IGCSE Mathematics: Algebraic Fractions Practice Questions
An algebraic fraction is simplified by factorising the numerator and denominator and cancelling common brackets. Nothing can be cancelled across an addition or subtraction, only across a product.
Topic 2.3 of Cambridge IGCSE Mathematics 0580 is an Extended topic that punishes careless cancelling more than anything else on the paper. Factorise first, then cancel whole brackets. The questions below apply that rule to simplifying, adding and solving.
What you need to know for Algebraic Fractions
- SimplifyingFactorise the numerator and the denominator completely, then cancel any bracket that appears in both.
- What can be cancelledOnly factors, meaning things being multiplied. A term that is part of an addition or subtraction cannot be cancelled on its own.
- Adding and subtractingFind a common denominator, usually the product of the two denominators, adjust both numerators, then combine and simplify the numerator.
- Multiplying and dividingMultiply numerators and denominators, cancelling common factors first. To divide, multiply by the reciprocal of the second fraction.
- Solving equations with fractionsMultiply every term on both sides by the lowest common denominator to clear the fractions, then solve as usual.
- Excluded valuesA denominator can never be zero, so values of x making a denominator zero are not valid solutions.
IGCSE Mathematics Algebraic Fractions questions and answers
4 exam-style questions written to the 0580 syllabus. Try each one on paper first, then open the worked answer to check your method against the marks.
Simplify fully (x2 minus 9) divided by (x2 + 7x + 12).
Show the worked answer
- Factorise the numerator. It is a difference of two squares: x2 minus 9 = (x + 3)(x minus 3).
- Factorise the denominator. Two numbers multiplying to 12 and adding to 7 are 3 and 4, so x2 + 7x + 12 = (x + 3)(x + 4).
- The fraction is now (x + 3)(x minus 3) over (x + 3)(x + 4).
- The bracket (x + 3) is a factor of both, so it cancels, leaving (x minus 3) divided by (x + 4).
Write 3 divided by (x + 1) plus 2 divided by (x minus 2) as a single fraction in its simplest form.
Show the worked answer
- The common denominator is the product of the two denominators: (x + 1)(x minus 2).
- Adjust each numerator: the first becomes 3(x minus 2) and the second becomes 2(x + 1).
- Expand the numerators: 3x minus 6, and 2x + 2. Adding gives 5x minus 4.
- The single fraction is (5x minus 4) divided by (x + 1)(x minus 2). The numerator does not factorise, so this is fully simplified.
Simplify (2x divided by 3) multiplied by (9 divided by 4x2).
Show the worked answer
- Multiplying fractions means multiplying numerators and multiplying denominators.
- Numerator: 2x times 9 = 18x. Denominator: 3 times 4x2 = 12x2.
- The fraction is 18x over 12x2. Divide the numbers by their HCF of 6: 18 divided by 6 = 3 and 12 divided by 6 = 2.
- Cancel one x from top and bottom: the answer is 3 divided by 2x.
Solve (x + 1) divided by 2, minus (x minus 3) divided by 5, equals 2.
Show the worked answer
- The lowest common denominator of 2 and 5 is 10. Multiply every term on both sides by 10 to clear the fractions.
- This gives 5(x + 1) minus 2(x minus 3) = 20.
- Expand carefully, applying the minus to both terms of the second bracket: 5x + 5 minus 2x + 6 = 20.
- Simplify to 3x + 11 = 20, so 3x = 9 and x = 3. Check by substituting back: (3 + 1) divided by 2 = 2, and (3 minus 3) divided by 5 = 0, and 2 minus 0 = 2.
Common mistakes in this topic
- Cancelling terms that are part of an addition rather than factors.
- Adding numerators without a common denominator.
- Sign errors when subtracting a bracketed numerator.
- Forgetting to factorise before attempting to simplify.
- Cancelling every occurrence of a letter regardless of its power.
Exam tips
- Factorise first, always. Nothing can be cancelled until both parts are products.
- When subtracting a fraction, put the whole numerator in brackets before expanding.
- For an equation, multiply every term by the lowest common denominator, including terms with no fraction.
- Check your solution by substituting back into the original equation.
- If the numerator of your final answer factorises, check whether it cancels further.
Practise 20 more questions like this, free
vStudyWise marks every answer instantly, tracks the topics you keep dropping marks on and turns them into a weekly study plan.
Algebraic Fractions FAQs
How do I simplify an algebraic fraction?
Factorise the numerator and the denominator completely, then cancel any bracket that appears in both. Cancelling before factorising is invalid, because you can only cancel factors, meaning parts that are multiplied, never terms that are added or subtracted.
Why can't I cancel the x squared terms in a fraction?
Cancelling is division, and dividing only works across a whole product. In an expression like x squared minus 9, the x squared is part of a subtraction, not a factor. Factorising it into (x + 3)(x minus 3) turns it into a product, at which point cancelling becomes valid.
How do I add two algebraic fractions?
Use the product of the denominators as the common denominator, multiply each numerator by the other denominator, then add the numerators over that common denominator. Expand and simplify the numerator, and check whether the result factorises and cancels.
How do I solve an equation containing algebraic fractions?
Multiply every term on both sides by the lowest common denominator, which clears all the fractions at once. Take care with signs when a bracketed numerator is being subtracted, then solve the resulting equation and substitute back to check.
Continue through the IGCSE Mathematics syllabus
Related IGCSE Mathematics topics
See all 41 IGCSE Mathematics practice topics ›
Written to the published Cambridge IGCSE Mathematics (0580) syllabus. Check your school entry code and syllabus year, because Core and Extended candidates are assessed on different content. Last reviewed 2026-08-12.