Algebraic Fractions

Algebraic fractions work exactly like numerical fractions — the only difference is that the numerators and denominators contain algebraic expressions instead of plain numbers. This topic sits on the Higher tier and draws together skills from factorising, expanding, and solving equations. AQA, Edexcel, and OCR all test algebraic fractions, whether as standalone simplification questions or embedded within equation-solving problems. On this page you will learn how to simplify algebraic fractions by cancelling common factors, how to add and subtract them using a common denominator, how to multiply and divide them, and how to solve equations that contain them. Strong factorising skills are essential here, so revisit Factorising Expressions if you need a refresher.

What Are Algebraic Fractions?

An algebraic fraction is a fraction where the numerator, the denominator, or both contain algebraic terms. Examples include x/3, (2x + 1)/(x − 4), and (x² − 9)/(x² + 5x + 6).

The rules for working with them mirror ordinary fraction rules:

The golden rule: you can only cancel factors, never individual terms. You cannot cancel the x in (x + 3)/x because x + 3 is not a product.

Step-by-Step Method

Simplifying

1. Factorise the numerator fully.
2. Factorise the denominator fully.
3. Cancel any factors that appear in both.

Adding and subtracting

1. Find the lowest common denominator (LCD). If the denominators are (x + 1) and (x − 2), the LCD is (x + 1)(x − 2).
2. Multiply each fraction so that both have the LCD.
3. Expand the numerators if needed.
4. Combine into a single fraction and simplify.

Solving equations containing algebraic fractions

1. Multiply every term by the LCD to eliminate all fractions.
2. Expand and simplify.
3. Solve the resulting equation (linear or quadratic).
4. Check that your solutions do not make any original denominator equal to zero (these would be excluded values).

Worked Example 1 — Simplifying

Question: Simplify (x² − 4)/(x² + 5x + 6).

Working:

Step 1: Factorise the numerator. x² − 4 = (x + 2)(x − 2) [difference of two squares].

Step 2: Factorise the denominator. x² + 5x + 6 = (x + 2)(x + 3).

Step 3: Cancel the common factor (x + 2).

Result: (x − 2)/(x + 3).

Answer: (x − 2)/(x + 3)

Worked Example 2 — Adding and Solving

Question: Solve 3/(x + 1) + 2/(x − 3) = 1.

Working:

Step 1: LCD = (x + 1)(x − 3). Multiply every term by the LCD.

3(x − 3) + 2(x + 1) = (x + 1)(x − 3)

Step 2: Expand both sides.

3x − 9 + 2x + 2 = x² − 2x − 3

5x − 7 = x² − 2x − 3

Step 3: Rearrange to zero.

0 = x² − 7x + 4

Step 4: Use the quadratic formula (a = 1, b = −7, c = 4).

Discriminant = 49 − 16 = 33.

x = (7 ± √33) / 2

Step 5: Check neither solution makes a denominator zero. x + 1 = 0 when x = −1; x − 3 = 0 when x = 3. Neither (7 + √33)/2 ≈ 6.37 nor (7 − √33)/2 ≈ 0.63 equals −1 or 3. Both are valid.

Answer: x = (7 + √33)/2 or x = (7 − √33)/2

Common Mistakes

• Cancelling terms instead of factors. In (x + 5)/x, you cannot cancel the x. You can only cancel when the numerator and denominator are fully factorised and share a common factor.
• Forgetting to multiply every term by the LCD. When clearing fractions in an equation, the right-hand side (even if it is just a number) must also be multiplied.
• Sign errors when expanding. In 2/(x − 3) becoming 2(x + 1)/[(x + 1)(x − 3)], make sure you expand 2(x + 1) correctly as 2x + 2, not 2x + 1.
• Not checking for excluded values. If your solution makes a denominator zero, it must be rejected. State this explicitly in your answer.
• Leaving the answer un-simplified. After adding fractions, check whether the resulting numerator and denominator share a common factor.

Exam Tips

1. Factorise before you cancel. Never try to cancel terms by crossing out parts of expressions. Always factorise fully first.
2. For simplification questions, show each factorisation step. Examiners want to see that you have factorised, not just written the final answer.
3. When solving equations with algebraic fractions, the resulting equation is often quadratic. Be ready to factorise or use the quadratic formula.
4. These questions often carry 4-5 marks on Higher papers. The marks are distributed across identifying the LCD, forming the equation, and solving correctly. Do not skip steps.

Practice Questions

Q1: Simplify (6x²)/(9x³).

Q2: Simplify (x² − x − 6)/(x² − 9).

Q3: Solve 5/(x − 2) − 3/(x + 1) = 1.

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Related Topics

• Factorising Expressions
• Solving Quadratic Equations — Quadratic Formula
• Completing the Square
• Solving Linear Equations

For a full list of Higher topics, see our GCSE Maths Topics Complete List.

Summary

• Algebraic fractions follow the same rules as numerical fractions.
• Simplify by factorising numerator and denominator, then cancelling common factors.
• Add or subtract by finding a common denominator and combining numerators.
• Multiply straight across; divide by flipping and multiplying.
• To solve equations containing algebraic fractions, multiply through by the LCD and solve the resulting equation.
• Always check that solutions do not make any denominator equal to zero.
• Strong factorising skills are the key to success with this topic.