Adding Algebraic Fractions Revision Notes for Leaving Cert Mathematics

Adding Algebraic Fractions

What are algebraic fractions?

Algebraic fractions are fractions that contain algebraic expressions (with variables like x, y, etc.) in either the numerator, denominator, or both. Examples include $\frac{3x}{4}$ , $\frac{x+2}{x-1}$ , and $\frac{2x-5}{3}$ .

Understanding algebraic fractions is essential because they follow the same fundamental principles as numerical fractions, but require additional algebraic manipulation skills.

The fundamental rule

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The key principle for adding algebraic fractions is exactly the same as for numerical fractions:

Algebraic fractions can be added or subtracted in the same way as numerical fractions.

This means we must have a common denominator before we can add or subtract the fractions.

Method for adding algebraic fractions

The systematic approach to adding algebraic fractions involves four clear steps that ensure accuracy and completeness.

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Four-Step Method for Adding Algebraic Fractions:

Step 1: Find the common denominator

For numerical denominators: Find the Least Common Multiple (LCM)
For algebraic denominators: Find the LCM of the algebraic expressions

Step 2: Convert each fraction Express each fraction using the common denominator by multiplying both numerator and denominator by the appropriate factor.

Step 3: Add or subtract Once the denominators are the same, add or subtract the numerators and keep the common denominator.

Step 4: Simplify Always simplify the final answer where possible.

Adding fractions with numerical denominators

When adding fractions like $\frac{3}{4} + \frac{2}{3}$ , we find the LCM of 4 and 3, which is 12.

$\frac{3}{4} + \frac{2}{3} = \frac{9}{12} + \frac{8}{12} = \frac{17}{12}$

This can be done more efficiently by using the direct method: $\frac{3}{4} + \frac{2}{3} = \frac{3(3) + 2(4)}{12} = \frac{9 + 8}{12} = \frac{17}{12}$

Similarly, for subtraction: $\frac{6}{7} - \frac{2}{3} = \frac{6(3) - 2(7)}{21} = \frac{18 - 14}{21} = \frac{4}{21}$

Worked examples

Example 1: Fractions with numerical denominators

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Worked Example: Numerical Denominators

Express as a single fraction: $\frac{4x - 3}{4} - \frac{x}{3}$

Solution: The LCM of 3 and 4 is 12.

$\frac{4x - 3}{4} - \frac{x}{3} = \frac{3(4x - 3) - 4(x)}{12}$

$= \frac{12x - 9 - 4x}{12} = \frac{8x - 9}{12}$

Example 2: Fractions with algebraic denominators

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Worked Example: Algebraic Denominators

Express as a single fraction: $\frac{5}{x + 3} - \frac{2}{x - 4}$

Solution: The LCM of $(x + 3)$ and $(x - 4)$ is $(x + 3)(x - 4)$ .

$\frac{5}{x + 3} - \frac{2}{x - 4} = \frac{5(x - 4) - 2(x + 3)}{(x + 3)(x - 4)}$

$= \frac{5x - 20 - 2x - 6}{(x + 3)(x - 4)} = \frac{3x - 26}{(x + 3)(x - 4)}$

Example 3: Mixed numerical and algebraic

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Worked Example: Mixed Denominators

Express as a single fraction: $\frac{2x + 3}{4} + \frac{x}{3}$

Solution: The LCM of 4 and 3 is 12.

$\frac{2x + 3}{4} + \frac{x}{3} = \frac{3(2x + 3) + 4(x)}{12}$

$= \frac{6x + 9 + 4x}{12} = \frac{10x + 9}{12}$

Example 4: Complex algebraic denominators

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Worked Example: Complex Algebraic Terms

Express as a single fraction: $\frac{1}{x + 3} + \frac{1}{x}$

Solution: The LCM of $(x + 3)$ and $x$ is $x(x + 3)$ .

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

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

Example 5: Addition with factorisation

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Worked Example: Factorization Required

Express as a single fraction: $\frac{4}{2x - 1} + \frac{3}{2x - 3}$

Solution: The LCM of $(2x - 1)$ and $(2x - 3)$ is $(2x - 1)(2x - 3)$ .

$\frac{4}{2x - 1} + \frac{3}{2x - 3} = \frac{4(2x - 3) + 3(2x - 1)}{(2x - 1)(2x - 3)}$

$= \frac{8x - 12 + 6x - 3}{(2x - 1)(2x - 3)} = \frac{14x - 15}{(2x - 1)(2x - 3)}$

Common exam traps and tips

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Exam Tip: Always check if your final answer can be simplified further by factoring.

Common Mistake: Forgetting to multiply both the numerator and denominator by the same factor when converting to a common denominator.

Memory Aid: Remember "LCD First" - find the Least Common Denominator first, then convert all fractions.

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Exam Strategy: When dealing with complex algebraic denominators, write out the LCM clearly before proceeding with the calculation.

Check Your Work: Substitute a simple value (like x = 1) into both your original expression and final answer to verify they give the same result.

Summary

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Key Points to Remember:

Algebraic fractions follow the same rules as numerical fractions - you must have a common denominator before adding or subtracting
Find the LCM first - for numerical denominators use standard LCM, for algebraic denominators multiply the different factors together
Convert all fractions to have the common denominator by multiplying both numerator and denominator by the appropriate factor
Add or subtract numerators only - the denominator stays the same once you have a common one
Always simplify your final answer by factoring and cancelling where possible

Adding Algebraic Fractions (Leaving Cert Mathematics): Revision Notes