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 exam boards.

What Is Adding Algebraic Fractions?

Adding algebraic fractions means combining two or more fractions whose numerators and/or denominators contain variables. You must rewrite them with a common denominator before you can add or subtract the numerators.

Key Formulas

Step-by-Step Method

1. Factorise each denominator if possible.
2. Find the lowest common denominator (LCD) — the simplest expression that both denominators divide into.
3. Multiply the numerator and denominator of each fraction so that both fractions have the LCD.
4. Add or subtract the numerators over the common denominator.
5. Expand and simplify the numerator by collecting like terms.
6. Factorise the result if possible, and cancel any common factors.

Worked Example 1 — Foundation Level

Question: Simplify 3/x + 2/x.

Working:

The denominators are already the same (both are x).

3/x + 2/x = (3 + 2)/x = 5/x.

Answer: 5/x

Worked Example 2 — Higher Level

Question: Write as a single fraction: 2/(x + 1) + 3/(x − 2).

Working:

• LCD = (x + 1)(x − 2).
• First fraction: 2(x − 2) / [(x + 1)(x − 2)].
• Second fraction: 3(x + 1) / [(x + 1)(x − 2)].
• Combine: [2(x − 2) + 3(x + 1)] / [(x + 1)(x − 2)].
• Expand numerator: 2x − 4 + 3x + 3 = 5x − 1.
• Result: (5x − 1) / [(x + 1)(x − 2)].

Answer: (5x − 1)/[(x + 1)(x − 2)]

Worked Example 3 — Exam Style

Question: Write as a single fraction in its simplest form: 4/(x + 3) − (x − 1)/(x² + 5x + 6).

Working:

• Factorise the second denominator: x² + 5x + 6 = (x + 2)(x + 3).
• LCD = (x + 2)(x + 3).
• First fraction: 4(x + 2) / [(x + 2)(x + 3)].
• Second fraction already has LCD: (x − 1) / [(x + 2)(x + 3)].
• Combine: [4(x + 2) − (x − 1)] / [(x + 2)(x + 3)].
• Expand numerator: 4x + 8 − x + 1 = 3x + 9.
• Factorise numerator: 3x + 9 = 3(x + 3).
• Cancel (x + 3): 3(x + 3) / [(x + 2)(x + 3)] = 3/(x + 2).

Answer: 3/(x + 2)

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.

Practice Questions

Q1 (Foundation): Simplify 1/x + 1/(2x).

Q2 (Higher): Write as a single fraction: 5/(x − 1) − 2/(x + 3).

/ [(x − 1)(x + 3)] = (5x + 15 − 2x + 2) / [(x − 1)(x + 3)] = (3x + 17)/[(x − 1)(x + 3)]]

Q3 (Higher): Write as a single fraction: 2/(x − 4) + 3/(x² − 16).

/ [(x + 4)(x − 4)] = (2x + 11)/[(x + 4)(x − 4)]]

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Related Topics

• Simplifying Algebraic Fractions
• Algebraic Fractions
• Adding and Subtracting Fractions

Summary

• Adding algebraic fractions requires a common denominator, just like numerical fractions.
• Factorise denominators first to find the LCD efficiently.
• Multiply each numerator by the factor needed to reach the LCD.
• Expand and simplify the numerator, then check for cancellation with the denominator.
• Always show clear working — method marks are available at every step.