Class Notes: Multiplying and Dividing Fractions

Objective

• Understand how to multiply and divide fractions using direct multiplication and the cancellation method.
• Learn the "keep, change, flip" rule for dividing fractions.

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Multiplying Fractions

Basic Rule

• Multiply the numerators together.
• Multiply the denominators together.

Example 1

• (\frac{2}{5} \times \frac{1}{3})
  • Numerators: (2 \times 1 = 2)
  • Denominators: (5 \times 3 = 15)
  • Answer: (\frac{2}{15})

Example 2 (Using Cancellation Method)

• (\frac{8}{15} \times \frac{5}{12})

Steps:

1. Factor numbers for easier simplification:
   • 8 = 4 × 2
   • 15 = 5 × 3
   • 5 remains 5
   • 12 = 4 × 3
2. Rewrite fractions with factors:
   • (\frac{4 \times 2}{5 \times 3} \times \frac{5}{4 \times 3})
3. Cancel common factors (cross-cancellation):
   • Cancel one 5 from numerator and denominator.
   • Cancel one 4 from numerator and denominator.
   • Cancel one 3 from numerator and denominator.
4. Multiply remaining numbers:
   • Numerators: 2 × 1 = 2 (but after final entry and more cancellations, only 1 may remain)
   • Denominators: 3 × 2 = 6
   • Final Answer: (\frac{1}{6})

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Dividing Fractions

Rule

• "Keep, Change, Flip":
  • Keep the first fraction.
  • Change division to multiplication.
  • Flip (take the reciprocal of) the second fraction.

Example 1

• (\frac{3}{8} \div \frac{9}{16})

Steps:

1. Apply "keep, change, flip":
   • (\frac{3}{8} \times \frac{16}{9})
2. Factor if possible:
   • 8 = 4 × 2
   • 16 = 4 × 4 or 2 × 2 × 2 × 2
   • 9 = 3 × 3
3. Cancel common factors:
   • Cancel a 4, a 2, and a 3 as appropriate.
4. Multiply remaining numbers:
   • Final Answer: (\frac{2}{3})

Example 2

• (\frac{12}{15} \div \frac{32}{35})

Steps:

1. "Keep, change, flip":
   • (\frac{12}{15} \times \frac{35}{32})
2. Factor:
   • 12 = 4 × 3
   • 15 = 5 × 3
   • 35 = 7 × 5
   • 32 = 8 × 4
3. Cancel common factors:
   • Cancel a 4, a 5, and a 3 as appropriate.
4. Multiply remaining numbers:
   • Final Answer: (\frac{7}{8})

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Summary / Key Points

• Multiplying Fractions: Multiply numerators and denominators directly. For large numbers, factorize and use cancellation before multiplying.
• Dividing Fractions: Use "keep, change, flip" to turn division into multiplication by the reciprocal, then solve as a multiplication problem.
• Cancellation: Breaking down numbers into factors helps simplify before multiplication, makes calculations easier, and often avoids large numbers.

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