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Algebraic Fractions

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Cambridge IGCSE · International Maths Extended

Algebraic Fractions

The big idea: Algebraic fractions follow exactly the same rules as ordinary number fractions — the only new skill is that you have to factorise first before you can simplify, combine, multiply, or cancel anything. If you can do fractions with numbers, and you can factorise, you can do this whole chapter.

Summary — What This Chapter Covers

Simplifying: factorise the top and bottom fully, then cancel any common factor (a single term or a whole bracket).
Adding & subtracting: find the lowest common denominator (LCD), rewrite each fraction over it, combine into one fraction, then check if it simplifies.
Multiplying: factorise everything first, cancel common factors across the two fractions, then multiply tops together and bottoms together.
Dividing: flip (reciprocate) the second fraction and turn ÷ into ×, then multiply as above.
Solving equations: either combine fractions into one and cross-multiply, or clear every denominator by multiplying both sides by each one in turn — then solve the resulting equation (often a quadratic or cubic).

1. Simplifying Algebraic Fractions

An algebraic fraction is just a fraction where the top (numerator) and/or bottom (denominator) contains letters, e.g. 2x / (x²+3x). To simplify one, you're doing the exact same thing you'd do with a numeric fraction like 15/20 = 3/4 — you find what's common to top and bottom, and cancel it. The only extra step with algebra is that the "common factor" is often hidden inside an expression, so your first job is always to factorise fully.

The Method

Step 1: Factorise the numerator fully.
Step 2: Factorise the denominator fully.
Step 3: Cancel any factor common to both.

Think of it like reducing 15/20: you spot that both are multiples of 5, and divide both by 5. Here, instead of spotting a common number, you spot a common bracket or term after factorising.

Cancelling a single term

Worked Example

Simplify x(5x − 1) / 4x

The x on top and the x on bottom are common factors — cancel them.

Result: (5x − 1) / 4

Cancelling a whole bracket

Worked Example

Simplify x(x + 2) / [(x + 2)(x − 1)]

The bracket (x + 2) appears on both top and bottom — cancel the whole bracket, not just part of it.

Result: x / (x − 1)

A trickier one — factorise before you can even see the common factor

Worked Example

Simplify (4x + 6) / (2x² − 7x − 15)

Factorise the top using 2 as a common factor: 2(2x + 3)

Factorise the bottom (a quadratic). Knowing the top gave (2x+3) is a big hint that this bracket appears again: (2x + 3)(x − 5)

Now the fraction is 2(2x + 3) / [(2x + 3)(x − 5)] — cancel (2x + 3)

Result: 2 / (x − 5)

Examiner Tip

When asked to simplify, always factorise top and bottom first. It's very likely one factor will match on both sides — if you're stuck factorising a tricky quadratic, look at what the other side factorised to. It often gives away the answer!

Common Mistake

only every single term 6x / (x + 1) cannot x x + 1 x

Practice Question 1

Simplify: (3x + 12) / (x² + 7x + 12)

Practice Question 2

Simplify: (x² − 9) / (2x + 6)

2. Adding & Subtracting Algebraic Fractions

1/2 + 1/5 common denominator

The 4-Step Method

What to Memorise

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Also in the full note

3. Multiplying & Dividing Algebraic Fractions
4. Solving Equations with Algebraic Fractions
Concepts Checklist
Exam Tips & Common Traps
Finding the LCD — don't just multiply blindly

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Algebraic Fractions

Algebraic Fractions

Summary — What This Chapter Covers

1. Simplifying Algebraic Fractions

Cancelling a single term

Cancelling a whole bracket

A trickier one — factorise before you can even see the common factor

2. Adding & Subtracting Algebraic Fractions

What to Memorise

More International Mathematics topics