Algebraic fractions follow the same rules as numerical fractions, but their denominators may rule out certain variable values. Simplify by factorising first: cancellation removes common factors, not separate terms within sums. To add or subtract fractions, use a common denominator. When rearranging a formula with the new subject in more than one place, collect those terms on one side and factor them. Keep any restrictions from the original denominators even when a factor cancels.
Learning objectives
- Simplify and add algebraic fractions while retaining excluded values.
- Rearrange a formula by collecting and factoring the required subject.
Key ideas
Record original denominator exclusions before cancelling.
Cancel common factors, never separate terms of a sum.
Use a common denominator to add or subtract.
If the subject appears twice, collect its terms, factor, then divide by a non-zero factor.
Worked examples
Example 1
Simplify the fraction and state its excluded values.
The original denominator must stay non-zero.
Answer
Example 2
Add the fractions using a common denominator.
Answer
Question styles to prepare for
Original practice guidance aligned to the specification, not a count of past-paper appearances or a prediction. The AO labels below describe the intended practice focus, not official AQA mark allocations.
AO1 · Methods
Simplify an algebraic fraction, combine fractions or rearrange a formula.
AO2 · Reasoning
Identify excluded values and explain why cancellation of terms is invalid.
AO3 · Problem solving
Build and solve an equation involving a rate or a rearranged relationship.
When a task asks you to explain, include a mathematical reason. When solving a problem, show how the information becomes a calculation, equation or diagram, then interpret your result.
Avoid this mistake
Cancelling a factor does not remove its original excluded value. Keep every original denominator non-zero.
Try it yourself
Attempt each question before opening its answer.
Check 1: Simplify 1/x + 1/(x + 1).
Answer: (2x + 1)/(x(x + 1)), where x ≠ 0, -1
Rewrite the numerators as x + 1 and x over the common denominator.
Check 2: Make a the subject of v = u + at.
Answer: a = (v - u)/t, where t ≠ 0
Subtract u before dividing by t.
Explain it in your own words
- Explain why cancelling x from (x + 2)/x is invalid.
- Factorise x² - 16 and state when (x² - 16)/(x - 4) is undefined.
- Rearrange y = 3x/(x + 2) to make x the subject.
Keep practising
Download the compact topic guide (PDF) · Practise this topic in Blitz · All higher notes
Specification source
AQA Algebra specification · Assessment objectives. Checked 16 October 2026. Topic references identify related content; these notes do not cover every specification statement.