Reciprocal Definition: A reciprocal of a fraction is what you get when you switch the numerator (top number) and the denominator (bottom number) of the fraction.
Example: Reciprocal of (\frac{1}{2}) is (\frac{2}{1}).
Multiplying a fraction by its reciprocal equals 1 (e.g., (\frac{1}{2} \times \frac{2}{1} = 1)).
Dividing Fractions Using Reciprocals
Reciprocal Trick: Instead of dividing by a fraction, multiply by its reciprocal.
Multiplying by a fraction is intuitive and simplifies the division process.
Key Concept: Dividing by a fraction (\frac{a}{b}) is equivalent to multiplying by its reciprocal (\frac{b}{a}).
Examples
Example 1: (\frac{3}{4} \div \frac{2}{7})
Re-write as a multiplication problem: Instead of dividing by (\frac{2}{7}), multiply by (\frac{7}{2}).
Multiply Fractions:
Multiply numerators: (3 \times 7 = 21).
Multiply denominators: (4 \times 2 = 8).
Result: (\frac{21}{8}).
Example 2: (\frac{15}{16} \div \frac{9}{22})
Re-write as a multiplication problem: Change to (\frac{15}{16} \times \frac{22}{9}).
Multiply Fractions:
Numerators: (15 \times 22 = 330).
Denominators: (16 \times 9 = 144).
Result: (\frac{330}{144}) (can be simplified).
Special Cases: Division of Complex Fractions
Complex Fractions: Sometimes, fractional division problems involve fractions as numerators and denominators.
Example: (\frac{\frac{2}{3}}{\frac{4}{5}}) can be rewritten by multiplying (\frac{2}{3}) by the reciprocal of (\frac{4}{5}), which is (\frac{5}{4}).
Result:
Multiply: (2 \times 5 = 10).
Multiply: (3 \times 4 = 12).
Answer: (\frac{10}{12}).
Conclusion
Key Takeaway: Dividing fractions is simplified to multiplying by the reciprocal.
Practice: Practice division exercises to reinforce the concept.
Further Learning
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