IGCSE Maths / Advanced algebra
Adding algebraic fractions revision notes
Edexcel International GCSE Maths (4MA1), chapter 23: Advanced algebra, specification point 2.2c. Free lesson.
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To add algebraic fractions, first make the denominators the same, then add the numerators. Leaving the denominator in brackets makes it easier to spot possible cancellation and gives the form expected in exams.
Why fractions need a common denominator
Fractions can only be added directly when they describe the same-sized parts. For example, thirds and quarters need to be changed into twelfths first. The denominator tells you the size of each part, so it stays the same when you add the numerators. For 2/3 + 1/4, the common denominator is 12. 2/3 = 8/12 and 1/4 = 3/12, so the answer is 11/12. Only the numerators are added. The denominator is not added.
How to find a common denominator with brackets
For algebraic fractions with different brackets underneath, use both brackets multiplied together as the common denominator. Each fraction must be multiplied by the factor its denominator is missing. Multiply its numerator and denominator by that factor, so the fraction keeps the same value. The common denominator for x + 1 and x − 2 is (x + 1)(x − 2). The fraction with denominator x + 1 is missing x − 2. The fraction with denominator x − 2 is missing x + 1.
How to add and tidy the numerators
Once the denominators match, combine the fractions by adding their numerators and keeping the common denominator unchanged. Then expand and collect like terms in the numerator. Leave the denominator in brackets. For 2/(x + 1) + 3/(x − 2), the numerator becomes 2(x − 2) + 3(x + 1). Expanding gives 2x − 4 + 3x + 3, which simplifies to 5x − 1. The result is (5x − 1)/((x + 1)(x − 2)).
What to avoid and how to present your answer
Do not add the denominators. Make them the same first, then leave that common denominator alone. Expanding the denominator gives an equivalent expression, but the bracketed form makes possible cancellation easier to see and is the expected presentation. Adding the denominators in 2/(x + 1) + 3/(x − 2) would incorrectly give 5/(2x − 1). The correct form is (5x − 1)/((x + 1)(x − 2)).
A quick example to check the method
Use the same steps when the brackets change. The common denominator is the two brackets multiplied, and each numerator is multiplied by the factor missing from its original denominator. For 3/(x + 2) + 4/(x − 1), the common denominator is (x + 2)(x − 1). The numerator becomes 3(x − 1) + 4(x + 2) = 3x − 3 + 4x + 8 = 7x + 5.