Simplify $\dfrac{x^2-9}{x^2+5x+6}$.

Factorise the top: $(x-3)(x+3)$. Factorise the bottom: $(x+2)(x+3)$. Cancel $(x+3)$: $\dfrac{x-3}{x+2}$. Answer: $\dfrac{x-3}{x+2}$

Write $\dfrac{2}{x}+\dfrac{3}{x+1}$ as a single fraction.

Common denominator $x(x+1)$. $\dfrac{2(x+1)+3x}{x(x+1)}$. $=\dfrac{5x+2}{x(x+1)}$. Answer: $\dfrac{5x+2}{x(x+1)}$

Solve $\dfrac{3}{x-1}-\dfrac{2}{x+1}=1$.

Multiply every term by $(x-1)(x+1)$: $3(x+1)-2(x-1)=(x-1)(x+1)$. $x+5=x^2-1$, so $x^2-x-6=0$. $(x-3)(x+2)=0$, giving $x=3$ or $x=-2$. Answer: $x=3$ or $x=-2$

Solve $\dfrac{5}{x+1} = \dfrac{3}{x-2}$.

M1 5(x − 2) = 3(x + 1) M1 5x − 10 = 3x + 3 (collects terms) A1 x = 6.5 oe Answer: $x=6.5$

IGCSE Math Revision Questions by topic Revision notes Past papers Student hub

Edexcel IGCSE Maths (4MA1) · Unit 2: Algebra Basics

Edexcel IGCSE 4MA1 Algebraic Fractions: exam questions and answers, notes and worked examples

Updated October 2026 · By Pete Bromfield

Algebraic fractions follow the same rules as number fractions. Simplify by factorising the top and bottom and cancelling whole factors; add or subtract over a common denominator; multiply straight across, and divide by multiplying by the reciprocal. To solve an equation with fractions, multiply every term by the common denominator to clear them.

Unit 2: Algebra Basics
Higher tier · calculator allowed
Free, no sign-in

Practise algebraic fractions in Unit 2 →

Notes
Formulas
Worked examples
Mistakes
FAQ

Algebraic Fractions notes

Simplifying

Factorise first, then cancel factors that appear on both the top and the bottom: $\frac{x^{2} - 4}{x^{2} + 3 x + 2} = \frac{(x - 2) (x + 2)}{(x + 1) (x + 2)} = \frac{x - 2}{x + 1}$ . You can only cancel a whole bracket, never a single term inside a sum.

Adding and subtracting

Use the lowest common denominator and multiply each numerator by the factor its denominator was missing: $\frac{3}{x} - \frac{2}{x + 2} = \frac{3 (x + 2) - 2 x}{x (x + 2)} = \frac{x + 6}{x (x + 2)}$ . Put the subtracted numerator in brackets so the minus sign reaches all of it.

Multiplying and dividing

Multiply the tops and multiply the bottoms, cancelling common factors first. To divide, multiply by the reciprocal of the second fraction: $\frac{2 x}{x + 3} \div \frac{4}{x^{2} - 9} = \frac{2 x}{x + 3} \times \frac{(x - 3) (x + 3)}{4} = \frac{x (x - 3)}{2}$ .

Solving equations with fractions

Multiply every term, on both sides, by the common denominator. $\frac{x}{2} + \frac{x}{3} = 10$ becomes $3 x + 2 x = 60$ , so $x = 12$ . When the unknown is in a denominator the result is often a quadratic; check that no solution makes a denominator zero.

Key formulas

Expanding Learn it: $(a + b)^{2} = a^{2} + 2 a b + b^{2}, (a - b)^{2} = a^{2} - 2 a b + b^{2}, a^{2} - b^{2} = (a + b) (a - b)$
Quadratic formula, $a x^{2} + b x + c = 0$ On the formula sheet: $x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$

On the formula sheet = printed on the formulae sheet in the 4MA1 Higher papers, so know where to find it. Learn it = not printed on the 4MA1 Higher formulae sheet, so learn it by heart. The whole 4MA1 formula sheet (free PDF) → · 4MA1 formulas to learn

Algebraic Fractions worked examples

Worked example 1: Simplifying an algebraic fraction

Simplify $\frac{x^{2} - 9}{x^{2} + 5 x + 6}$ .

Solution

Factorise the top: $(x - 3) (x + 3)$ .
Factorise the bottom: $(x + 2) (x + 3)$ .
Cancel $(x + 3)$ : $\frac{x - 3}{x + 2}$ .

4 steps

Tip: Only cancel whole factors, never single terms.

Worked example 2: Adding algebraic fractions

Write $\frac{2}{x} + \frac{3}{x + 1}$ as a single fraction.

Solution

Common denominator $x (x + 1)$ .
$\frac{2 (x + 1) + 3 x}{x (x + 1)}$ .
$= \frac{5 x + 2}{x (x + 1)}$ .

4 steps

Tip: Multiply each numerator by the missing factor.

Worked example 3: Equation with algebraic fractions

Solve $\frac{3}{x - 1} - \frac{2}{x + 1} = 1$ .

Solution

Multiply every term by $(x - 1) (x + 1)$ : $3 (x + 1) - 2 (x - 1) = (x - 1) (x + 1)$ .
$x + 5 = x^{2} - 1$ , so $x^{2} - x - 6 = 0$ .
$(x - 3) (x + 2) = 0$ , giving $x = 3$ or $x = - 2$ .

4 steps · 1 to fill in

Tip: Check neither answer makes a denominator zero.

Free account: +5 practice questions on every topic, with instant feedback, and full worked answers on the free questions. Free, no card. Join free →

Algebraic Fractions exam-style questions

Try each question before you open the answer (one has its full worked solution open). The mark schemes use Edexcel-style marks: M for method, A for accuracy (after the method mark), B for an independent result.

Question 1 · easy · 3 marks

Solve $\frac{x + 4}{3} - \frac{x - 2}{5} = 2$ .

Show the answer and mark scheme

M1 multiplies all terms by 15 (or common denominator)
M1 5(x + 4) − 3(x − 2) = 30 with correct signs
A1 x = 2

Answer: $x = 2$

Question 2 · medium · 3 marks

Solve $\frac{5}{x + 1} = \frac{3}{x - 2}$ .

Full worked solution Free, no sign-in

M1 5(x − 2) = 3(x + 1)
M1 5x − 10 = 3x + 3 (collects terms)
A1 x = 6.5 oe

Answer: $x = 6.5$

Every question here has a full worked solution. See them all →

Question 3 · medium · 3 marks

Solve $\frac{4}{x} + \frac{3}{2 x} = \frac{11}{2}$ .

Show the answer and mark scheme

M1 common denominator 2x, e.g. 8/(2x) + 3/(2x)
M1 11/(2x) = 11/2 or 22 = 11x oe
A1 x = 1

Answer: $x = 1$

Question 4 · hard · 4 marks

Solve $\frac{3}{x + 2} + \frac{2}{x - 1} = 1$ . Give your solutions correct to 3 significant figures.

Show the answer and mark scheme

M1 3(x − 1) + 2(x + 2) = (x + 2)(x − 1)
M1 rearranges to x² − 4x − 3 = 0
M1 correct substitution into quadratic formula or completing the square, (x − 2)² = 7
A1 x = 4.65, x = −0.646 (accept 4.645…, −0.6457…)

Answer: $x = 4.65, x = - 0.646$

Question 5 · hard · 3 marks

Solve $\frac{x}{x - 3} - \frac{2}{x + 1} = 1$ .

Show the answer and mark scheme

M1 x(x + 1) − 2(x − 3) = (x − 3)(x + 1)
M1 expands and cancels x² correctly, e.g. −x + 6 = −2x − 3
A1 x = −9

Answer: $x = - 9$

Question 6 · hard · 3 marks

Solve $\frac{6}{x + 3} = x - 2$ .

Show the answer and mark scheme

M1 (x − 2)(x + 3) = 6
M1 x² + x − 12 = 0 and factorises (x + 4)(x − 3)
A1 x = 3, x = −4

Answer: $x = 3, x = - 4$

Question 7 · very hard · 4 marks

Solve $\frac{2}{x - 1} - \frac{1}{x + 2} = \frac{1}{2}$ .

Show the answer and mark scheme

M1 2[2(x + 2) − (x − 1)] = (x − 1)(x + 2) oe
M1 x² − x − 12 = 0 oe
M1 (x − 4)(x + 3) = 0 or correct formula
A1 x = 4, x = −3

Answer: $x = 4, x = - 3$

Question 8 · very hard · 5 marks

(a) Show that the equation $\frac{x + 1}{2 x - 3} + \frac{3}{x} = 2$ can be written as $3 x^{2} - 13 x + 9 = 0$ . (3) (b) Hence solve $\frac{x + 1}{2 x - 3} + \frac{3}{x} = 2$ , giving your solutions correct to 3 significant figures. (2)

Show the answer and mark scheme

M1 (a) x(x + 1) + 3(2x − 3) = 2x(2x − 3)
M1 x² + 7x − 9 = 4x² − 6x
A1 3x² − 13x + 9 = 0 (cso)
M1 (b) x = (13 ± √(169 − 108))/6 = (13 ± √61)/6
A1 x = 3.47, x = 0.865 (accept 3.468…, 0.8649…)

Answer: $x = 3.47, x = 0.865$

Common algebraic fractions mistakes

Cancelling terms, not factors. In $\frac{x + 6}{x + 2}$ nothing cancels: the $x$ s are terms in a sum, not factors.
Multiplying only the fractions. When you clear fractions, multiply every term on both sides by the common denominator, including terms that are not fractions.
A minus sign that only reaches the first term. In $\frac{3}{x} - \frac{x - 1}{x + 2}$ the top is $3 (x + 2) - x (x - 1)$ : keep the second numerator in brackets so the minus reaches every term.
Keeping a solution that breaks the equation. Check every solution in the original equation. A value that makes any denominator zero must be rejected.

Where students lose marks across 4MA1, from the examiner reports →

Algebraic Fractions FAQ

How do I add algebraic fractions with different denominators?

Find the lowest common denominator (often the two denominators multiplied together), rewrite each fraction over it by multiplying its numerator by the missing factor, then add the numerators and simplify.

Why do some algebraic fraction equations turn into quadratics?

When the unknown is in a denominator, multiplying through by the denominators multiplies x by x, so an x² term appears. Rearrange to = 0 and solve by factorising or with the quadratic formula, which is on the 4MA1 Higher formulae sheet.

Can I use a calculator for algebraic fractions in 4MA1?

Yes. A calculator may be used in all four 4MA1 papers, so both Higher papers (1H and 2H) allow one. Most marks are for the method you write down, so show your working.

How long are the 4MA1 Higher papers?

Papers 1H and 2H are 2 hours each and 100 marks each, worth 50% each. Each tier has its own formulae sheet in the exam.

Next steps

Algebra revision notes · 4MA1 formula sheet · Taking Cambridge 0580 instead? Cambridge 0580 algebraic fractions questions

Every question and worked example here is original, written for IGCSE Math Revision rather than copied from Pearson papers, and every result was re-solved independently before publishing.

Practise algebraic fractions with your progress saved

Save your answers and track your progressA free account remembers the questions you have done and brings you back to this page.

Join free

Already have an account? Sign in

Edexcel IGCSE 4MA1 Algebraic Fractions: exam questions and answers, notes and worked examples

Try it first

Algebraic Fractions notes

Simplifying

Adding and subtracting

Multiplying and dividing

Solving equations with fractions

Key formulas

Algebraic Fractions worked examples

Worked example 1: Simplifying an algebraic fraction

Solution

Worked example 2: Adding algebraic fractions

Solution

Worked example 3: Equation with algebraic fractions

Solution

Algebraic Fractions exam-style questions

Question 1 · easy · 3 marks

Question 2 · medium · 3 marks

Full worked solution Free, no sign-in

Question 3 · medium · 3 marks

Question 4 · hard · 4 marks

Question 5 · hard · 3 marks

Question 6 · hard · 3 marks

Question 7 · very hard · 4 marks

Question 8 · very hard · 5 marks

Common algebraic fractions mistakes

Algebraic Fractions FAQ

Next steps

Practise algebraic fractions with your progress saved

More Unit 2 topics: Algebra Basics