Lecture Notes: Understanding and Simplifying Fractions

Introduction to Fractions

• Definition: Fractions represent parts of a whole number.
  • Examples: one half (1/2), three quarters (3/4).
• Components of a Fraction:
  • Numerator: The number on top, indicating how many parts you have.
  • Denominator: The number on the bottom, indicating how many total parts make up a whole.

Example:

• A pizza divided into 4 parts, giving 3 parts to someone.
  • Fraction: 3/4 (three quarters of the pizza).
  • Remaining: 1/4 (one quarter).

Simplifying Fractions

• Objective: Reduce the numerator and denominator to the smallest possible numbers.
• Method:
  • Divide both the numerator and denominator by the same number.

Example:

• Fraction: 12/18
  • Divide by 6: 12 ÷ 6 = 2 and 18 ÷ 6 = 3
  • Simplified Fraction: 2/3
• Alternative Method:
  • Divide by smaller numbers incrementally (e.g., divide by 2, then by 3).

Key Points in Simplification

• Equivalence: Simplified fractions represent the same quantity as their original form.
  • Example: 12/18, 6/9, and 2/3 are all equivalent.
• Visual Representation:
  • Fractions can be visually represented (e.g., as rings divided into sections) to illustrate equivalence.

Practice Problems

• Simplify 4/12:
  • Quick Approach: Divide by 4 → Result: 1/3
  • Step-by-Step: Divide by 2 → 2/6, then divide by 2 again → 1/3
• Simplify 20/25:
  • Only common factor is 5.
  • Divide by 5 → Result: 4/5 (four fifths)

Conclusion

• Simplifying fractions is about finding equivalent, smaller fractions.
• Important to divide both numerator and denominator by the same number.
• Practice makes perfect in identifying the common factors quickly.