Lecture on Fractions: Addition, Subtraction, Multiplication, and Division

Adding Fractions

• Example 1:

  • Add (\frac{3}{5} + \frac{4}{7})
  • Common denominator: 35 (5 x 7)
  • Calculation:
    • (3 \times 7 = 21)
    • (4 \times 5 = 20)
    • (21 + 20 = 41)
  • Result: (\frac{41}{35})

• Example 2:

  • Subtract (\frac{7}{8} - \frac{2}{9})
  • Common denominator: 72 (8 x 9)
  • Calculation:
    • (7 \times 9 = 63)
    • (8 \times 2 = 16)
    • (63 - 16 = 47)
  • Result: (\frac{47}{72})

Adding and Subtracting Multiple Fractions

• Example:

  • Calculate (\frac{3}{4} + \frac{5}{3} - \frac{7}{2})
  • Identify Least Common Denominator (LCD): 12
  • Multiplication for each fraction to get the same denominator:
    • (\frac{3}{4} \to \frac{9}{12} (\times 3))
    • (\frac{5}{3} \to \frac{20}{12} (\times 4))
    • (\frac{7}{2} \to \frac{42}{12} (\times 6))
  • Combine numerators:
    • (9 + 20 - 42 = -13)
  • Result: (-\frac{13}{12})

• Practice Example:

  • Calculate (\frac{8}{5} - \frac{2}{3} + \frac{9}{4})
  • Use common denominator: 60
  • Multiplication:
    • (\frac{8}{5} \to \frac{96}{60} (\times 12))
    • (\frac{2}{3} \to \frac{40}{60} (\times 20))
    • (\frac{9}{4} \to \frac{135}{60} (\times 15))
  • Combine numerators:
    • (96 - 40 + 135 = 191)
  • Result: (\frac{191}{60})

Multiplying Fractions

• Process: Multiply numerators and denominators across fractions.

• Example:

  • Multiply (\frac{3}{5} \times \frac{7}{2})
  • Result: (\frac{21}{10})

• Simplifying Larger Numbers:

  • Example: Multiply (\frac{24}{30} \times \frac{45}{27})
  • Break numbers down:
    • 24 = 6 x 4, 27 = 9 x 3
    • 45 = 9 x 5, 30 = 6 x 5
  • Cancel common factors (5, 9, 6)
  • Result: (\frac{4}{3})

• Practice Example:

  • Multiply (\frac{56}{77} \times \frac{35}{40})
  • Simplify:
    • 56 = 8 x 7, 77 = 11 x 7
    • 35 = 7 x 5, 40 = 8 x 5
  • Cancel common factors (8, 7, 5)
  • Result: (\frac{7}{11})

Dividing Fractions

• Keep, Change, Flip Method:

  • "Keep" the first fraction, "Change" division to multiplication, "Flip" the second fraction.

• Example:

  • Divide (\frac{8}{5} \div \frac{12}{7})
  • Keep: (\frac{8}{5}), Change: (\div \to \times), Flip: (\frac{7}{12})
  • Simplify before multiplying:
    • Cancel common factor (4)
    • Result: (\frac{14}{15})

• Another Example:

  • Divide (\frac{4}{3} \div \frac{9}{5})
  • Result: (\frac{20}{27})

• Complex Example:

  • Divide (\frac{36}{54} \div \frac{64}{48})
  • Simplify using Keep, Change, Flip:
    • Cancel common factors (9, 16, 4)
    • Simplified Result: (\frac{1}{2})