GCSE MATHS • HIGHER TIER
GCSE Simplifying Algebraic Fractions Made Simple
Learn how to simplify algebraic fractions by factorising expressions, applying index laws, identifying common factors and cancelling factors correctly.
Clear method
Exam practice
Worked proofs
Proof in Four Moves
1
Factorise the Numerator
Factorise the numerator if possible.
2
Factorise the Denominator
Factorise the denominator if possible.
3
Identify Common Factors
Look for factors that appear in both the numerator and denominator.
4
Cancel Common Factors
Cancel only factors that are common to both parts of the fraction.
5
Write the Simplified Answer
Rewrite the remaining expression clearly.
6
Check Again
Make sure no further simplification is possible.
What You’ll Learn
Understand Algebraic Fractions
Recognise fractions that contain algebraic expressions in the numerator, denominator or both.
Simplify Numerical and Algebraic Factors
Reduce number factors and variable factors correctly.
Use Factorisation
Factorise expressions before cancelling common factors.
Apply Index Laws
Use laws of indices to simplify variable powers in fractions.
Cancel Factors Correctly
Understand why only factors can be cancelled, not terms that are being added or subtracted.
Avoid Common GCSE Mistakes
Learn the errors students often make and how to prevent them.
Key Representations
Factorise First
Cancel Factors, Not Terms
Simplify Number Factors
Use Index Laws
Check the Final Form
More Worked Examples
1
Common Numerical Factor
Simplify 6x/9.
Both numbers share a factor of 3. 6x/9 = 2x/3
Explanation: The fraction is simplified in the same way as a numerical fraction.
2
Numbers and Variables
Simplify 8x²/12x.
Step 2: Simplify the variables. x²/x = x
3
Using Index Rules
Simplify x²/x.
4
Coefficients and Powers
Simplify 12x²/18x.
5
Difference of Two Squares
Simplify (x² − 9)/(x − 3).
x² − 9 = (x − 3)(x + 3)
Step 2: Cancel the common factor (x − 3).
6
Common Factor in the Numerator
Simplify (x² + 5x)/x.
x(x + 5)
Cancel the common factor x.
7
Factorising 2x
Simplify (2x² + 6x)/(2x).
2x(x + 3)
Cancel the common factor 2x.
8
Difference of Two Squares
Simplify (x² − 4)/(x − 2).
(x − 2)(x + 2)
Cancel the common factor (x − 2).
- Common GCSE Mistakes
- Cancelling terms instead of factors.
- Incorrectly trying to cancel the x in (x + 2)/x.
- Forgetting to factorise an expression before cancelling
- Not simplifying the numerical coefficients fully.
- Misapplying index laws when simplifying powers.
- Stopping before the fraction is in its simplest form.
- Exam Tips
- Look for a common factor before doing anything else.
- If you see a quadratic expression, check whether it can be factorised.
- Remember the difference of two squares: a² − b² = (a − b)(a + b).
- Cancel complete factors only.
- Use index laws carefully for powers of the same variable.
- Write each factorisation step clearly to avoid accidental cancellation errors.
- Check your final answer to see whether any further simplification is possible.
Skills Used in Algebraic Proof
Ready to Practice?
Printable Worksheet
Use a printable worksheet to practise simplifying algebraic fractions independently.
Interactive Quiz
Work through algebraic-fraction questions involving common factors, powers and factorisation.
Exam-Style Questions
Apply the same techniques to questions written in GCSE exam style.
Watch the Simplifying Algebraic Fractions Tutorial
Step-by-step walkthroughs of algebraic proofs with clear methods and exam tips.
Frequently Asked Questions
An algebraic fraction is a fraction containing algebraic expressions in the numerator, denominator or both.
Only common factors can be cancelled. Terms that are being added or subtracted cannot be cancelled individually.
Factorising rewrites an expression as a product of factors. This can reveal common factors that can then be cancelled.
No. The x in the numerator is part of a sum, so it is a term rather than a common factor of the entire numerator.
Index laws help simplify powers of the same variable. For example, x³/x² = x.
The supplied revision guide identifies Simplifying Algebraic Fractions as an important Higher Tier GCSE Algebra topic.
