Algebraic Fractions
Algebraic fractions are fractions where the numerator, denominator or both contain algebraic expressions, such as (x^2 - 4)/(x + 2). At GCSE Higher you need to simplify them by factorising, and add, subtract, multiply or divide them using the same rules as numerical fractions, often as part of solving an equation.
Before you start
Make sure you're comfortable with these topics first:
Method
- Factorise the numerator and denominator of each fraction fully wherever possible.
- Cancel any common factors that appear in both the numerator and denominator (never cancel individual terms that are added or subtracted).
- For multiplying fractions, multiply numerators together and denominators together, cancelling common factors first.
- For dividing, turn the second fraction upside down (find its reciprocal) and multiply.
- For adding or subtracting, find a common denominator (usually the product of the two denominators), rewrite each fraction over it, then combine the numerators.
- If solving an equation with algebraic fractions, multiply every term by the common denominator to clear the fractions before solving.
Worked example
Simplify fully (x^2 - 16) / (x^2 + x - 12).
- Factorise the numerator using the difference of two squares: x^2 - 16 = (x - 4)(x + 4).
- Factorise the denominator: x^2 + x - 12 = (x + 4)(x - 3).
- Write the fraction as [(x - 4)(x + 4)] / [(x + 4)(x - 3)].
- Cancel the common factor (x + 4) from the numerator and denominator.
- State the final answer: (x - 4) / (x - 3).
Practice questions
Try each question, then tap to reveal the answer.
Exam-style questions
Written in the style of a GCSE Maths exam paper, with a full mark scheme.
Simplify fully (x^2 + x - 6) / (x + 3).
Simplify fully (x^2 - 9) / (x^2 - x - 12).
Solve (x + 2)/4 + (x - 1)/3 = 3. Give your answer as a fraction in its simplest form.
Free printable worksheet
Want more practice on paper? Download the algebraic fractions worksheet pack - 8 pages of exam-style questions with a full mark scheme. No sign-up, no email wall - just the PDF, free for personal and classroom use.
Build a full practice pack.
This topic is one of hundreds in the library - pick the ones a student needs and generate a printable PDF in minutes.