Simplification of Fractions Revision Notes for Grade 10 NSC Matric Mathematics

Simplification of Fractions

Basic fraction operations with algebraic expressions

Before tackling complex algebraic fractions, let's review the fundamental rules for working with fractions that contain variables.

Key fraction operations

Multiplication of fractions: $\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$ where $b \neq 0$ and $d \neq 0$

Addition of fractions with same denominators: $\frac{a}{b} + \frac{c}{b} = \frac{a+c}{b}$ where $b \neq 0$

Division of fractions: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}$ where $b \neq 0$ , $c \neq 0$ , and $d \neq 0$

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Important rule: Dividing by a fraction is equivalent to multiplying by its reciprocal. This is a fundamental concept that will be used frequently in algebraic fraction problems.

Understanding algebraic fractions

Algebraic fractions are fractions where the numerator, denominator, or both contain algebraic expressions. To simplify these fractions effectively, you must first factorise both the numerator and denominator completely.

When fractions become undefined

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A fraction becomes undefined when its denominator equals zero. Always identify these restriction values before beginning your simplification process.

For example, in the fraction $\frac{x^2 + 3x}{x + 3}$ , if $x = -3$ , then the denominator becomes zero, making the fraction undefined.

Step-by-step approach to simplifying algebraic fractions

Method 1: Single fraction simplification

Step 1: Factorise both the numerator and denominator completely

Step 2: Identify any common factors between numerator and denominator

Step 3: Cancel the common factors

Step 4: Write your final simplified answer

Worked examples

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Worked Example 1: Basic factorisation and cancellation

Question: Simplify $\frac{ax - b + x - ab}{ax^2 - abx}$ where $x \neq 0$ and $x \neq b$

Solution:

Step 1: Use grouping to factorise the numerator and identify the common factor $ax$ in the denominator

$\frac{(axe - ab) + (x - b)}{ax^2 - abx} = \frac{a(x - b) + (x - b)}{ax(x - b)}$

Step 2: Take out the common factor $(x - b)$ in the numerator

$= \frac{(x - b)(a + 1)}{ax(x - b)}$

Step 3: Cancel the common factor $(x - b)$ from numerator and denominator

$= \frac{a + 1}{ax}$

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Worked Example 2: Division of algebraic fractions

Question: Simplify $\frac{x^2 - x - 2}{x^2 - 4} \div \frac{x^2 + x}{x^2 + 2x}$ where $x \neq 0$ , $x \neq \pm 2$

Solution:

Step 1: Factorise the numerators and denominators

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

Step 2: Change division to multiplication by the reciprocal

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

Step 3: Cancel common factors and write the final answer

$= 1$

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Worked Example 3: Addition and subtraction of algebraic fractions

Question: Simplify $\frac{x - 2}{x^2 - 1} + \frac{x^2}{x - 2} - \frac{x^3 + x - 4}{x^2 - 4}$ where $x \neq \pm 2$

Solution:

Step 1: Factorise all denominators

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

Step 2: Find the lowest common denominator: $(x + 2)(x - 2)$

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

Step 3: Combine as a single fraction

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

Step 4: Expand and simplify the numerator

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

Step 5: Factor out common terms and write the final answer

$\frac{2(x^2 + 1)}{(x + 2)(x - 2)}$

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Worked Example 4: Complex fraction addition and subtraction

Question: Simplify $\frac{2}{x^2 - x} + \frac{x^2 + x + 1}{x^3 - 1} - \frac{x}{x^2 - 1}$ where $x \neq 0$ and $x \neq \pm 1$

Solution:

Step 1: Factorise numerators and denominators

$\frac{2}{x(x - 1)} + \frac{x^2 + x + 1}{(x - 1)(x^2 + x + 1)} - \frac{x}{(x - 1)(x + 1)}$

Step 2: Find the common denominator and combine fractions

$\frac{2(x + 1) + x(x + 1) - x^2}{x(x - 1)(x + 1)}$

Step 3: Simplify the numerator and write the final answer

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

Key exam tips

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Essential strategies for success:

Always factorise first before attempting any simplification
Identify restriction values where denominators become zero
When dividing fractions, multiply by the reciprocal
For addition/subtraction, find the lowest common denominator
Check your final answer by substituting a simple value for the variable
Show all working clearly - partial marks are often awarded for correct methods

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Key Points to Remember:

Factorisation is essential - always factorise numerators and denominators completely before simplifying
Division of fractions means multiplying by the reciprocal of the second fraction
Common factors in numerator and denominator can be cancelled, but never cancel terms that are added or subtracted
Undefined fractions
occur when denominators equal zero - always state these restrictions
Lowest common denominators are needed when adding or subtracting fractions with different denominators

Simplification of Fractions (Grade 10 NSC Matric Mathematics): Revision Notes