Dividing Fractions Lecture Notes

Key Concepts

Division of Fractions: To divide fractions, convert the division into a multiplication problem by flipping the second fraction (finding the reciprocal).
Reciprocal: The reciprocal of a fraction is obtained by swapping the numerator and the denominator.
Multiplying Fractions: After converting to multiplication, follow the standard multiplication rules.
Reducing Fractions: Look for ways to simplify or reduce fractions both before and after multiplication.

Convert to Multiplication
- Keep the first fraction.
- Flip the second fraction (find the reciprocal).
- Change the operation from division to multiplication.
Multiply Fractions
- Multiply numerators with numerators and denominators with denominators.
- Example: (\frac{2}{3} \times \frac{8}{5} = \frac{16}{15})
Check for Reduction
- Simplify the resulting fraction if possible.
- Improper fractions can be left as they are for ease of use.

Problem: (\frac{2}{3} \div \frac{5}{8})
Solution:
- Flip (\frac{5}{8}) to (\frac{8}{5}).
- Multiply: (\frac{2}{3} \times \frac{8}{5} = \frac{16}{15}).
- Improper fraction, leave it as is.

Problem: Involving signs.
Solution:
- Negative and positive fraction; result will be negative.
- Example calculation: (8 \times 8 = 64), (3 \times 7 = 21).
- Result: (\frac{64}{21}) (negative).

Problem: (\frac{3}{5} \div -\frac{2}{1})
Solution:
- Consider (\frac{3}{5}) times the reciprocal (-\frac{1}{2}).
- Multiply: (3 \times -1 = -3), (5 \times 2 = 10).
- Result: (-\frac{3}{10}).

Problem: (-\frac{3}{1} \div \frac{12}{5})
Solution:
- Reciprocal is (\frac{5}{12}) for (\frac{12}{5}).
- Simplify: (3 \div 3 = 1), (12 \div 3 = 4).
- Multiply: (-1 \times 5 = -5), (1 \times 4 = 4).
- Result: (-\frac{5}{4}).

Sign Consideration: Pay attention to positive and negative signs when multiplying.
Improper Fractions: Often left in their improper form, especially in advanced mathematics, to facilitate further calculations.