Multiplying & Dividing Algebraic Fractions (Edexcel GCSE Maths)

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Written by: Naomi C

Reviewed by: Dan Finlay

Multiplying & Dividing Algebraic Fractions

How do I multiply algebraic fractions?

STEP 1
Simplify both fractions first by fully factorising
- E.g. $\frac{x}{3 x + 6} \times \frac{2 x + 4}{x + 7} = \frac{x}{3 (x + 2)} \times \frac{2 (x + 2)}{x + 7}$
STEP 2
Cancel any common factors on top and bottom (from either fraction)
- E.g. $\frac{x}{3 (x + 2)} \times \frac{2 (x + 2)}{x + 7} = \frac{x}{3} \times \frac{2}{x + 7}$
STEP 3
Multiply the tops together
Multiply the bottoms together
- E.g. $\frac{2 x}{3 (x + 7)}$
STEP 4
Check for any further factorising and cancelling
- E.g. $\frac{2 x}{3 (x + 7)}$ 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 $\div \frac{a}{b}$ becomes $\times \frac{b}{a}$
- E.g. $\frac{3 x - 12}{x} \div \frac{2 x + 8}{x + 3} = \frac{3 x - 12}{x} \times \frac{x + 3}{2 x + 8}$

Then follow the same rules for multiplying two fractions

Worked Example

Divide $\frac{x + 3}{x - 4}$ by $\frac{2 x + 6}{x^{2} - 16}$ , giving your answer as a simplified fraction.

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

$\frac{x + 3}{x - 4} \div \frac{2 x + 6}{x^{2} - 16} = \frac{x + 3}{x - 4} \times \frac{x^{2} - 16}{2 x + 6}$

Factorise all numerators and denominators to see which factors cancel out
You need to use the difference of two squares, $x^{2} - 4^{2} = (x - 4) (x + 4)$

$\frac{x + 3}{x - 4} \times \frac{x^{2} - 16}{2 x + 6} = \frac{x + 3}{x - 4} \times \frac{(x - 4) (x + 4)}{2 (x + 3)}$

Multiply the remaining numerators and denominators together

$\frac{1 \times (x + 4)}{1 \times 2} = \frac{x + 4}{2}$

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

$\frac{x + 4}{2}$ is already in its simplest form

$\frac{x + 4}{2}$

Last updated: 1 July 2024

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