How to Simplify Algebraic Fractions

Simplifying algebraic fractions works just like cancelling number fractions: factorise the top and the bottom, then cancel any factor they share. The golden rule for Edexcel IGCSE Maths is that you can only cancel whole factors, never single terms. This guide shows you how to factorise, cancel correctly, and write the result in its simplest form, with worked examples and free auto-marked practice below.

Prior Knowledge This page builds on taking out a common factor and finding the highest common factor.
⇩ Jump to Practice Questions ⇩

How to simplify an algebraic fraction

It is exactly like cancelling a number fraction, but you factorise first. The golden rule: only cancel whole factors, never single terms.

  1. Factorise the numerator and the denominator fully.
  2. Cancel any factor that appears in both the top and the bottom.
  3. Write what is left, and simplify any leftover numbers too.
Cancel factors, not terms

In \(\dfrac{x(x+5)}{x}\) the \(x\) is a factor of the whole top, so it cancels to \(x+5\). In \(\dfrac{x+5}{x}\) the \(x\) is NOT a factor of the top, so nothing cancels.

Factorise first
\(x^2+5x = x(x+5)\)
Write the top and bottom as products before you cancel anything.
Cancel a common factor
\(\dfrac{x(x+5)}{x}=x+5\)
A factor in both the top and bottom cancels to 1.
Tidy the numbers
\(\dfrac{6x}{9}=\dfrac{2x}{3}\)
Simplify any number fraction that is left at the end.

Worked examples

💡 Example 1: a common variable factor

Simplify \(\dfrac{x^2 + 5x}{x}\).

\[ \begin{array}{rcl} \dfrac{x^2 + 5x}{x} &=& \dfrac{x(x+5)}{x} \\ &=& x+5 \end{array} \]
What is happening?

Factorise the top: \(x^2+5x=x(x+5)\). Now \(x\) is a factor of the whole top and the whole bottom, so it cancels, leaving \(x+5\).

💡 Example 2: a common number factor

Simplify \(\dfrac{2x + 2y}{2z}\).

\[ \begin{array}{rcl} \dfrac{2x + 2y}{2z} &=& \dfrac{2(x+y)}{2z} \\ &=& \dfrac{x+y}{z} \end{array} \]
What is happening?

The number \(2\) is a factor of the whole top and the whole bottom, so it cancels. A fraction is left because nothing else is common to all three letters.

💡 Example 3: a common bracket cancels to a number

Simplify \(\dfrac{5a - 5b}{a - b}\).

\[ \begin{array}{rcl} \dfrac{5a - 5b}{a - b} &=& \dfrac{5(a-b)}{a-b} \\ &=& 5 \end{array} \]
What is happening?

Factorise the top: \(5a-5b=5(a-b)\). The bracket \((a-b)\) is in both the top and bottom, so it cancels to \(1\), leaving just the number \(5\).

💡 Example 4: factor a bracket, then tidy the numbers

Simplify \(\dfrac{4x^2 - 4xy}{6x - 6y}\).

\[ \begin{array}{rcl} \dfrac{4x^2 - 4xy}{6x - 6y} &=& \dfrac{4x(x-y)}{6(x-y)} \\ &=& \dfrac{4x}{6} \\ &=& \dfrac{2x}{3} \end{array} \]
What is happening?

Factorise both parts: the top is \(4x(x-y)\), the bottom is \(6(x-y)\). Cancel the common bracket \((x-y)\), then simplify the leftover number fraction \(\dfrac{4x}{6}\) to \(\dfrac{2x}{3}\). Always tidy the leftover numbers at the end.

🔑 Key Points

  • Factorise the top and bottom fully before you cancel anything.
  • Only cancel whole factors that multiply the entire top and bottom (a number, a letter, or a bracket).
  • A common bracket like \((a-b)\) cancels to \(1\), so the answer can be just a number.
  • Simplify any leftover number fraction (for example \(\dfrac{6x}{9}=\dfrac{2x}{3}\)).

⚠️ Common Pitfalls

  • Cancelling terms, not factors: in \(\dfrac{x+5}{x}\) you cannot cancel the \(x\).
  • Cancelling before factorising, so the common factor is not visible yet.
  • Leaving a number that still cancels (\(\dfrac{6x}{9}\) is not finished).
  • Sign slips: \(\dfrac{3-x}{x-3} = -1\), because \(3-x = -(x-3)\).
⇩ Practise now ⇩

Confident with cancelling? The same skill powers solving equations that contain fractions.

Next: Equations with Fractions →

Simplifying Algebraic Fractions: Practice Rooms

Randomly generated, auto-marked practice in simplifying algebraic fractions for Edexcel IGCSE Maths. Every fraction is simplified by cancelling a common factor: Room 1 cancels a common factor (a letter or a number); Room 2 cancels a common bracket (sometimes leaving just a number, sometimes a fraction); Room 3 mixes every type. Every grid gives 16 fresh questions. Give every answer in its simplest form, using / for a fraction; it renders live as real maths. Each correct answer adds to your global streak.

✓ Correct 0
✗ Re-attempts 0
🔥 Streak 0
🏆 Best 0

Answer in simplest form, for example x+5, (x+y)/z, 5 or 2x/3. An equivalent but un-cancelled answer (such as the original fraction) is marked wrong. Order and spacing do not matter.