Simplify and operate with algebraic fractions
Factor before cancelling, use common denominators, and retain restrictions from the original denominators.
Use this pathway for Cambridge IGCSE International Mathematics 0607, 2028–2030. Core and Extended objectives are labelled in each lesson; Extended adds requirements beyond Core. Paper 1 and Paper 2 are non-calculator. Papers 3–6 require a graphic display calculator (GDC). These original lessons use self-checked practice; they are not a Cambridge grade, full timed paper or teacher review.
Before you begin
- Check whether your school has entered you for Core or Extended. Core and Extended share most Number ideas, but some notation and methods are Extended only.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which expression equals 3x/6?
What you will learn
- Simplify one-step algebraic fractions for Core
- Factorise and simplify rational expressions for Extended
- Operate on algebraic fractions while respecting non-zero denominators
Cancel common factors, not terms
A common factor multiplying the whole numerator and denominator can cancel: 6x/9 = 2x/3. A term inside a sum cannot be cancelled: (x + 2)/x is not 2.
Factorise first. (x² − 9)/(x − 3) = x + 3 only when x ≠ 3; cancellation does not restore a value excluded by the original denominator.
Choose the fraction operation
For addition and subtraction, use a common denominator. For multiplication, multiply numerators and denominators then simplify. To divide, multiply by the reciprocal.
Extended candidates manipulate more involved rational expressions. Keep restrictions from every original denominator and check factors before cancelling.
Pause and explain
Can x be cancelled from (x + 4)/x?
Worked example
Simplify (x² − 9)/(x² + 5x + 6).
Show the worked solution
- Factor the numerator: (x − 3)(x + 3).
- Factor the denominator: (x + 2)(x + 3).
- Cancel x + 3, but the original denominator excludes x = −2 and x = −3.
Answer (x − 3)/(x + 2), where x ≠ −2, −3.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Simplify 6x/9 and 1/x + 2/(3x), stating restrictions.
Give me a hint
Reduce the first fraction, then use 3x as a common denominator.
Compare my reasoning
- 6x/9 = 2x/3.
- 1/x = 3/(3x), so the sum is 5/(3x).
- The original denominator requires x ≠ 0.
2x/3; 5/(3x), x ≠ 0.
Look for these in your work
- Common factor cancelled
- Common denominator used
- Restriction stated
Simplify (x² + 5x + 6)/(x² − 4) and state excluded values.
Give me a hint
Factor both quadratics, then cancel only the shared factor.
Compare my reasoning
- Numerator = (x + 2)(x + 3).
- Denominator = (x − 2)(x + 2).
- Cancel x + 2; the original denominator excludes x = ±2.
(x + 3)/(x − 2), x ≠ ±2.
Look for these in your work
- Both expressions factored
- Only common factor cancelled
- Original restrictions retained
A rate is (x² − 4)/(x − 2) units per cycle. Simplify it and explain why x = 2 remains excluded.
Give me a hint
Factor as a difference of squares and compare with the original denominator.
Compare my reasoning
- x² − 4 = (x − 2)(x + 2).
- For x ≠ 2, cancellation gives x + 2.
- At x = 2 the original quotient divides by zero.
x + 2 units per cycle, for x ≠ 2.
Look for these in your work
- Factorisation correct
- Factor cancelled
- Excluded input preserved
Common mistakes
- Cancelling one term from a sum
- Cancelling before factorising
- Dividing without reciprocating
- Forgetting excluded denominator values
What can you explain now?
Simplify 6a/9.
Compare with the explanation
2a/3
Divide top and bottom by 3.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Why can factors cancel but terms in a sum cannot, and why do restrictions survive simplification?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.