Algebraic Fractions — AQA GCSE Maths Revision Notes
What you'll learn
This topic covers fractions that contain algebra in the top, the bottom, or both. By the end of this guide you should be able to simplify an algebraic fraction by cancelling, add and subtract them using a common denominator, and multiply and divide them.
You should also be able to factorise quadratics inside fractions before simplifying, solve equations containing algebraic fractions, and recognise the one cancelling error that accounts for most lost marks in this topic.
The organising idea is that you may only cancel factors, never terms. A factor is something the whole of the top and the whole of the bottom are multiplied by; a term is something merely added or subtracted. In (3x)/(3y) the 3 is a factor of each part, so it cancels. In (x + 3)/(x + 5) nothing cancels at all, because the x is added, not multiplied — crossing out the x values there is simply wrong, and it changes the value of the expression. This is why factorising comes first in almost every question: factorising is what turns terms into factors so that cancelling becomes legal.
Key terms and definitions
Algebraic fraction — a fraction with algebra in the numerator, the denominator, or both.
Factor — an expression that the whole of another expression is multiplied by.
Term — a part of an expression separated from the rest by a plus or minus sign.
Common denominator — a shared denominator used when adding or subtracting.
Reciprocal — a fraction turned upside down.
Lowest terms — the form of a fraction in which top and bottom share no common factor.
Core concepts
Cancelling: factors only
Cancelling means dividing the top and the bottom by the same thing, so that thing must divide the whole of each.
In (6x)/(9x) both parts share a factor of 3x, so the fraction simplifies to 2/3.
In (x + 4)/(x + 7) nothing can be cancelled. The x is a term, not a factor, and crossing it out gives 4/7, which is a completely different expression — substituting x = 1 gives 5/8 for the original but 4/7 for the "simplified" version, which settles the matter.
If you are unsure whether something is a factor, ask whether the expression could be written as that thing multiplied by a bracket. If it could, it is a factor.
Simplifying by factorising
Since only factors cancel, the routine is always the same: factorise the top, factorise the bottom, then cancel.
For (x² + 3x)/x, factorising the top gives x(x + 3), and the x cancels, leaving x + 3.
For (x² − 9)/(x + 3), the top is a difference of two squares, factorising to (x + 3)(x − 3). The bracket (x + 3) now appears on both top and bottom, so it cancels and the answer is x − 3.
Notice how useless the original expression looked before factorising. Nothing cancelled until the terms had been turned into factors, which is the pattern to expect.
For (x² + 5x + 6)/(x² + 4x + 4), factorise both: the top gives (x + 2)(x + 3) and the bottom gives (x + 2)², so cancelling one (x + 2) leaves (x + 3)/(x + 2).
Multiplying
Multiply the numerators together and the denominators together, then simplify. No common denominator is needed.
Cancelling before multiplying keeps the expressions small and is almost always worth doing. In (2/x) × (x/5), the x cancels immediately, leaving 2/5 with no expansion required.
Where brackets are involved, cancel whole brackets against whole brackets, never parts of them.
Dividing
Dividing by a fraction means multiplying by its reciprocal: keep the first, change the sign to multiply, and flip the second.
So (a/b) ÷ (c/d) = (a/b) × (d/c) = ad/bc.
After flipping, treat it exactly as a multiplication — cancel first, then multiply.
Adding and subtracting
Here a common denominator is needed, exactly as with numerical fractions.
Where the denominators share nothing, their product is the common denominator. For 1/x + 1/y, use xy: the fractions become y/(xy) and x/(xy), so the answer is (x + y)/(xy).
For 2/x + 3/(x + 1), the common denominator is x(x + 1), giving 2(x + 1)/(x(x + 1)) + 3x/(x(x + 1)), which combines to (2x + 2 + 3x)/(x(x + 1)) and simplifies to (5x + 2)/(x(x + 1)).
The numerators are expanded and collected; the denominator is usually left in factorised form, since that is tidier and makes any further cancelling visible.
Subtraction and the sign trap
When subtracting, the whole of the second numerator is subtracted, so it needs a bracket.
For 5/x − (x + 2)/x, the answer is (5 − (x + 2))/x, which is (5 − x − 2)/x, giving (3 − x)/x.
Writing (5 − x + 2)/x is the standard error: the minus sign must be applied to every term in the second numerator, not just the first. Putting the bracket in as you write the line prevents it entirely.
Denominators that are quadratics
Factorise every denominator first. The lowest common denominator is then the product of the distinct factors, each taken once.
For 1/(x² − 1) + 1/(x + 1), factorising gives 1/((x + 1)(x − 1)) + 1/(x + 1). The common denominator is (x + 1)(x − 1), which the first fraction already has, so only the second needs converting, by multiplying top and bottom by (x − 1).
Building the common denominator from unfactorised quadratics leads to far heavier algebra than necessary.
Equations with algebraic fractions
To solve an equation containing algebraic fractions, multiply every term by the common denominator to clear the fractions, then solve as usual.
Multiplying only some terms is the common slip, and the resulting equation is then not equivalent to the original.
Check at the end that no solution makes a denominator zero — such a value must be rejected, because division by zero is undefined.
Worked examples
Example 1: Simplifying by factorising
Simplify (x² − 4)/(x² + 4x + 4).
The top is a difference of two squares: (x + 2)(x − 2).
The bottom factorises to (x + 2)(x + 2).
One (x + 2) is common to both, so cancelling it leaves (x − 2)/(x + 2).
No further cancelling is possible, because the remaining brackets are different and the terms inside them cannot be crossed out individually.
Example 2: Adding with different denominators
Write 3/(x − 1) + 2/x as a single fraction.
The denominators share nothing, so the common denominator is x(x − 1).
Convert each: 3x/(x(x − 1)) and 2(x − 1)/(x(x − 1)).
Add the numerators: 3x + 2x − 2 = 5x − 2.
The answer is (5x − 2)/(x(x − 1)), with the denominator left factorised.
Example 3: Dividing
Simplify (x² − 9)/(2x) ÷ (x + 3)/(4).
Keep, change, flip: (x² − 9)/(2x) × 4/(x + 3).
Factorise the difference of two squares: (x + 3)(x − 3)/(2x) × 4/(x + 3).
The bracket (x + 3) cancels, and 4 divided by 2 gives 2.
The answer is 2(x − 3)/x.
Multiplying everything out before cancelling would have produced a cubic on top and a great deal of unnecessary work.
Example 4: Solving an equation containing fractions
Solve 3/x + 1/2 = 2.
The denominators are x and 2, so the common denominator is 2x. Multiply every term by it.
The first term gives 6, the second gives x, and the right-hand side gives 4x.
So 6 + x = 4x, which rearranges to 6 = 3x, giving x = 2.
Check in the original: 3/2 + 1/2 = 2. ✓
Note that x = 2 does not make any denominator zero, so it is a valid solution. Had the answer come out as x = 0, it would have had to be rejected.
When an expression will not simplify
Not every algebraic fraction simplifies, and recognising that is part of the skill.
(x + 5)/(x + 2) is already in its lowest terms: the brackets are different, and the terms inside them cannot be cancelled individually.
Likewise (x² + 1)/(x + 1) does not simplify, because x² + 1 has no real factors — it is a sum of two squares, not a difference, so the factorising trick that works on x² − 1 does not apply here.
Spending time trying to force a cancellation that does not exist is a common way to lose time in an exam. If the top and bottom share no bracket after both have been fully factorised, the expression is finished.
Common mistakes and how to avoid them
Cancelling terms instead of factors. Nothing cancels in (x + 4)/(x + 7). Test a doubtful cancellation by substituting a number into both versions.
Cancelling before factorising. Factorise first; the cancelling only becomes possible afterwards.
Forgetting the bracket when subtracting. The minus sign applies to every term in the second numerator.
Finding a common denominator for a multiplication. Only addition and subtraction need one.
Expanding the denominator unnecessarily. Leave it factorised so that further cancelling stays visible.
Multiplying only some terms when clearing fractions in an equation. Every term must be multiplied.
Keeping a solution that makes a denominator zero. Such values must be rejected.
Exam technique for "Algebraic Fractions"
Factorise everything in sight as your first written line. Method marks are routinely awarded for correct factorising even when the cancelling afterwards goes wrong.
Write the common denominator down explicitly before converting either fraction, so the examiner can see where the converted numerators came from.
Put brackets around a numerator you are subtracting as you write it, not afterwards.
Cancel whole brackets, and say so — striking through matching brackets makes the step visible.
Leave the final denominator in factorised form unless the question asks otherwise.
When solving an equation, state the common denominator you multiplied through by, and check no solution makes a denominator zero.
Quick revision summary
Only factors cancel, never terms. In (x + 4)/(x + 7) nothing cancels; in (x² − 9)/(x + 3) everything does, once the top is factorised to (x + 3)(x − 3).
So factorise first — factorising is what converts terms into factors and makes cancelling legal.
To multiply: multiply tops and bottoms, cancelling beforehand wherever possible. To divide: keep, change, flip, then treat it as a multiplication.
To add or subtract: find a common denominator — the product of the distinct factors once each denominator is factorised — convert both fractions, then combine the numerators.
When subtracting, bracket the whole second numerator so the minus reaches every term inside it.
To solve an equation with algebraic fractions, multiply every term by the common denominator, then reject any solution that makes a denominator zero.