IGCSE Maths algebraic fractions: simplify, add and solve

Where algebraic fractions sit in each syllabus

The key ideas

An algebraic fraction is a fraction with an expression on top, underneath, or both, such as (x + 1)/(x - 3). Everything you know about 6/8 = 3/4 still applies. You simplify by dividing top and bottom by a common factor, you add by finding a common denominator, and you divide by multiplying by the reciprocal.

The single most important rule is that you can only cancel factors, never terms. In (x + 6)/6 the 6 on top is a term, joined by a plus sign, so nothing cancels. In 6(x + 1)/6 the 6 is a factor, multiplying the whole top, so it cancels to x + 1. That is why almost every algebraic fraction question starts with factorising: common factors, the difference of two squares a^2 - b^2 = (a + b)(a - b), and quadratics such as x^2 - 5x + 6 = (x - 2)(x - 3).

To add or subtract, use the lowest common denominator, which for two different linear denominators is usually their product. Put every numerator over that denominator, expand carefully, and keep the subtracted numerator in a bracket. To solve an equation containing fractions, multiply every term by every denominator to clear them, then solve the equation you are left with. Always check that no answer makes a denominator zero.

Worked example 1: simplify (x^2 - 2x)/(x^2 - 5x + 6)

1. Factorise the numerator by taking out the common factor x: x^2 - 2x = x(x - 2).
2. Factorise the denominator. You need two numbers that multiply to 6 and add to -5, which are -2 and -3: x^2 - 5x + 6 = (x - 2)(x - 3).
3. Write the fraction as x(x - 2) / ((x - 2)(x - 3)).
4. Cancel the common factor (x - 2) from top and bottom.
5. Answer: x/(x - 3). Check with x = 5: the original is (25 - 10)/(25 - 25 + 6) = 15/6 = 2.5, and 5/(5 - 3) = 2.5. The two agree.

Worked example 2: write (3x + 1)/(x + 2) - (x - 2)/(x - 1) as a single fraction

1. The common denominator is (x + 2)(x - 1).
2. Multiply each numerator by the denominator it is missing: (3x + 1)(x - 1) - (x - 2)(x + 2), all over (x + 2)(x - 1).
3. Expand the first product: (3x + 1)(x - 1) = 3x^2 - 3x + x - 1 = 3x^2 - 2x - 1.
4. Expand the second product: (x - 2)(x + 2) = x^2 - 4. Keep it in a bracket because it is being subtracted.
5. Subtract: 3x^2 - 2x - 1 - (x^2 - 4) = 3x^2 - 2x - 1 - x^2 + 4 = 2x^2 - 2x + 3.
6. Answer: (2x^2 - 2x + 3)/((x + 2)(x - 1)). The numerator does not factorise, so this is fully simplified. Check with x = 2: the original is 7/4 - 0 = 1.75, and (8 - 4 + 3)/(4 × 1) = 7/4. Correct.

Worked example 3: solve 2/(x + 2) + 3/(2x - 1) = 1

1. Multiply every term by (x + 2)(2x - 1): 2(2x - 1) + 3(x + 2) = (x + 2)(2x - 1).
2. Expand: 4x - 2 + 3x + 6 = 2x^2 - x + 4x - 2, so 7x + 4 = 2x^2 + 3x - 2.
3. Collect everything on one side: 0 = 2x^2 - 4x - 6. Divide by 2: x^2 - 2x - 3 = 0.
4. Factorise: (x - 3)(x + 1) = 0, so x = 3 or x = -1.
5. Check both: x = 3 gives 0.4 + 0.6 = 1, and x = -1 gives 2/1 + 3/(-3) = 2 - 1 = 1. Neither makes a denominator zero, so both answers stand.

Common mistakes that cost marks

• Cancelling terms: (x + 3)/(x + 7) does not simplify to 3/7. Only whole factors cancel.
• Losing the minus sign: when subtracting a numerator such as (x - 2)(x + 2), the whole expansion must be subtracted, which changes the sign of every term.
• Multiplying out the denominator too early. Leave it factorised; examiners accept (x + 2)(x - 1) and it makes checking for cancelling easier.
• Forgetting to multiply the right-hand side when clearing fractions. In example 3 the 1 becomes (x + 2)(2x - 1), not 1.
• Factorising quadratics with a leading coefficient wrongly, for example writing 2x^2 + 3x as 2x(x + 3) instead of x(2x + 3).
• Not simplifying the final answer when the question says 'in its simplest form'.

Exam technique and how a tutor helps

Algebraic fractions are usually worth 3 to 5 marks and often sit inside a longer question: a fraction equation that becomes a quadratic, or a simplification that feeds a later part. Write every line. Method marks are given for the correct common denominator and for a correct expanded numerator even if a later slip spoils the answer. On Cambridge Paper 2 there is no calculator, so the factorising has to be secure by hand. Edexcel 4MA1 papers allow a calculator throughout, but it does not help with the algebra itself.

Substituting a simple value such as x = 2 or x = 5 into the original and your answer is the fastest check available, and it takes under a minute. If the two values differ, the error is in your algebra, not the question.

In one-to-one lessons a tutor watches the student work line by line on the shared whiteboard and spots the exact step that breaks, which is nearly always factorising or a sign. The fix is a short set of targeted questions on that step, repeated across a few lessons until it is automatic, followed by mixed past-paper questions where the fraction is hidden inside a bigger problem.

Self-check: can you do these?

• Simplify (2x^2 + 3x)/(4x^2 - 9). (Answer: x/(2x - 3))
• Write 1/(x - 2) + (x + 1)/(x - 3) as a single fraction. (Answer: (x^2 - 5)/((x - 2)(x - 3)))
• Simplify (3a/4) ÷ (9a/10). (Answer: 5/6)
• Solve x/(2x + 1) = 4. (Answer: x = -4/7)
• Explain in one sentence why (x + 6)/6 is not equal to x + 1.