Who this is for: Cambridge IGCSE Mathematics (0580/0607) students who want algebraic fractions — simplifying, adding, subtracting and solving fractional equations — to become a reliable source of marks instead of a topic they only half-remember.
What query it owns: how to understand and revise algebraic fractions in Cambridge IGCSE Mathematics.
Why this is safe: this page owns the algebraic fractions revision-guide angle, while Tutopiya’s Algebraic Fractions subtopic page owns the learning resource and the free algebraic fractions quiz owns the practice.

Algebraic fractions combine fraction arithmetic with algebra. In Cambridge IGCSE Mathematics (0580/0607) you must factorise numerators and denominators, find common denominators, and solve equations where the unknown appears in the denominator. Marks are lost on cancelled factors and sign slips. This guide explains the subtopic, the exam wording, and where to practise.

Key takeaways

• Factorise first — numerators and denominators before cancelling or combining.
• Only cancel factors, never terms added or subtracted across a fraction bar.
• To add or subtract, find the lowest common denominator and write equivalent fractions.
• When solving equations, multiply through by the LCD to clear denominators — then check for invalid values.

What are algebraic fractions in Cambridge IGCSE Maths?

An algebraic fraction has polynomials in the numerator, denominator or both — for example (x² − 4)/(x + 2). In Cambridge IGCSE Mathematics you simplify by factorising and cancelling common factors, combine fractions using a common denominator, and solve equations by clearing fractions. Extended papers may include slightly harder factorisation.

Read the notes on Tutopiya’s Algebraic Fractions subtopic page before attempting questions.

The core ideas you must master

Idea	What it means	How the exam uses it
Simplifying	Cancel common factors after factorising	”Simplify (x² − 9)/(x + 3)“
Adding / subtracting	Common denominator, then combine numerators	”Express 2/(x + 1) + 3/(x − 2) as a single fraction”
Solving equations	Multiply by LCD; solve resulting equation	”Solve 3/(x − 1) = 2”
Restrictions	Values that make denominator zero	”State the value of x that must be excluded”
Factorising	Needed before cancelling	Links to the Factorisation subtopic

How to simplify an algebraic fraction — step by step

1. Factorise the numerator and denominator completely.
2. Identify common factors in numerator and denominator.
3. Cancel those factors — write ÷ signs in working if it helps.
4. Leave the answer as a simplified fraction; do not expand unnecessarily.
5. Note excluded values if the question asks (denominator ≠ 0 before cancelling).

Test yourself with the free Algebraic Fractions quiz.

How to add or subtract algebraic fractions

1. Factorise each denominator.
2. Find the lowest common denominator (LCD).
3. Write each fraction with the LCD — multiply numerator and denominator as needed.
4. Combine numerators over the single denominator.
5. Simplify the result if possible.

Algebraic fractions in past-paper wording: command words that matter

Command word / phrase	What the question wants	Typical stem
Simplify	Factorise and cancel	”Simplify (2x² − 8)/(x − 2).”
Express as a single fraction	Add or subtract then simplify	”Express 1/x + 2/(x + 1) as a single fraction.”
Solve the equation	Clear denominators; solve	”Solve 5/(x + 2) = 3.”
Show that	Prove the given simplified form	”Show that (x² − 1)/(x + 1) = x − 1.”
Write down the value that must be excluded	State when denominator = 0	”Write down the value of x that must be excluded.”

Worked exam-style stems (how to answer the wording)

1. “Simplify (x² − 9)/(x + 3).” Numerator: (x − 3)(x + 3). Cancel (x + 3) → x − 3. Reward: factorisation, correct cancellation.
2. “Express 2/(x − 1) − 1/(x + 2) as a single fraction.” LCD = (x − 1)(x + 2). Combine → (x + 5)/((x − 1)(x + 2)) after simplification. Reward: common denominator, correct numerator.
3. “Solve 4/(x + 1) = 2.” Multiply both sides by (x + 1): 4 = 2(x + 1) → x = 1. Check: denominator ≠ 0. Reward: clearing fractions, solution.
4. “Show that (x² − 4)/(x − 2) simplifies to x + 2 for x ≠ 2.” Factorise numerator: (x − 2)(x + 2). Cancel (x − 2) → x + 2. State x ≠ 2 because the original denominator is zero there. Reward: factorisation, cancellation, excluded value stated.
5. “Write down the value of x that must be excluded from 3/(x + 5) + 1/(x − 1).” Set each denominator to zero: x = −5 or x = 1. Reward: both excluded values listed — a common 1-mark stem before a longer simplify question.

When you can recognise the wording instantly, work the full set on the Algebra topical past-paper questions and the Algebraic Fractions quiz to lock the method in.

How algebraic fractions connect to the rest of Algebra

Simplifying depends on Factorisation and Simplifying Algebraic Expressions. Use the Cambridge IGCSE Maths resource hub to revisit weak skills before tackling harder fractions.

Common mistakes students make

• Cancelling terms instead of factors — e.g. cancelling x in (x + 3)/x.
• Forgetting to factorise fully before simplifying.
• Sign errors when subtracting numerators over a common denominator.
• After solving, not checking whether the answer makes a denominator zero.
• Expanding the denominator when the mark scheme wants factorised form.

When you need more support

If algebraic fractions keep costing marks, strengthen Factorisation first, then retake the Algebraic Fractions quiz. A Cambridge IGCSE Maths tutor can fix the method quickly.

Frequently asked questions

Can I cancel x in (x + 2)/x? No — x is a term added to 2 in the numerator, not a factor of the whole numerator. Factorise first; only cancel common factors.

What is the first step when adding algebraic fractions? Factorise each denominator and find the lowest common denominator before writing equivalent fractions.

What does excluded value mean? The value of the variable that makes any denominator zero before simplifying — the expression is undefined there.

How do I revise algebraic fractions effectively? Master factorisation, then practise simplify → combine → solve in that order. Use the algebraic fractions quiz after each stage.

Ready to master Cambridge IGCSE Maths algebraic fractions?

Start with the Algebraic Fractions subtopic page, then book a free trial with a Cambridge IGCSE Maths specialist.

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