebraic fractions?

    E.g. fraction numerator 3 x over denominator x plus 5 end fraction plus fraction numerator 4 x plus 2 over denominator x end fraction.

    The process for adding two algebraic fractions is:

    1. Find the lowest common denominator, x open parentheses x plus 5 close parentheses

    2. Write each fraction as an equivalent fraction over the lowest common denominator, fraction numerator 3 x over denominator open parentheses x plus 5 close parentheses end fraction cross times x over x plus fraction numerator 4 x plus 2 over denominator x end fraction cross times fraction numerator open parentheses x plus 5 close parentheses over denominator open parentheses x plus 5 close parentheses end fraction equals fraction numerator 3 x squared over denominator x open parentheses x plus 5 close parentheses end fraction plus fraction numerator open parentheses x plus 5 close parentheses open parentheses 4 x plus 2 close parentheses over denominator x open parentheses x plus 5 close parentheses end fraction

    3. Add the numerators, fraction numerator 3 x squared plus open parentheses x plus 5 close parentheses open parentheses 4 x plus 2 close parentheses over denominator x open parentheses x plus 5 close parentheses end fraction.

    4. Simplify, fraction numerator 7 x squared plus 22 x plus 10 over denominator x open parentheses x plus 5 close parentheses end fraction.

  • What is the process for subtracting two algebraic fractions?

    E.g. fraction numerator x plus 6 over denominator x end fraction minus fraction numerator 3 over denominator x minus 1 end fraction.

    The process for subtracting two algebraic fractions is:

    1. Find the lowest common denominator, x open parentheses x minus 1 close parentheses

    2. Write each fraction as an equivalent fraction over the lowest common denominator, fraction numerator x plus 6 over denominator x end fraction cross times fraction numerator open parentheses x minus 1 close parentheses over denominator open parentheses x minus 1 close parentheses end fraction minus fraction numerator 3 over denominator x minus 1 end fraction cross times x over x equals fraction numerator open parentheses x plus 6 close parentheses open parentheses x minus 1 close parentheses over denominator x open parentheses x minus 1 close parentheses end fraction minus fraction numerator 3 x over denominator x open parentheses x minus 1 close parentheses end fraction

    3. Subtract the numerators, fraction numerator open parentheses x plus 6 close parentheses open parentheses x minus 1 close parentheses minus 3 x over denominator x open parentheses x minus 1 close parentheses end fraction.

    4. Simplify, fraction numerator x squared plus 2 x minus 6 over denominator x open parentheses x minus 1 close parentheses end fraction.

  • What is the process for multiplying algebraic fractions?

    E.g.fraction numerator 6 x over denominator x plus 1 end fraction cross times fraction numerator 2 x minus 4 over denominator 3 x squared minus 6 end fraction.

    To multiply algebraic fractions:

    1. Simplify both fractions first by factorising and cancelling common factorsfraction numerator 6 x over denominator x plus 1 end fraction cross times fraction numerator 2 open parentheses x minus 2 close parentheses over denominator 3 x open parentheses x minus 2 close parentheses end fraction equals fraction numerator 6 x over denominator x plus 1 end fraction cross times fraction numerator 2 over denominator 3 x end fraction.

    2. Multiply the numerators together, 6 x cross times 2 equals 12 x.

    3. Multiply the denominators together, open parentheses x plus 1 close parentheses cross times 3 x equals 3 x squared plus 3 x.

    4. Check for further factorising and cancelling, fraction numerator 12 x over denominator 3 x squared plus 3 x end fraction equals fraction numerator 12 x over denominator 3 x open parentheses x plus 1 close parentheses end fraction equals fraction numerator 4 over denominator x plus 1 end fraction.

  • What is the reciprocal of an algebraic fraction.

    The reciprocal of an algebraic fraction is the fraction 'flipped', i.e. with the original denominator divided by the numerator.

    For example the reciprocal of fraction numerator x minus 4 over denominator x squared plus 3 end fraction is fraction numerator x squared plus 3 over denominator x minus 4 end fraction.

  • What is the process for dividing algebraic fractions?

    E.g. fraction numerator 2 x over denominator 7 x minus 1 end fraction divided by fraction numerator x over denominator x plus 3 end fraction.

    To divide algebraic fractions:

    1. Find the reciprocal of the second fraction and replace divided by with cross times, fraction numerator 2 x over denominator 7 x minus 1 end fraction cross times fraction numerator x plus 3 over denominator x end fraction.

    2. Then follow the rules for multiplying two algebraic fractions.

    Note that this is the same thing you do when dividing normal fractions.

  • What are the two main methods for solving equations with algebraic fractions?

    E.g. fraction numerator 2 over denominator x plus 1 end fraction plus fraction numerator x over denominator x minus 3 end fraction equals 2.

    The two main methods for solving equations with algebraic fractions are:

    1. Adding/subtract the fractions first,fraction numerator 2 open parentheses x minus 3 close parentheses plus x open parentheses x plus 1 close parentheses over denominator open parentheses x plus 1 close parentheses open parentheses x minus 3 close parentheses end fraction equals 2.

      Then solve the resulting equation.

    2. Multiply everything by the common denominator to eliminate fractions first, 2 open parentheses x minus 3 close parentheses plus x open parentheses x plus 1 close parentheses equals 2 open parentheses x plus 1 close parentheses open parentheses x minus 3 close parentheses.
      Then solve the resulting equation.

  • What are the steps to solve an equation with algebraic fractions by multiplying through by the common denominator?

    E.g. fraction numerator 1 over denominator 2 x end fraction plus fraction numerator 7 x plus 5 over denominator 2 x end fraction equals 4.

    The steps to solve an equation with algebraic fractions by multiplying through by the common denominator are:

    1. Multiply every term by the common denominator, 1 plus 7 x plus 5 equals 8 x.

    2. Expand any brackets and collect like terms, 7 x plus 6 equals 8 x.

    3. Rearrange and solve the equation, 6 equals x.