Math Antics: Dividing Fractions

Introduction to Reciprocals

• Definition: A reciprocal is a fraction where the numerator and denominator are swapped.
  • Example: Reciprocal of ( \frac{1}{2} ) is ( \frac{2}{1} ).
• Property: Multiplying a fraction by its reciprocal equals 1.

Dividing Fractions Using Reciprocals

• Key Concept: Instead of dividing by a fraction, multiply by its reciprocal.
  • This works because fractions represent division.
  • Multiplying by ( \frac{1}{2} ) is the same as dividing by 2.
  • Similarly, multiplying by ( \frac{2}{1} ) is the same as dividing by 2.
• Avoid Common Mistakes:
  • Only take the reciprocal of the second fraction (the divisor).

Example Problems

Example 1

• Problem: ( \frac{3}{4} \div \frac{2}{7} )
  • Solution: Re-write as ( \frac{3}{4} \times \frac{7}{2} )
  • Multiply tops: ( 3 \times 7 = 21 )
  • Multiply bottoms: ( 4 \times 2 = 8 )
  • Answer: ( \frac{21}{8} )

Example 2

• Problem: ( \frac{15}{16} \div \frac{9}{22} )
  • Solution: Re-write as ( \frac{15}{16} \times \frac{22}{9} )
  • Multiply tops: ( 15 \times 22 = 330 )
  • Multiply bottoms: ( 16 \times 9 = 144 )
  • Answer: ( \frac{330}{144} )
  • Note: Simplify if necessary.

Special Case: Complex Fractions

• Concept: Fractions can be made up of other fractions, appearing complex due to multiple fraction lines.
• Solution: Re-write as a simpler multiplication problem by taking the reciprocal of the bottom fraction.
  • Example: ( \frac{\frac{2}{3}}{\frac{4}{5}} )
    • Re-write as: ( \frac{2}{3} \times \frac{5}{4} )
    • Multiply tops: ( 2 \times 5 = 10 )
    • Multiply bottoms: ( 3 \times 4 = 12 )

Conclusion

• Dividing fractions is simplified by multiplying by reciprocals.
• Practice problems to reinforce understanding.

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