Multiplying Fractions and Simplifying

Key Concepts

• Multiplying Fractions: Multiply numerators together and denominators together.
• Simplification: The product needs to be in simplified form or lowest terms.
  • Simplification can occur before or after multiplying.

Examples

Example 1: (\frac{1}{4} \times \frac{3}{5})

• Multiplication:
  • Numerator: (1 \times 3 = 3)
  • Denominator: (4 \times 5 = 20)
• Simplification:
  • Check for common factors between numerator and denominator.
  • No common factors other than 1, so (\frac{3}{20}) is already in lowest terms.
• Meaning:
  • (\frac{3}{20}) is (\frac{1}{4}) of (\frac{3}{5}) or one-fourth of a copy of (\frac{3}{5}).
  • Model by partitioning (\frac{3}{5}) into four equal parts and shading 3 out of 20 total parts.

Example 2: (\frac{1}{8} \times \frac{4}{5})

• Multiplication:
  • Numerator: (1 \times 4 = 4)
  • Denominator: (8 \times 5 = 40)
• Simplification:
  • Check for common factors:
    • (4) and (8) share a common factor (4).
  • Prime Factorization Method:
    • (4 = 2 \times 2)
    • (8 = 2 \times 2 \times 2)
    • Simplify: (\frac{2}{2} = 1)
  • Simplified Result: (\frac{1}{10})
  • Alternative Simplification:
    • Recognize common factor: (4) in both numerator and denominator.
    • Simplify (\frac{4}{8}) to (\frac{1}{2}).
  • Result: (1 \times 1 = 1) and (2 \times 5 = 10).

Simplification Techniques

• Prime Factorization to identify and cancel common factors.
• Recognizing common factors and reducing before multiplying.

Conclusion

• Multiplying fractions involves multiplying across and simplifying.
• Simplifying before multiplying can simplify the process and ensure the product is in lowest terms.