Algebraic Fractions

Summary

Algebraic fractions involve operations like addition, subtraction, multiplication, and division with fractions that contain algebraic expressions.

• Lowest Common Multiple (LCM) — the smallest multiple that is exactly divisible by each denominator.
  Example: The LCM of x and 2y is 2xy.
• Addition and Subtraction — requires finding the LCM of the denominators and adjusting them to be the same before performing the operation on the numerators.
  Example: To add $\frac{1}{x}$ and $\frac{1}{2y}$, find the LCM which is 2xy.
• Multiplication and Division — do not require the denominators to be the same; simply multiply or divide the numerators and denominators.
  Example: $\frac{3}{4} \times \frac{2}{3} = \frac{6}{12}$.
• Simplification — involves factorizing the numerator and denominator and cancelling out common factors.
  Example: Simplify $\frac{6x}{9}$ by cancelling the common factor 3 to get $\frac{2x}{3}$.

Exam Tips

Key Definitions to Remember

• Lowest Common Multiple (LCM)
• Factorization

Common Confusions

• Forgetting to find the LCM for addition and subtraction
• Trying to make denominators the same for multiplication and division

Typical Exam Questions

• How do you add $\frac{1}{x}$ and $\frac{1}{2y}$?
  Find the LCM of x and 2y, adjust the fractions, then add.
• Simplify $\frac{6x}{9}$.
  Factorize and cancel common factors to get $\frac{2x}{3}$.
• What is the result of $\frac{3}{4} \times \frac{2}{3}$?
  Multiply to get $\frac{6}{12}$ and simplify to $\frac{1}{2}$.

What Examiners Usually Test

• Ability to find and use the LCM for addition and subtraction
• Correct simplification of algebraic fractions
• Understanding when to factorize and cancel common terms