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Algebraic Fractions

FoundationHigherAQAEdexcelOCR

Practise algebraic fractions with this free Foundation and Higher GCSE Maths worksheet from Virtus Academy. You'll work through simplifying and calculating with algebraic fractions, building confidence for your exam, and every question comes with worked solutions in the mark scheme. Suitable for AQA, Edexcel and OCR. Factorise the top and bottom first so you can cancel common factors.

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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.

Foundation

WorksheetFoundation · PDFMark SchemeFoundation · PDF

Higher

WorksheetHigher · PDFMark SchemeHigher · PDF

Topic overview

An algebraic fraction has an expression in the numerator, the denominator, or both. The rules are the same as for numerical fractions, but factorising usually comes first.

Simplifying means cancelling a factor common to the top and bottom. Crucially, you can only cancel factors, never individual terms. In \(\frac{x+3}{3}\) nothing cancels, because the \(3\) on top is added rather than multiplied.

Adding and subtracting still need a common denominator, and multiplying and dividing work exactly as with numbers. Factorising first often reveals a common bracket that cancels, turning an intimidating expression into a short one.

Revision notes

Simplifying by cancelling factors

Factorise the numerator and denominator, then cancel any bracket or term appearing in both.

For \(\frac{x^2 + 5x}{x}\): factorise the top to \(x(x + 5)\), then cancel the \(x\), leaving \(x + 5\).

Only factors cancel

A term joined by addition or subtraction cannot be cancelled.

In \(\frac{x + 4}{4}\) the fours do not cancel, because the top is a sum. Cancelling them would give \(x\), which is wrong — test with \(x = 4\): the true value is \(2\), not \(4\).

Multiplying and adding

Multiply straight across, cancelling first where possible. To add, find a common denominator as usual.

For \(\frac{1}{x} + \frac{2}{3}\), the common denominator is \(3x\), giving \(\frac{3}{3x} + \frac{2x}{3x} = \frac{3 + 2x}{3x}\).

Key points

  • Factorise before attempting to cancel.
  • Only factors cancel, never individual terms.
  • A common bracket on top and bottom can be cancelled.
  • Multiplying works straight across.
  • Adding needs a common denominator.
  • Test with a number if you are unsure a cancellation is valid.

Worked examples

Example 1

Simplify \(\dfrac{x^2 + 3x}{x}\).

Working

\[\frac{x(x + 3)}{x}\]factorise the numerator
\[x + 3\]cancel the common factor of x

Example 2

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

Working

\[\frac{2(x + 3)}{4}\]factorise the numerator
\[\frac{x + 3}{2}\]cancel the common factor of 2

Example 3

Explain why \(\dfrac{x + 5}{5}\) does not simplify to \(x\).

Working

\[\text{The 5 on top is added, not multiplied}\]only factors can be cancelled
\[\text{Test } x = 5: \frac{10}{5} = 2 \text{, not } 5\]substituting shows the cancellation is invalid

Common mistakes

  • Cancelling terms rather than factors.

    In (x + 4)/4 the fours do not cancel. Factorise first and cancel only whole factors.

  • Not factorising before cancelling.

    Common factors are often hidden until both parts are factorised.

  • Forgetting a common denominator when adding.

    Algebraic fractions add exactly like numerical ones, so the denominators must match.

  • Cancelling only part of a bracket.

    The whole bracket must match on top and bottom to cancel.

Exam tips

  • Factorise the numerator and denominator fully before cancelling.
  • Ask whether the thing you are cancelling is multiplied or added.
  • Substitute a number to test a cancellation you are unsure about.
  • Leave the answer fully simplified.

Key terms

Algebraic fraction
A fraction with an expression on the top or bottom.
Cancel
To divide the top and bottom by a common factor.
Factor
Something multiplied, which can be cancelled.
Common denominator
A shared bottom needed for adding fractions.

Related topics

Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.