Class Note: Dividing Fractions

Objective

Learn how to divide a fraction by another fraction using the "keep, change, flip" method, including simplification and cancellation techniques.

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Key Steps for Dividing Fractions

1. Keep, Change, Flip Rule

   • Keep the first fraction as it is.
   • Change the division sign to multiplication.
   • Flip (take the reciprocal of) the second fraction.

2. Simplify where possible:

   • Before or after multiplying, look for opportunities to cancel terms to make multiplication and final answers simpler.

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Examples

Example 1

Problem:
(\frac{4}{7} \div \frac{3}{7})

Steps:

• Keep: (\frac{4}{7})
• Change: Division to multiplication
• Flip: (\frac{7}{3})
• Multiply: (\frac{4}{7} \times \frac{7}{3})
• Cancel (7):
  (\frac{4}{\cancel{7}} \times \frac{\cancel{7}}{3} = \frac{4}{3})
• Answer: (\frac{4}{3})

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Example 2

Problem:
(\frac{4}{9} \div \frac{2}{3})

Steps:

• Keep: (\frac{4}{9})
• Change: Division to multiplication
• Flip: (\frac{3}{2})
• Multiply: (\frac{4}{9} \times \frac{3}{2})
• Multiply across: Numerators: (4 \times 3 = 12), Denominators: (9 \times 2 = 18)
• Simplify:
  (\frac{12}{18}) (Divide both by 6)
• (\frac{12}{6} = 2), (\frac{18}{6} = 3)
• Answer: (\frac{2}{3})

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Example 3

Problem:
(\frac{4}{7} \div \frac{8}{21})

Steps:

• Keep: (\frac{4}{7})
• Change: Division to multiplication
• Flip: (\frac{21}{8})
• Multiply: (\frac{4}{7} \times \frac{21}{8})
• Factor: (21 = 7 \times 3), (8 = 4 \times 2)
• Cancel (4) and (7):
  • (\frac{\cancel{4}}{7} \times \frac{21}{\cancel{8}}) becomes
  • (\frac{1}{7} \times \frac{21}{2}) (after cancellation)
  • Or, after full reduction, simply multiply:
    (\frac{3}{2})
• Answer: (\frac{3}{2})

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Summary

• To divide fractions, remember "keep, change, flip".
• Always look to simplify fractions by cancelling common factors before multiplying.
• This method reduces large numbers and simplifies answers efficiently.