Divide by multiplying by the reciprocal
To divide algebraic fractions, flip the second fraction and multiply; then cancel.
Add/subtract over a common denominator
Use a common denominator, combine the numerators carefully (mind the signs when subtracting), then simplify.
Factorise top and bottom, then cancel
Factorise the numerator and denominator fully (watch for the difference of two squares) and cancel only common factors — never individual terms.
Drawn from real examiner reports.
Sign errors when subtracting a fraction
Forgetting that subtracting a fraction means subtracting all of its numerator, e.g. mishandling , which should become with both signs flipped.
A commonly reported examiner note records this on a recent higher-tier paper.
Cancelling terms instead of factors
Cancelling an or a number that is only part of a sum (e.g. cancelling the in ) instead of cancelling a whole bracket. Only common factors may be cancelled.
A commonly reported examiner note records this across recent higher-tier papers.
Not multiplying every term when solving
When clearing denominators to solve, students multiply some terms but not others (e.g. forgetting to multiply a term on the right-hand side by the common denominator).
A commonly reported examiner note records this on a recent higher-tier paper.
Difference of squares with a coefficient
Treating as drops the factor of — it is . Pull out the common factor first (or factor as and then take the out of each bracket).
On a recent higher-tier paper $16x^2-36$ was frequently factorised as $(2x-3)(2x+3)$, losing the factor of 4.
Dropping the x when combining fractions
Two classic slips: writing the final answer over a constant denominator (e.g. dividing top and bottom by but only on the top), and not bracketing a subtracted numerator so the second sign is not flipped.
A commonly reported examiner note records answers that dropped the $x$ in the denominator, e.g. giving $\frac{31-14x}{12}$; and a student who wrote $8x-12-(7+8x)$ as $8x-12-7+8x$ and lost the answer to the sign.
Factorise first — even on a hard question
On a tough algebraic-fractions question, factorising any quadratics (and spotting a difference of two squares) usually unlocks cancelling and earns method marks.
Factorise every quadratic first, then cancel
On a "simplify" or "show that" fraction finale, factorise every numerator and denominator first. Cancelling common brackets collapses the expression; multiplying them out instead produces an unmanageable quartic that rarely recovers.
Bracket a subtracted numerator
When subtracting one fraction from another, wrap the second numerator in a bracket and flip every sign inside it: , and . This single habit removes the most common sign error in the topic.
Factorise top and bottom fully, then cancel common brackets: You may cancel a whole factor like — never cancel a term inside a sum (you cannot cancel the or the ).
Factorise 16x² − 36 fully.
Write as a single fraction.
