What you'll learn
Algebraic fractions are fractions with algebra in the numerator or denominator. In this guide you will learn how to simplify them by cancelling, how to add and subtract them using a common denominator, how to multiply and divide them, and how to handle quadratics inside fractions. These higher-tier skills appear in algebra and equation questions at GCSE.
Key terms and definitions
Algebraic fraction — a fraction whose numerator and/or denominator contains algebra.
Common denominator — a shared denominator used to add or subtract fractions.
Simplify — cancel common factors to write a fraction in its lowest terms.
Reciprocal — the fraction turned upside down, used when dividing.
Factor — an expression that divides exactly into another.
Core concepts
Simplifying algebraic fractions
To simplify, factorise the numerator and denominator, then cancel common factors. For example, (x² + 3x)/(x) = x(x + 3)/x = x + 3. Only cancel factors, never individual terms across a + or − sign. With quadratics, factorise fully first: (x² − 9)/(x + 3) = (x + 3)(x − 3)/(x + 3) = x − 3.
Multiplying fractions
To multiply algebraic fractions, multiply the numerators and multiply the denominators, then simplify. It is often easier to cancel common factors first (cross-cancelling), then multiply. For example, (2/x) × (x/5) = 2x/5x = 2/5.
Dividing fractions
To divide, multiply by the reciprocal of the second fraction (turn it upside down and multiply). For example, (a/b) ÷ (c/d) = (a/b) × (d/c) = ad/bc. Then simplify.
Adding and subtracting fractions
To add or subtract, write both fractions over a common denominator, then combine the numerators. For example, 1/x + 1/y = y/(xy) + x/(xy) = (x + y)/(xy). Take care with signs when subtracting: subtract the whole numerator.
Fractions with quadratics
When denominators are quadratics, factorise them to find the common denominator more easily. The lowest common denominator is the product of the distinct factors. Always factorise first, then build equivalent fractions.
Worked examples
Example 1: Simplifying
Simplify (x² − 4)/(x² + 2x).
Factorise: (x + 2)(x − 2) / x(x + 2) = cancel (x + 2) → (x − 2)/x.
Example 2: Adding
Simplify 3/x + 2/x.
Same denominator: (3 + 2)/x = 5/x.
Example 3: Dividing
Work out (x/4) ÷ (x/8).
Multiply by the reciprocal: (x/4) × (8/x) = 8x/4x = 2.
Common mistakes and how to avoid them
Cancelling terms instead of factors. Only cancel common factors, after factorising — never across + or −.
Forgetting the common denominator. You must have the same denominator before adding or subtracting.
Sign errors when subtracting. Subtract the entire numerator; use brackets.
Not flipping the second fraction when dividing. Dividing means multiply by the reciprocal.
Leaving the answer unfactorised. Simplify fully at the end.
Exam technique for Algebraic Fractions
Factorise everything first to spot common factors.
Cancel only common factors, not terms.
Use a common denominator to add or subtract, watching signs.
Multiply by the reciprocal to divide.
Simplify the final answer completely.
Quick revision summary
Algebraic fractions contain algebra in the numerator or denominator. To simplify, factorise top and bottom, then cancel common factors — never cancel individual terms across a + or − ((x² − 9)/(x + 3) = x − 3). To multiply, multiply numerators and denominators (cancelling first where possible); to divide, multiply by the reciprocal of the second fraction. To add or subtract, write both over a common denominator, combine the numerators (subtracting the whole numerator carefully, using brackets), and simplify (1/x + 1/y = (x + y)/(xy)). When denominators are quadratics, factorise them to find the lowest common denominator. The classic errors are cancelling terms instead of factors, forgetting the common denominator, sign slips when subtracting, and not flipping the divisor. Factorise first, cancel only factors, use common denominators with care over signs, multiply by the reciprocal to divide, and always simplify fully.