Simplifying Algebraic Fractions | Edexcel IGCSE Maths

Simplifying algebraic fractions is a core Edexcel IGCSE Maths skill: you factorise the top and bottom, then cancel the common factors (never single terms). This page shows how to cancel monomials using the index laws, multiply and divide fractions with the keep, change, flip method, add and subtract over a common denominator, and factorise a quadratic before cancelling. Each method has a worked example, and below them is a set of free, auto-marked practice questions so you can build the clear, mark-winning structure examiners expect.

Prior Knowledge You should be confident with the index laws and with factorising quadratic expressions, because simplifying an algebraic fraction nearly always comes down to factorising first, then cancelling.
⇩ Jump to Practice Questions ⇩

What is an algebraic fraction?

An algebraic fraction is just a fraction with letters in the top, the bottom, or both, such as \(\dfrac{8x^2y}{12xy^3}\), \(\dfrac{x+5}{3}\) or \(\dfrac{x^2-9}{x+3}\). You simplify it exactly the way you simplify a number fraction: factorise the numerator and denominator, then cancel the factors that appear on both. The single most important rule is that you can only cancel factors (things multiplied), never single terms in a sum.

The four moves for simplifying algebraic fractions

Cancel \(\dfrac{x^m}{x^n}=x^{\,m-n}\) Divide the numbers, subtract the powers of each matching letter.
Multiply \(\dfrac{a}{b}\times\dfrac{c}{d}=\dfrac{ac}{bd}\) Multiply tops together and bottoms together, then cancel.
Divide \(\dfrac{a}{b}\div\dfrac{c}{d}=\dfrac{a}{b}\times\dfrac{d}{c}\) Keep, change, flip: turn divide into multiply by the reciprocal.
Add / Subtract \(\dfrac{a}{b}\pm\dfrac{c}{d}=\dfrac{ad\pm bc}{bd}\) Build a common denominator; bracket the tops when subtracting.

Worked examples

💡 Cancel common factors

Simplify \(\dfrac{18x^3y}{24x y^4}\).

\[ \begin{array}{rcl} \dfrac{18x^3y}{24xy^4} &=& \dfrac{18}{24}\cdot\dfrac{x^3}{x}\cdot\dfrac{y}{y^4} \\[4pt] &=& \dfrac{3}{4}\cdot x^{2}\cdot \dfrac{1}{y^{3}} \\[4pt] &=& \dfrac{3x^{2}}{4y^{3}} \end{array} \]
What is happening?

Reduce the numbers first \(\left(\tfrac{18}{24}=\tfrac34\right)\), then subtract powers: \(x^{3-1}=x^2\) and \(y^{1-4}=y^{-3}\), which drops to the bottom.

💡 Multiply, then cancel

Simplify \(\dfrac{6m^2}{5n}\times\dfrac{15n^3}{8m}\).

\[ \begin{array}{rcl} \dfrac{6m^2}{5n}\times\dfrac{15n^3}{8m} &=& \dfrac{90m^2n^3}{40mn} \\[4pt] &=& \dfrac{9}{4}\,m\,n^{2} \\[4pt] &=& \dfrac{9mn^{2}}{4} \end{array} \]
What is happening?

Multiply tops and bottoms, then cancel: \(\tfrac{90}{40}=\tfrac94\), \(m^{2-1}=m\), \(n^{3-1}=n^2\).

💡 Divide (keep, change, flip)

Simplify \(\dfrac{4a}{9b}\div\dfrac{10a^2}{3b}\).

\[ \begin{array}{rcl} \dfrac{4a}{9b}\div\dfrac{10a^2}{3b} &=& \dfrac{4a}{9b}\times\dfrac{3b}{10a^2} \\[4pt] &=& \dfrac{12ab}{90a^2b} \\[4pt] &=& \dfrac{2}{15a} \end{array} \]
What is happening?

Flip the second fraction and multiply. Then cancel \(\tfrac{12}{90}=\tfrac{2}{15}\), one \(a\) and the \(b\), leaving a single \(a\) on the bottom.

💡 Subtract with brackets

Simplify \(\dfrac{x+4}{6}-\dfrac{2x-1}{9}\).

\[ \begin{array}{rcl} \dfrac{x+4}{6}-\dfrac{2x-1}{9} &=& \dfrac{3(x+4)-2(2x-1)}{18} \\[4pt] &=& \dfrac{3x+12-4x+2}{18} \\[4pt] &=& \dfrac{14-x}{18} \end{array} \]
What is happening?

Common denominator \(18\). Bracket each top before expanding so the subtraction flips every sign in the second bracket \(\left(-2(2x-1)=-4x+2\right)\).

💡 Factorise first, then cancel

Simplify \(\dfrac{x^2+x-6}{x-2}\).

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

The top will not cancel as it stands: \(x^2\), \(x\) and \(6\) are terms, not factors. Factorise the quadratic into two brackets, spot the bracket that matches the bottom, and cancel the whole bracket \((x-2)\).

⚠️ Do not miss: difference of two squares

If the top (or bottom) is one square minus another square, it always factorises into two brackets. This is the case most students miss, because there is no middle \(x\) term to prompt the factorising:

\(a^2 - b^2 = (a-b)(a+b)\)

So a fraction such as \(\dfrac{x^2-25}{x+5}\) is not stuck. Factorise the top and cancel the matching bracket:

\[ \begin{array}{rcl} \dfrac{x^2-25}{x+5} &=& \dfrac{(x-5)(x+5)}{x+5} \\[4pt] &=& x-5 \end{array} \]

Watch for the giveaway: two terms, both perfect squares, joined by a minus. That is your signal to split it into \((\,)(\,)\).

Cancel factors, never terms

✗ Wrong
\(\dfrac{x^2+x-6}{x-2}\;\ne\; x^2+x+3\)

You cannot cancel the \(x\) or the \(6\): they are added on, not multiplied.

✓ Right
\(\dfrac{(x-2)(x+3)}{x-2}\;=\; x+3\)

Factorise into brackets first; the matching bracket is a factor, so it cancels.

Golden rule: if the top or bottom is a sum or difference, factorise it before you cancel anything.

🔑 Key points

  • Factorise top and bottom, then cancel matching factors.
  • Cancelling monomials: divide the numbers, subtract the powers \(\left(x^m/x^n=x^{m-n}\right)\).
  • Divide by a fraction with keep, change, flip.
  • Add or subtract over a common denominator; bracket every numerator.
  • Leave the answer fully simplified, with no common factor remaining.

⚠️ Common pitfalls

  • Cancelling a single term out of a sum, e.g. crossing the \(x\) in \(\dfrac{x+3}{x}\).
  • Forgetting the bracket when subtracting, so only the first sign of the second top changes.
  • Adding denominators instead of finding a common one.
  • Stopping early and leaving a fraction that still has a common factor.
⇩ Practice now ⇩

Three auto-marked rooms take you from cancelling and the index laws, through multiplying, dividing, adding and subtracting, up to factorising before you cancel.

Next topic → Equations with Roots & Powers

Simplifying Algebraic Fractions: Practice Rooms

These IGCSE Maths practice rooms drill simplifying algebraic fractions. Room 1 covers cancelling common factors and the index laws; Room 2 covers multiplying, dividing, adding and subtracting; Room 3 is factorising before you cancel, from common factors up to factorising quadratics, ending with a fraction that leaves brackets on the top and bottom. In Rooms 1 and 2 each skill owns two columns, an easier column then a harder one: in Room 1, columns 1 and 2 are cancelling while columns 3 and 4 are the index laws; in Room 2, columns 1 and 2 are multiply and divide while columns 3 and 4 are add and subtract. Type each answer in its simplest form using / for division and ^ for powers; as you type, your answer is shown in the box as real maths, with a proper stacked fraction and raised indices.

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

Give every answer fully simplified. Rooms 1 and 2: a single term such as 2x^2/3y or 5xy^2 (no + or − in the final form). Room 3: a single term such as x+5, a^2/b or 7, or a fraction of brackets such as (x-1)/(x-2). Type a / and it displays as a stacked fraction.