Lecture Notes: Adding, Subtracting, Multiplying, and Dividing Fractions

Adding Fractions

• Example: ( \frac{3}{5} + \frac{4}{7} )
  • Find a common denominator by multiplying denominators: ( 5 \times 7 = 35 ).
  • Multiply numerators by the opposite denominator:
    • ( 3 \times 7 = 21 )
    • ( 5 \times 4 = 20 )
  • Add the results: ( 21 + 20 = 41 ).
  • Result: ( \frac{41}{35} ).

Subtracting Fractions

• Example: ( \frac{7}{8} - \frac{2}{9} )
  • Find a common denominator: ( 8 \times 9 = 72 ).
  • Calculate modified numerators:
    • ( 7 \times 9 = 63 )
    • ( 8 \times 2 = 16 )
  • Subtract: ( 63 - 16 = 47 ).
  • Result: ( \frac{47}{72} ).

Adding/Subtracting Three Fractions

• Example: ( \frac{3}{4} + \frac{5}{3} - \frac{7}{2} )
  • Find the least common multiple (LCM) of denominators (2, 3, 4):
    • LCM is 12.
  • Adjust each fraction:
    • ( \frac{3 \times 3}{4 \times 3} = \frac{9}{12} )
    • ( \frac{5 \times 4}{3 \times 4} = \frac{20}{12} )
    • ( \frac{7 \times 6}{2 \times 6} = \frac{42}{12} )
  • Combine numerators: ( 9 + 20 - 42 = -13 ).
  • Result: ( \frac{-13}{12} ).

Multiplying Fractions

• Example: ( \frac{3}{5} \times \frac{7}{2} )

  • Multiply straight across: ( 3 \times 7 = 21 ) and ( 5 \times 2 = 10 ).
  • Result: ( \frac{21}{10} ).

• Simplifying Large Numbers:

  • Break down numbers to simplify before multiplying.
  • Example: Multiply ( \frac{24}{27} \times \frac{45}{30} )
    • Breakdown: 24 = 6*4, 27 = 9*3, 45 = 9*5, 30 = 6*5.
    • Cancel common factors: Result is ( \frac{4}{3} ).

Dividing Fractions

• Example: ( \frac{8}{5} \div \frac{12}{7} )

  • Apply "Keep, Change, Flip":
    • Keep ( \frac{8}{5} ), change division to multiplication, flip the second fraction: ( \frac{7}{12} ).
    • Simplify: Cancel common factors before multiplying.
    • Result: ( \frac{14}{15} ).

• Complex Division:

  • Example: ( \frac{36}{54} \div \frac{64}{48} )
    • Simplify using the "Keep, Change, Flip" principle.
    • Breakdown: 36 = 9*4, 54 = 9*6, 48 = 16*3, 64 = 16*4.
    • Cancel common factors: Result is ( \frac{1}{2} ).