Simplifying algebraic fractions Revision Notes for AQA GCSE Further Maths

Simplifying algebraic fractions

What are algebraic fractions?

Algebraic fractions are fractions that contain algebraic expressions (with variables like x, y, a, etc.) in the numerator, denominator, or both. The good news is that these fractions follow exactly the same rules as regular numerical fractions that you already know!

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The beauty of algebraic fractions is that if you've mastered working with numerical fractions, you already have the foundation needed to tackle algebraic ones. The only difference is that instead of working with just numbers, you're working with expressions that contain variables.

Just like with numerical fractions, algebraic fractions involve two main pairs of operations:

Multiplication (×) and division (÷)
Addition (+) and subtraction (−)

Basic principles for simplifying algebraic fractions

The key to simplifying algebraic fractions is to look for common factors that appear in both the numerator and denominator. Once you identify these common factors, you can cancel them out to simplify the fraction.

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Step-by-step approach:

Step 1: Factor completely - Break down both the numerator and denominator into their factors

Step 2: Identify common factors - Look for terms that appear in both parts

Step 3: Cancel the common factors - Remove matching factors from top and bottom

Step 4: Write the simplified result - Express your final answer in its simplest form

Worked examples

Example 1: Basic numerical and algebraic simplification

Let's start with some fundamental examples to build your confidence:

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Worked Example: Basic Simplification

(i) Simplifying $\frac{18}{24}$ :

Factor the numerator: $18 = 6 \times 3$
Factor the denominator: $24 = 6 \times 4$
Cancel the common factor of 6: $\frac{6 \times 3}{6 \times 4} = \frac{3}{4}$

(ii) Simplifying $\frac{2x + 2}{3x + 3}$ :

Factor the numerator: $2x + 2 = 2(x + 1)$
Factor the denominator: $3x + 3 = 3(x + 1)$
Cancel the common factor $(x + 1)$ : $\frac{2(x + 1)}{3(x + 1)} = \frac{2}{3}$

(iii) Simplifying $\frac{a^2 - a - 6}{a^2 - 8a + 15}$ :

Factor the numerator: $a^2 - a - 6 = (a - 3)(a + 2)$
Factor the denominator: $a^2 - 8a + 15 = (a - 3)(a - 5)$
Cancel the common factor $(a - 3)$ : $\frac{(a - 3)(a + 2)}{(a - 3)(a - 5)} = \frac{a + 2}{a - 5}$

Example 2: Multiplying and dividing algebraic fractions

When multiplying fractions, multiply the numerators together and multiply the denominators together. For division, multiply by the reciprocal of the second fraction.

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Worked Example: Multiplication and Division

(i) Multiplying $\frac{2}{3} \times \frac{9}{14}$ :

Multiply numerators: $2 \times 9 = 18$
Multiply denominators: $3 \times 14 = 42$
Simplify: $\frac{18}{42} = \frac{3}{7}$

(ii) Dividing $\frac{3}{4} \div \frac{9}{16}$ :

Change division to multiplication: $\frac{3}{4} \times \frac{16}{9}$
Multiply: $\frac{3 \times 16}{4 \times 9} = \frac{48}{36} = \frac{4}{3}$

(iii) More complex algebraic multiplication: For expressions like $\frac{3a^2b}{2c} \times \frac{4c^3}{9ab}$ , multiply the fractions and then simplify by cancelling common factors.

Example 3: Adding and subtracting algebraic fractions

When adding or subtracting fractions, you need a common denominator. This is often the least common multiple of the individual denominators.

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Worked Example: Addition and Subtraction

(i) Adding $\frac{2}{3} + \frac{3}{4}$ :

Find common denominator: LCM of 3 and 4 is 12
Convert fractions: $\frac{2}{3} = \frac{8}{12}$ and $\frac{3}{4} = \frac{9}{12}$
Add: $\frac{8}{12} + \frac{9}{12} = \frac{17}{12}$

(ii) Adding algebraic fractions like $\frac{5x}{6} + \frac{x}{4}$ :

Find common denominator: LCM of 6 and 4 is 12
Convert: $\frac{5x}{6} = \frac{10x}{12}$ and $\frac{x}{4} = \frac{3x}{12}$
Add: $\frac{10x}{12} + \frac{3x}{12} = \frac{13x}{12}$

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For more complex expressions involving brackets in denominators, you need to be especially careful with finding the correct common denominator. Take your time to identify all the factors involved.

Common mistakes to avoid

Be particularly careful with these frequent errors:

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Mistake 1: Incorrect cancellation

Wrong approach: Trying to cancel terms that are not common factors

Example: In $\frac{2x + 2}{3x + 3}$ , you cannot simply cancel the x terms or the number terms separately

Correct approach: Factor first, then cancel complete common factors

$\frac{2x + 2}{3x + 3} = \frac{2(x + 1)}{3(x + 1)} = \frac{2}{3}$

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Mistake 2: Forgetting to factor completely

Wrong approach: Only partially factoring expressions before attempting to simplify

Correct approach: Always factor both numerator and denominator completely before looking for common factors to cancel

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Mistake 3: Arithmetic errors with signs

When working with negative terms, be extra careful with your signs throughout the calculation process. Double-check each step to ensure you haven't accidentally changed a positive to negative or vice versa.

Key techniques to remember

Understanding these fundamental techniques will help you approach any algebraic fraction with confidence:

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Always factor first - Before attempting any simplification, factor both the numerator and denominator completely. This is your foundation for success.

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Look for patterns - Common factoring patterns include difference of squares, perfect square trinomials, and common factor extraction. Recognising these patterns will speed up your work significantly.

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Check your work - Substitute a simple value (like $x = 1$ ) into both your original expression and simplified answer to verify they give the same result. This is a powerful way to catch errors.

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Work systematically - Don't rush; take each step methodically to avoid errors. It's better to work slowly and correctly than quickly and incorrectly.

Remember!

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Key Points to Remember:

Algebraic fractions follow the same rules as numerical fractions - if you can work with regular fractions, you can handle algebraic ones too!
Factor completely before simplifying - this is the most crucial step and prevents most common errors
Only cancel common factors - you can only cancel terms that appear as factors in both the numerator and denominator
Find common denominators for addition and subtraction - just like with numerical fractions, you need the same denominator before you can add or subtract
Check your final answer - make sure it's in its simplest form and that you haven't made any sign errors along the way

Simplifying algebraic fractions (AQA GCSE Further Maths): Revision Notes