Algebraic Fractions
Fractions with letters in them. The rules are exactly the same as for numerical fractions — the difficulty is that you must usually factorise first, and that sign errors when subtracting are extremely easy to make.
1. Simplifying — cancel factors, never terms
Factorise the numerator and denominator, then cancel common FACTORS.
Example: (x² − 9)/(x + 3)
- Factorise the top: (x + 3)(x − 3)
- Cancel the common bracket (x + 3)
- = x − 3
Example: (x² + 5x + 6)/(x² + 2x)
- Top: (x + 2)(x + 3); bottom: x(x + 2)
- Cancel (x + 2)
- = (x + 3)/x
This is the rule everything depends on: you can only cancel FACTORS — things being MULTIPLIED — never individual terms.
In (x + 3)/(x + 5) nothing cancels. The x’s are being added to something, not multiplied, so crossing them out is wrong. Likewise in (2x + 4)/2 you cannot cancel the 2 with the 2x alone — factorise first: 2(x + 2)/2 = x + 2.
A quick test: if you can’t put a bracket around it as a multiplied piece, you can’t cancel it.
Simplify fully. Leaving a fraction that still has a common factor loses the mark — tutors flagged this specifically.
2. Adding and subtracting
Same as numerical fractions: you need a COMMON DENOMINATOR.
Example: 2/x + 3/y
- Common denominator xy
- = 2y/xy + 3x/xy = (2y + 3x)/xy
Example: 3/(x+1) + 2/(x+2)
- Common denominator (x+1)(x+2)
- = [3(x+2) + 2(x+1)] / [(x+1)(x+2)]
- = (3x + 6 + 2x + 2) / [(x+1)(x+2)]
- = (5x + 8) / [(x+1)(x+2)]
The common denominator is usually the PRODUCT of the denominators when they share no factors. If they do share a factor, use the lowest common multiple — e.g. for 1/2x and 1/4x use 4x, not 8x².
Cross-multiplication is not how you add fractions. It is for solving an equation of the form a/b = c/d. Using it to add was a documented error.
Subtracting — use brackets
Example: 5/(x−1) − 2/(x+3)
- Common denominator (x−1)(x+3)
- = [5(x+3) − 2(x−1)] / [(x−1)(x+3)]
- Expand carefully: 5x + 15 − 2x + 2
- = (3x + 17) / [(x−1)(x+3)]
Put the second numerator in brackets when subtracting. The minus applies to every term: −2(x − 1) gives −2x +2, not −2x − 2. This is the highest-frequency error in the topic, and it was flagged in lessons as the reason to write brackets every time.
3. Multiplying and dividing
Multiplying — factorise, cancel, then multiply straight across:
(x + 2)/(x − 1) × (x − 1)/(x + 5)
- Cancel (x − 1)
- = (x + 2)/(x + 5)
Cancel before you multiply. Expanding first creates large expressions that then need factorising again.
Dividing — flip the second fraction and multiply:
(x + 1)/x ÷ (x + 1)/3 = (x + 1)/x × 3/(x + 1) = 3/x
Flip the second fraction only. Forgetting to invert at all, or inverting the wrong one, were both recorded errors.
No common denominator is needed for multiplying or dividing — only for adding and subtracting.
4. Solving equations with algebraic fractions
Multiply every term by the denominators to clear the fractions.
Example: x/2 + x/3 = 10
- Multiply every term by 6: 3x + 2x = 60
- 5x = 60 → x = 12
Example: 3/(x+1) = 2/x
- Cross-multiply (valid here, because it’s an equation): 3x = 2(x + 1)
- 3x = 2x + 2 → x = 2
Example producing a quadratic: 1/x + 1/(x+1) = 1
- Multiply through by x(x+1): (x+1) + x = x(x+1)
- 2x + 1 = x² + x
- 0 = x² − x − 1 → solve with the quadratic formula
Multiply EVERY term, including any that are already whole numbers. Missing one is the standard error.
Check for invalid solutions. A value making a denominator zero must be rejected — if x = −1 appeared as a solution above, it would be invalid because 1/(x+1) would be undefined.
5. “Show that” questions
These give you the answer and ask for the working.
Every step must be visible. As one tutor put it: don’t do any steps in your head — marks are given for the steps. Since the target is printed in the question, the working is the entire answer.
Work from one side towards the given result, and don’t skip the expansion or the cancellation.
6. Mistakes that cost marks
Cancelling terms instead of factors — the biggest error here.
Not factorising first.
Failing to simplify fully.
Not using a common denominator for addition or subtraction.
Choosing a wrong or unnecessarily large common denominator.
Omitting brackets when subtracting, so signs go wrong.
Using cross-multiplication to add fractions.
Forgetting to flip when dividing, or flipping the wrong fraction.
Multiplying only some terms when clearing fractions in an equation.
Keeping a solution that makes a denominator zero.
Skipping steps in a “show that” question.
Frequently asked questions
How do I simplify an algebraic fraction? Factorise the top and bottom, then cancel common factors.
Can I cancel the x in (x + 3)/(x + 5)? No. You can only cancel factors, and x is being added here, not multiplied.
How do I add algebraic fractions? Find a common denominator — usually the product of the two — convert both, then add the numerators.
Why do I need brackets when subtracting? Because the minus sign applies to every term of the second numerator.
Do I need a common denominator to multiply? No — only for adding and subtracting.
How do I divide algebraic fractions? Flip the second fraction and multiply.
How do I solve an equation with fractions? Multiply every term by the denominators to clear them, then solve as normal.
When can I cross-multiply? Only when you have an equation of the form a/b = c/d — never when adding.
Why do I need to check my solutions? Because a value that makes a denominator zero is not valid.
Quick revision checklist
- I factorise before simplifying
- I cancel only factors, never terms
- I simplify fully
- I find a common denominator before adding or subtracting
- I use the lowest common denominator where the denominators share factors
- I put the second numerator in brackets when subtracting
- I expand carefully, watching every sign
- I cancel before multiplying
- I flip the second fraction when dividing
- I clear fractions by multiplying every term when solving
- I reject solutions that make a denominator zero
- I show every step in “show that” questions
These notes cover algebraic fractions in the Cambridge IGCSE Mathematics (0580) syllabus, and are written for Grade 9–11 / Year 10–11 students. They are based on teaching patterns observed across a large set of one-to-one IGCSE Maths lessons, with particular attention to the errors students make most often and the wording examiners reward. Always check the current syllabus and formula list for your own exam series.
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