Class Notes: Multiplying Three Fractions (Cancellation Method)

Objective

Understand how to multiply three fractions together efficiently using cancellation before full multiplication.

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Key Steps

1. Multiply Across:

   • When multiplying fractions, multiply the numerators together and multiply the denominators together.
   • Example: (\frac{a}{b} \times \frac{c}{d} \times \frac{e}{f} = \frac{a \times c \times e}{b \times d \times f})

2. Cancellation Before Multiplying:

   • Always cancel (reduce) identical numbers from the top (numerators) and bottom (denominators) before multiplying.
   • This simplifies the work and avoids dealing with large numbers.

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Example 1

Multiply: (\frac{1}{3} \times \frac{5}{7} \times \frac{2}{5})

• Cancellation: 5 cancels in numerator and denominator.
• Multiply Remaining: (1 \times 2 = 2) (top), (3 \times 7 = 21) (bottom)
• Final Answer: (\frac{2}{21})

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Example 2

Multiply: (\frac{8}{6} \times \frac{12}{15} \times \frac{3}{4})

• Break Down Larger Numbers:
  • 8 = 4 × 2
  • 6 = 3 × 2
  • 12 = 4 × 3
  • 15 = 5 × 3
• Cancellation: Cancel 5, 4, 3, and possibly additional pairs.
• Result: After all cancellations, only a 1 remains in the numerator and 4 in the denominator.
• Final Answer: (\frac{1}{4})

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Practice Problems

1. (\frac{4}{7} \times \frac{14}{15} \times \frac{25}{32})

• Break Down Numbers:
  • 14 = 7 × 2
  • 15 = 5 × 3
  • 25 = 5 × 5
  • 32 = 4 × 8 (further, 8 = 4 × 2)
• Cancellation: Cancel 7, 5, 4, 2
• Leftover: Numerator (5), Denominator (3 × 4 = 12)
• Final Answer: (\frac{5}{12})

2. (\frac{8}{9} \times \frac{3}{4} \times \frac{15}{16})

• Break Down Numbers:
  • 8 = 4 × 2
  • 9 = 3 × 3
  • 15 = 5 × 3
  • 16 = 8 × 2 (or 4 × 4)
• Cancellation: Cancel pairs of 2, 3, 4
• Leftover: Numerator (5), Denominator (8)
• Final Answer: (\frac{5}{8})

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Tips

• Always look for ways to factor numbers in numerator and denominator to maximize cancellation.
• Simplifying before multiplying makes calculations and final answers much easier.

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Further Learning & Examples

• For more examples and video explanations, refer to the instructor's website or YouTube channel (search Organic Chemistry Tutor).
• Request topics like dividing fractions or other math concepts for more help.

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