Free GCSE Maths lesson: Algebra
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Free Lessons → GCSE / Key Stage 4 → Maths → Algebraic Fractions

Lesson 39 · GCSE / Key Stage 4 · Maths · Algebra

Algebraic Fractions

Learn how to simplify, combine and solve with fractions that contain algebra, using the same fraction rules you already know.

Qualification: GCSE Key Stage 4 Subject: Maths Strand: Algebra

Lesson overview

An extension algebra topic where fraction rules and factorising meet.

Algebraic fractions are fractions with letters in the numerator, denominator, or both. GCSE questions may ask you to simplify them, multiply or divide them, add or subtract them, state excluded values, or solve equations that contain them.

QualificationGCSE Mathematics
Key stageKey Stage 4
StrandAlgebra
Tier Higher
Calculator status Calculator and non-calculator
Exam-board status Higher-tier algebra skill

What you will learn

  • How algebraic fractions behave like ordinary fractions.
  • How to spot denominator restrictions.
  • How to simplify by factorising and cancelling common factors.
  • How to multiply and divide algebraic fractions.
  • How to add and subtract using a common denominator.
  • How to solve simple equations that include algebraic fractions and reject impossible denominator values.
  • How to simplify quadratic algebraic fractions while keeping every original denominator restriction.

Key facts before you start

Algebraic fraction A fraction with algebra in the numerator, denominator or both.
Restrictions Values that make a denominator zero are not allowed.
Common denominator Add or subtract by rewriting fractions over a common denominator.
Factor first Factorising often reveals what can cancel safely.

Algebraic fractions infographic

Before viewing

Before viewing, find the denominator and say which value would make it zero.

Infographic explaining GCSE Maths algebraic fractions, including factorising first, denominator restrictions, cancelling common factors, common denominators and excluded value checks.
Use this visual to factorise first, cancel only common factors, build common denominators and check excluded values.Download visual

After viewing

After viewing, simplify one algebraic fraction and state any excluded value.

Why this matters

Algebraic fractions appear in advanced GCSE algebra because they combine several important skills: factorising, equivalent fractions, expanding brackets and solving equations.

The key idea is calm and familiar: do not cancel terms just because they look similar. Cancel common factors only after the numerator and denominator have been factorised.

Prior knowledge

You should already be comfortable with:

  • simplifying number fractions,
  • multiplying and dividing fractions,
  • using common denominators,
  • factorising expressions,
  • solving linear equations.

Clear explanation

Denominators cannot be zero

A fraction is undefined if its denominator is zero. For 5x − 3, the denominator cannot be 0, so x cannot be 3. If a factor cancels later, the original excluded value still matters.

Simplifying algebraic fractions

Factorise first, then cancel common factors. A factor is something multiplied by the rest of the expression.

6x²9x = 6 × x × x9 × x Cancel the common factor 3x. 6x²9x = 2x3, where x ≠ 0

Factorising before cancelling

If the numerator or denominator has more than one term, factorise it before cancelling.

x² − 9x + 3 = (x − 3)(x + 3)x + 3 Cancel the common factor x + 3. x² − 9x + 3 = x − 3, where x ≠ −3

Adding algebraic fractions

To add or subtract, make a common denominator first, then combine the numerators.

2x + 35 = 105x + 3x5x 2x + 35 = 10 + 3x5x, where x ≠ 0

Dividing algebraic fractions

To divide by a fraction, multiply by its reciprocal, then factorise and cancel common factors.

3x5 ÷ 6x25 = 3x5 × 256x Cancel the common factors x, 3 and 5. 3x5 ÷ 6x25 = 52, where x ≠ 0

Solving equations with algebraic fractions

Multiply every term by the common denominator to clear the fractions, then solve the equation.

x3 + 2 = 7 Multiply every term by 3: x + 6 = 21 x = 15

Always check the original denominators after solving. If your answer would make any original denominator equal zero, that answer must be rejected.

Worked examples

Example 1: Simplify by cancelling a common factor

Simplify 8a12.

8a12 = 2a3
Reveal answer

Answer: 2a3.

Example 2: Factorise before cancelling

Simplify x² + 5xx.

x² + 5xx = x(x + 5)x Cancel the common factor x.
Reveal answer

Answer: x + 5, where x ≠ 0.

Example 3: Multiply algebraic fractions

Simplify 3x4 × 89.

3x × 84 × 9 = 24x36
Reveal answer

Answer: 2x3.

Example 4: Add algebraic fractions

Simplify 1x + 23.

1x = 33x and 23 = 2x3x
Reveal answer

Answer: 3 + 2x3x, where x ≠ 0.

Example 5: Subtract with bracketed numerators

Simplify x + 43 − 2x.

x(x + 4)3x − 63x x² + 4x − 63x
Reveal answer

Answer: x² + 4x − 63x, where x ≠ 0.

Example 6: Solve and check restrictions

Solve 4x − 2 = 1.

x cannot be 2 because the original denominator would be zero. 4 = x − 2 x = 6
Reveal answer

Answer: x = 6. This is allowed because 6 − 2 ≠ 0.

Quick checks

Choose an answer, then check your thinking.

1. What must be true for 4x?

2. Which step should usually come before cancelling in x² − 4x − 2?

3. To add 1x and 12, what do you need first?

Practice questions

Question 1

Simplify 15x20.

Reveal answer and marking guidance

Answer: 3x4.

Marking: Divide numerator and denominator by the common factor 5.

Question 2

Simplify 12a²18a.

Reveal answer and marking guidance

Answer: 2a3, where a ≠ 0.

Marking: Cancel the common factor 6a from top and bottom.

Question 3

Simplify x² − 16x + 4.

Reveal answer and marking guidance

Answer: x − 4, where x ≠ −4.

Marking: Factorise x² − 16 as (x − 4)(x + 4), then cancel the common factor x + 4.

Question 4

Simplify 5x6 × 1225.

Reveal answer and marking guidance

Answer: 2x5.

Marking: Multiply to get 60x over 150, then simplify by dividing by 30.

Question 5

Simplify 3x + 25.

Reveal answer and marking guidance

Answer: 15 + 2x5x, where x ≠ 0.

Marking: Use common denominator 5x: 3/x becomes 15/(5x), and 2/5 becomes 2x/(5x).

Question 6

Solve 6x + 1 = 2.

Reveal answer and marking guidance

Answer: x = 2, with x ≠ −1.

Marking: Multiply both sides by x + 1 to get 6 = 2(x + 1), then solve 6 = 2x + 2.

Question 7

Simplify x + 24 + 3x.

Reveal answer and marking guidance

Answer: x² + 2x + 124x, where x ≠ 0.

Marking: Use common denominator 4x. The first numerator becomes x(x + 2), so keep brackets until it expands to x² + 2x.

Question 8

Solve x + 35 = 2x − 13.

Reveal answer and marking guidance

Answer: x = 147 = 2.

Marking: Multiply by 15 to get 3(x + 3) = 5(2x − 1). Expand to 3x + 9 = 10x − 5, then solve 14 = 7x.

Question 9

Simplify x² + 3xx² − 9.

Reveal answer and marking guidance

Answer: xx − 3, where x ≠ 3 and x ≠ −3.

Marking: Factorise to x(x + 3)(x − 3)(x + 3), then cancel the common factor x + 3. Keep both original restrictions because x² − 9 cannot be zero.

Question 10

Simplify x² − x − 12x² − 16, stating any values that x cannot take.

Reveal answer and marking guidance

Answer: x + 3x + 4, where x ≠ 4 and x ≠ −4.

Marking: Factorise to (x − 4)(x + 3)(x − 4)(x + 4), then cancel the common factor x − 4. Keep both original restrictions because x² − 16 cannot be zero.

Answers and marking guidance

The exact practice answers are hidden under each question so you can try first. For algebraic fractions, marks usually come from factorising before cancelling, keeping original denominator restrictions in mind, building a valid common denominator, and using brackets around multi-term numerators. If you solve an equation with fractions, show the line where every term has been multiplied by the common denominator and check the answer against the original denominators.

Common mistakes

  • Cancelling terms instead of factors: in x + 3x, the x in the denominator is not a factor of the whole numerator.
  • Forgetting restrictions: if a denominator contains x, say when x cannot make the original denominator zero.
  • Combining fractions without a common denominator: add or subtract only after the denominators match.
  • Losing brackets: when multiplying a whole numerator, keep expressions like 2(x + 5) in brackets until expanded.
  • Only multiplying one side of an equation: clearing fractions means multiplying every term on both sides.

Extension challenge

Simplify x² + 7x + 12x² − 9, stating any values that x cannot take.

Reveal answer

Answer: x + 4x − 3, where x ≠ 3 and x ≠ −3.

Factorise the numerator to (x + 3)(x + 4) and the denominator to (x − 3)(x + 3). Cancel the common factor x + 3, but keep both original denominator restrictions.

Exam-board guidance

Algebraic fractions are assessed across GCSE Maths boards as more demanding algebra. The same method marks matter everywhere: factorise, use common denominators, bracket whole numerators, keep denominator restrictions visible, reject impossible values and write each algebra step clearly.

AQA GCSE Maths

Focus on the shared method, notation and checking habit; wording and context can vary by route.

OCR GCSE Maths

Show clear method steps and use precise notation; question wording may vary by route.

Pearson Edexcel GCSE Maths

Keep working visible and check units or notation; contexts may vary by route.

Eduqas GCSE Maths

Use the same core skill in practical contexts, with clear reasoning and final checks.

WJEC Wales

Connect the method to the context and state what the result means.

CCEA GCSE Maths

Make each method step visible and keep notation, units and final checks clear.

Next lesson

Next, move into Geometry and Measures with Angles, Lines and Parallel Lines.

Previous lesson: Functions Next lesson: Angles and parallel lines