Algebraic Fractions: Simplifying, Multiplying and Dividing
An algebraic fraction is a fraction whose numerator or denominator contains algebraic expressions, such as (x + 3)/(x^2 - 9). Simplifying one means factorising the numerator and denominator fully and cancelling any common factor, while multiplying and dividing algebraic fractions follows the same rules as multiplying and dividing ordinary numerical fractions. This is a core AQA Further Maths skill.
Before you start
Make sure you're comfortable with these topics first:
Method
- Factorise the numerator and denominator of each fraction fully, using common factors, the difference of two squares or quadratic factorising as needed.
- Cancel any factor that appears in both the numerator and denominator; never cancel individual terms that are not full factors.
- When multiplying two fractions, factorise both fractions first, then cancel common factors across the whole calculation before multiplying what remains.
- When dividing by a fraction, turn the second fraction upside down (find its reciprocal) and multiply instead.
- Multiply the remaining numerators together and the remaining denominators together to leave a single fraction.
- Check the final fraction cannot be simplified any further before writing your answer.
Worked example
Simplify fully (x^2 + x - 6)/(x^2 - 9).
- Factorise the numerator: x^2 + x - 6 = (x + 3)(x - 2).
- Factorise the denominator using the difference of two squares: x^2 - 9 = (x - 3)(x + 3).
- Rewrite the fraction as [(x + 3)(x - 2)] / [(x - 3)(x + 3)].
- Cancel the common factor of (x + 3) from the numerator and denominator.
- Final answer: (x - 2)/(x - 3).
Practice questions
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Q1Simplify fully (6x^2y)/(9xy^3).Show answer
Answer: 2x/(3y^2) (divide the coefficients by 3 and cancel matching powers of x and y)
Q2Simplify fully (3x + 12)/(x^2 + 4x).Show answer
Answer: 3/x (factorise to 3(x+4)/[x(x+4)] and cancel (x+4))
Q3Simplify fully (x^2 - 25)/(x^2 + 2x - 15).Show answer
Answer: (x - 5)/(x - 3) (cancel the shared factor (x+5))
Q4Simplify fully (2x/(5y)) x (15y^2/(4x^2)).Show answer
Answer: 3y/(2x) (multiply then simplify the numerical coefficient to 3/2)
Q5Simplify fully [(x^2 - 9)/(x + 4)] / [(x - 3)/(2x + 8)].Show answer
Answer: 2x + 6, or 2(x + 3) (invert and cancel (x+4) and (x-3))
Q6Simplify fully [(x^2 + 6x + 8)/(x^2 - 16)] x [(x^2 - 4x)/(x^2 + 5x + 6)].Show answer
Answer: x/(x + 3) (cancel (x+2), (x+4) and (x-4))
Exam-style questions
Written in the style of a GCSE Further Maths exam paper, with a full mark scheme.
Simplify fully (4x - 28)/(x^2 - 7x).
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Simplify fully (x^2 - 36)/(x^2 + x - 30).
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A rectangular tile has area (x^2 - 16)/(x + 2) cm^2 and length (x - 4)/(x^2 - 4) cm. Find an expression for the width of the tile in its simplest form.
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See real GCSE Further Maths past-paper questions, with official mark schemes →
Free printable worksheet
Want more practice on paper? Download the algebraic fractions: simplifying, multiplying and dividing worksheet pack - 10 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
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