Lecture Notes: Multiplying and Dividing Fractions

Multiplying Fractions

• Basic Rule: Multiply the numerators together and the denominators together.
  • Example: (\frac{7}{15} \times \frac{3}{4})
    • Multiply numerators: (7 \times 3 = 21)
    • Multiply denominators: (15 \times 4 = 60)
    • Result: (\frac{21}{60})
  • Simplification: Divide both top and bottom by common factor (e.g., 3)
    • Simplified Result: (\frac{7}{20})

Example 2

• (\frac{4}{7} \times \frac{9}{5})
  • Result: (\frac{36}{35})
  • Cannot be simplified further.

Multiplying Mixed Numbers

• Convert mixed numbers to improper fractions first.
  • Example: Multiply (\frac{4}{5}) by (2\frac{3}{4})
    • Convert: (2\frac{3}{4} = \frac{11}{4})
    • Multiply: (\frac{4}{5} \times \frac{11}{4} = \frac{44}{20})
    • Simplify: (\frac{11}{5})

Multiplying Fractions < 1

• Result is smaller than both fractions.
  • Example: (\frac{1}{2} \times \frac{1}{3} = \frac{1}{6})

Dividing Fractions

• Use the reciprocal of the second fraction and multiply.
  • Example: (\frac{3}{4} \div \frac{5}{9})
    • Reciprocal: (\frac{9}{5})
    • Multiply: (\frac{3}{4} \times \frac{9}{5} = \frac{27}{20})

Example 2

• (\frac{2}{3} \div \frac{4}{5})
  • Reciprocal: (\frac{5}{4})
  • Multiply: (\frac{2}{3} \times \frac{5}{4} = \frac{10}{12})
  • Simplify: (\frac{5}{6})

Dividing with Mixed Numbers

• Convert mixed numbers to improper fractions first.
  • Example: (3\frac{1}{2} \div \frac{2}{5})
    • Convert: (3\frac{1}{2} = \frac{7}{2})
    • Reciprocal: (\frac{5}{2})
    • Multiply: (\frac{7}{2} \times \frac{5}{2} = \frac{35}{4})
    • Convert to Mixed Number: (8\frac{3}{4})

• Summary: This lecture covered the fundamental procedures for multiplying and dividing fractions, including dealing with mixed numbers and simplification steps. Understanding these techniques is crucial for effectively working with fractional values.