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iGCSE (2021 Edition)

13.04 Simplifying algebraic fractions

Lesson

An algebraic fraction is a fraction where the numerator and denominator are algebraic expressions. To simplify a numeric fraction we cancel any common factors in the numerator and denominator. We use the same method to simplify algebraic fractions.

Worked example

Simplify $\frac{4x^2-16x}{6x-24}$4x216x6x24.

Think: To simplify the fraction we want to cancel any common factors in the numerator and denominator. To find these common factors, we first want to factorise the expressions in the numerator and denominator.

Do: In $4x^2-16x$4x216x there is a common factor of $2x$2x. In $6x-24$6x24 there is a common factor of $3$3.

$\frac{4x^2-16x}{6x-24}$4x216x6x24 $=$= $\frac{2x\left(2x-8\right)}{3\left(2x-8\right)}$2x(2x8)3(2x8)

Factorising the numerator and denominator

$=$= $\frac{2x}{3}$2x3

Cancelling $\left(2x-8\right)$(2x8) from the numerator and denominator

Since there are no more common factors between $2x$2x and $3$3, we conclude that $\frac{4x^2-16x}{6x-24}=\frac{2x}{3}$4x216x6x24=2x3.

Reflect: It's worth fully factorising the numerator and denominator first in case there are any common factors. In some cases we might have to use other factorisation techniques to do this.

Summary

An algebraic fraction is a fraction where the numerator and denominator are algebraic expressions.

To simplify an algebraic fraction, first factorise the expressions in the numerator and denominator and then cancel any common factors in the numerator and denominator.

Practice questions

Question 1

Simplify: $\frac{64mn^2}{40m^2n}$64mn240m2n

Question 2

Factorise and simplify: $\frac{5u}{5uv-30uw}$5u5uv30uw

Question 3

Factorise and simplify: $\frac{m^2-16}{m-4}$m216m4

Outcomes

0607C2.9

Algebraic fractions: simplification, addition or subtraction of fractions with integer denominators, multiplication or division of two simple fractions.

0607E2.9

Algebraic fractions: simplification, including use of factorisation, addition or subtraction of fractions with linear denominators or single term, multiplication or division and simplification of two fractions.

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