Algebraic fractions
Section: Algebra and graphs 1 | Syllabus: Cambridge IGCSE Mathematics (0580)
Adding, Subtracting, Multiplying and Dividing Algebraic Fractions
Extended Only
Algebraic fractions follow exactly the same rules as numerical fractions - a common denominator is needed to add or subtract, but not to multiply or divide.
- To add or subtract, rewrite each fraction with a common denominator (multiplying the denominators together always works), then combine the numerators
- To multiply, multiply the numerators together and the denominators together
- To divide, multiply by the reciprocal of the second fraction
Worked Example: Adding and Subtracting Algebraic Fractions
- Question: Simplify.
(i) x/4 + (x-3)/5 (ii) 3x/2 - 2(x-1)/3- Step 1 (i): Common denominator 20: (5x/20) + (4(x-3)/20) = (5x+4x-12/20)
- Answer (i): (9x - 12)/20
- Step 1 (ii): Common denominator 6: (9x/6) - (4(x-1)/6) = (9x-4x+4/6)
- Answer (ii): (5x + 4)/6
Worked Example: Multiplying and Dividing Algebraic Fractions
- Question: Simplify.
(i) 2a/5 × 15a/8 (ii) 2a/5 ÷ 15a/8- Step 1 (i): Multiply numerators and denominators: (2a × 15a/5 × 8) = (30a^2/40)
- Answer (i): 3a²/4
- Step 1 (ii): Multiply by the reciprocal of the second fraction: (2a/5) × (8/15a) = (16a/75a)
- Answer (ii): 16/75
Common Mistakes
MistakeAdding or subtracting numerators and denominators directly without finding a common denominator, e.g. treating x/4 + (x-3)/5 as (2x-3)/9
Fixa common denominator is always needed for addition or subtraction - multiplying the two denominators together (4 × 5 = 20) always works
MistakeForgetting to distribute a subtracted numerator across every term, e.g. writing 9x - 4(x-1) as 9x - 4x - 1
Fixthe minus sign applies to the whole bracket: 9x - 4(x-1) = 9x - 4x + 4, since subtracting -1 gives +4
Factorising and Simplifying Rational Expressions
Extended Only
An algebraic fraction can only be simplified by cancelling factors that are multiplied together - so the numerator and denominator must be fully factorised first, before anything can be cancelled.
- Factorise the numerator and denominator completely
- Cancel any factor that appears in both the numerator and the denominator
Worked Example: Factorising and Simplifying a Rational Expression
- Question: Simplify.
(x^2-x-6/x^2-9)- Step 1: Factorise the numerator: numbers with product -6 and sum -1 are -3 and 2, giving x^2-x-6 = (x-3)(x+2)
- Step 2: Factorise the denominator as a difference of two squares: x^2-9 = (x-3)(x+3)
- Step 3: Cancel the common factor (x-3): ((x-3)(x+2)/(x-3)(x+3)) = (x+2/x+3)
- Answer: (x+2)/(x+3)
Worked Example: Simplifying with a Common Factor Denominator
- Question: Simplify.
(2x^2-6x/x^2-9)- Step 1: Factorise the numerator by extracting the common factor: 2x^2-6x = 2x(x-3)
- Step 2: Factorise the denominator: x^2-9 = (x-3)(x+3)
- Step 3: Cancel the common factor (x-3): (2x(x-3)/(x-3)(x+3)) = (2x/x+3)
- Answer: 2x/(x+3)
Common Mistakes
MistakeCancelling terms that are added or subtracted rather than multiplied, e.g. cancelling the x² in (2x²-6x)/(x²-9) directly without factorising first
Fixonly a common factor (something multiplied) can be cancelled - the numerator and denominator must be fully factorised into brackets first
MistakeLeaving the denominator unfactorised, missing that it shares a common factor with the numerator
Fixalways check the denominator for a difference of two squares, a common factor, or a quadratic that splits into two brackets before concluding nothing cancels
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