GCSE Maths Algebraic Fractions Notes
Algebraic fractions follow the same rules as number fractions — but factorising first is what unlocks simplifying and cancelling.
Key concepts
- Simplifying
- Factorise the numerator and denominator, then cancel any common bracket.
- Common denominator
- To add or subtract, rewrite each fraction over the same denominator, usually the product of the two.
- Solving fraction equations
- Multiply every term by the denominator(s) to clear the fractions, then solve as usual.
Key formulas
Add fractions
a/b + c/d = (ad + bc)/(bd)
Multiply fractions
a/b × c/d = ac/bd
Divide by multiplying by the reciprocal.
Worked example
Simplify (x² − 9) / (x² + 4x + 3).
- 1Factorise the numerator: x² − 9 = (x + 3)(x − 3).
- 2Factorise the denominator: x² + 4x + 3 = (x + 3)(x + 1).
- 3Cancel the common factor (x + 3).
Answer: (x − 3)/(x + 1)
Common mistakes
✗ Cancelling terms instead of factors, e.g. crossing out x in (x + 2)/x.
✓ Only cancel a factor common to the whole top and bottom, not individual terms.
✗ Forgetting a common denominator when adding.
✓ Rewrite both fractions over the same denominator before combining numerators.
✗ Losing a sign when subtracting fractions.
✓ Put the second numerator in brackets and subtract the whole thing.
Quick check
Try these, then reveal the answers.
Simplify 6x / 3x².Reveal answer
2/x
Write x/2 + x/3 as one fraction.Reveal answer
5x/6
Factorise x² − 25.Reveal answer
(x + 5)(x − 5)
Exam tips
- Factorise first — almost every higher algebraic-fraction question needs it.
- State any values of x that make a denominator zero if the question asks for restrictions.
- When solving, multiplying through by the denominators removes fractions in one clean step.
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Frequently asked questions
Why can't I cancel single terms in a fraction?+
Cancelling is only valid for factors that multiply the entire numerator and denominator, not terms that are added or subtracted.
Are algebraic fractions on foundation GCSE?+
Basic simplifying can appear, but adding, subtracting and solving algebraic fractions are higher-tier topics.
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