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Algebraic FractionsCambridge IGCSE Maths: Revision notes

Section 1

How do you add and subtract algebraic fractions?

Algebraic fractions behave exactly like numerical fractions: to add or subtract, first find a common denominator, then combine the numerators.

For example: x3+x−42\frac{x}{3} + \frac{x-4}{2}

  1. Common denominator is 6
  2. Rewrite: 2x6+3(x−4)6\frac{2x}{6} + \frac{3(x-4)}{6}
  3. Combine: 2x+3x−126=5x−126\frac{2x + 3x - 12}{6} = \frac{5x - 12}{6}
Key termscommon denominator
Common mistake

When subtracting, put the whole numerator in brackets before expanding, e.g. x3−x−42\frac{x}{3} - \frac{x-4}{2} becomes 2x−3(x−4)6\frac{2x - 3(x-4)}{6} — forgetting the brackets loses the sign on −4-4.

Section 2

How do you multiply and divide algebraic fractions?

To multiply, multiply numerators together and denominators together, cancelling common factors first where possible.

3a4×9a10=27a240\frac{3a}{4} \times \frac{9a}{10} = \frac{27a^2}{40}

To divide, multiply by the reciprocal (flip the second fraction):

3a4÷9a10=3a4×109a=30a36a=56\frac{3a}{4} \div \frac{9a}{10} = \frac{3a}{4} \times \frac{10}{9a} = \frac{30a}{36a} = \frac{5}{6}

Key termsreciprocal
Exam tip

Cancel common factors before multiplying out fully — this keeps numbers smaller and reduces arithmetic errors.

Section 3

How do you factorise and simplify rational expressions?

A rational expression is simplified by factorising both numerator and denominator fully, then cancelling any common factor.

x2−2xx2−5x+6=x(x−2)(x−2)(x−3)=xx−3\frac{x^2 - 2x}{x^2 - 5x + 6} = \frac{x(x-2)}{(x-2)(x-3)} = \frac{x}{x-3}

  1. Factorise the numerator
  2. Factorise the denominator
  3. Cancel any bracket that appears in both

Only whole factors can be cancelled — never cancel individual terms within a bracket.

Key termsrational expressionfactorise
Common mistake

You cannot cancel the xx in x+2x\frac{x+2}{x} to get 22 — cancellation only works on whole common factors, not terms added within a bracket.

Section 4

How do you solve fractional equations?

To solve an equation containing algebraic fractions, multiply every term by the common denominator to clear the fractions, then solve the resulting equation.

For example, solve x3+x−42=5\frac{x}{3} + \frac{x-4}{2} = 5:

  1. Multiply throughout by 6: 2x+3(x−4)=302x + 3(x-4) = 30
  2. Expand: 2x+3x−12=302x + 3x - 12 = 30
  3. Simplify: 5x=425x = 42
  4. Solve: x=425=8.4x = \frac{42}{5} = 8.4

With an algebraic denominator (e.g. 1x+1x+2=1\frac{1}{x} + \frac{1}{x+2} = 1), multiply by both denominators and check the solution doesn't make a denominator zero.

Key termsfractional equation
Exam tip

State clearly: 'multiply both sides by [denominator]' — this method step earns the M1 mark even if a later arithmetic slip occurs.

Must Know

  • To add/subtract algebraic fractions, find a common denominator first, using brackets around subtracted numerators
  • To multiply, multiply numerators and denominators; to divide, multiply by the reciprocal
  • Factorise numerator and denominator fully before cancelling common bracket factors
  • Never cancel individual terms — only whole common factors
  • Solve fractional equations by multiplying through by the common denominator to clear fractions

That's the notes covered.

Carry on to the next subtopic.