Free GCSE Maths lesson: Algebra
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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

GCSE specification fit

A higher-tier 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 guidanceMostly Higher

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.

Why this matters

Algebraic fractions appear in higher 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
Answer: 2a3.

Example 2: Factorise before cancelling

Simplify x² + 5xx.

x² + 5xx = x(x + 5)x Cancel the common factor x.
Answer: x + 5, where x ≠ 0.

Example 3: Multiply algebraic fractions

Simplify 3x4 × 89.

3x × 84 × 9 = 24x36
Answer: 2x3.

Example 4: Add algebraic fractions

Simplify 1x + 23.

1x = 33x and 23 = 2x3x
Answer: 3 + 2x3x, where x ≠ 0.

Example 5: Subtract with bracketed numerators

Simplify x + 43 − 2x.

x(x + 4)3x − 63x x² + 4x − 63x
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
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 higher-demand 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

Factorise first, cancel only common factors, and keep any excluded values from the original denominator even if a factor cancels later.

OCR GCSE Maths

When adding or subtracting algebraic fractions, build one common denominator before combining numerators and use brackets around any multi-term numerator.

Pearson Edexcel GCSE Maths

After multiplying by a common denominator, bracket the whole numerator if it has more than one term, then expand or simplify carefully.

Eduqas GCSE Maths

Show the factorising line before cancelling so the common factor is visible; cancelling separate terms is not valid algebra.

WJEC Wales

Keep restrictions, common denominators and simplified final expressions visible because algebraic-fraction marks often reward each tidy method line.

CCEA GCSE Maths

In non-calculator units, use exact factorising, common denominators and cancellation; avoid decimal approximations for algebraic fractions.

Next lesson

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

Previous lesson: Functions Next lesson: Angles and parallel lines