Sheet № 144 · Higher only · AQA · Edexcel · OCR
Adding Algebraic Fractions –
Adding and subtracting algebraic fractions follows the same principle as adding numerical fractions — you need a common denominator. The challenge is that the denominators are algebraic expressions, so finding the lowest common denominator requires more thought. This Higher-only topic is worth 3–4 marks and comes up regularly across all e
§Key definitions
Question:
Simplify 3/x + 2/x.
Answer:
(5x − 1)/[(x + 1)(x − 2)]
Q1 (Foundation):
Simplify 1/x + 1/(2x).
Q2 (Higher):
Write as a single fraction: 5/(x − 1) − 2/(x + 3).
Q3 (Higher):
Write as a single fraction: 2/(x − 4) + 3/(x² − 16).
§Formulas to memorise
a/b + c/d = (ad + bc)/(bd)
Find the lowest common denominator (LCD), adjust each fraction, then combine the numerators
Factorise each denominator — if possible.
Find the lowest common denominator (LCD) — the simplest expression that both denominators divide into.
Multiply the numerator and denominator of each fraction — so that both fractions have the LCD.
Add or subtract the numerators — over the common denominator.
Expand and simplify — the numerator by collecting like terms.
Factorise the result — if possible, and cancel any common factors.
Worked example
Simplify 3/x + 2/x.
Working:
⚠ Common mistakes
- ✗Forgetting to multiply the numerator when adjusting for the LCD. If you change the denominator, you must change the numerator by the same factor.
- ✗Sign errors when subtracting. In expressions like − (x − 1), the minus distributes to give −x + 1, not −x − 1. Use brackets to avoid this.
- ✗Not simplifying the final answer. Always check whether the numerator factorises and whether anything cancels with the denominator.
✦ Exam tips
- →Factorise denominators first — this often reveals that the LCD is simpler than you expect.
- →Keep the denominator in factorised form. Only expand the numerator.
- →Show each step clearly. Examiners award method marks for finding the LCD, adjusting fractions, and combining.
- →If the question says "simplify fully," check for cancellation at the end.