Algebraic fractions

An algebraic fraction is simply a fraction which contains a variable in the numerator, denominator, or both.

Examples of algebraic fractions

• \frac{x+2}{3}
• \frac{5}{y-4}
• \frac{2a+1}{b-3}
• \frac{m+1}{n-2} + \frac{3}{n-2}
• \frac{4x}{x^2 - 9}

Simplifying algebraic fractions

To simplify an algebraic fraction, we follow similar steps to simplifying numerical fractions:

1. Find the highest common factor (HCF) of the numerator and denominator.
2. Divide both the numerator and denominator by the HCF.
3. Write the simplified fraction.

Examples

Simplify \frac{6x}{9}

• The HCF of 6x and 9 is 3.
• Divide both the numerator and denominator by 3:
  • \frac{6x \div 3}{9 \div 3} = \frac{2x}{3}
• Answer: \frac{2x}{3}

Simplify \frac{4y^2 + 8y}{12y}

• The HCF of 4y^2 + 8y and 12y is 4y.
• Divide both the numerator and denominator by 4y:
  • \frac{(4y^2 + 8y) \div 4y}{12y \div 4y} = \frac{y + 2}{3}
• Answer: \frac{y + 2}{3}

Simplify \frac{3a^2 - 6a}{9a}

• The HCF of 3a^2 - 6a and 9a is 3a.
• Divide both the numerator and denominator by 3a:
  • \frac{(3a^2 - 6a) \div 3a}{9a \div 3a} = \frac{a - 2}{3}
• Answer: \frac{a - 2}{3}

Simplify \frac{5x^2 + 10x}{15x}

• The HCF of 5x^2 + 10x and 15x is 5x.
• Divide both the numerator and denominator by 5x:
  • \frac{(5x^2 + 10x) \div 5x}{15x \div 5x} = \frac{x + 2}{3}
• Answer: \frac{x + 2}{3}