Class Note: Subtracting Three Fractions with Unlike Denominators

Objective

Learn how to subtract three fractions with different denominators by finding a common denominator.

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Key Steps

1. Find a Common Denominator

• Multiply all denominators together (not always the least common denominator [LCD], but it will work).
• Example: For denominators 6, 5, 4:
  • (6 \times 5 = 30)
  • (30 \times 4 = 120)
  • Common denominator = 120

2. Adjust Each Fraction

• For each fraction, multiply the numerator and denominator by the product of the other two denominators:
  • First fraction: Multiply by (5 \times 4 = 20)
  • Second fraction: Multiply by (6 \times 4 = 24)
  • Third fraction: Multiply by (6 \times 5 = 30)
• Ensures all fractions have the same denominator.

3. Calculate Equivalent Fractions

• After adjustment, all denominators = 120.
• Example for clarification:
  • If numerators become: 100, 48, 30 (from calculations as per lecture)
  • Subtract: (100 - 48 = 52), then (52 - 30 = 22)
  • Fraction becomes: (\frac{22}{120})

4. Simplify the Final Fraction

• Check if numerator and denominator share a common factor.
• In this case: (22 = 11 \times 2, 120 = 60 \times 2), so cancel out 2.
• Result: (\frac{11}{60}) (simplest form).

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Example 2

Subtract:
[ \frac{7}{9} - \frac{1}{3} - \frac{2}{5} ]

1. Find LCD

• Multiplying all denominators: (9 \times 3 \times 5 = 135)
• However, since 3 is a factor of 9, we can use (9 \times 5 = 45)
• LCD = 45

2. Adjust Each Fraction

• (\frac{7}{9} \rightarrow \frac{35}{45}) ((7 \times 5), (9 \times 5))
• (\frac{1}{3} \rightarrow \frac{15}{45}) ((1 \times 15), (3 \times 15))
• (\frac{2}{5} \rightarrow \frac{18}{45}) ((2 \times 9), (5 \times 9))

3. Subtract Numerators

• (35 - 15 = 20)
• (20 - 18 = 2)
• Result: (\frac{2}{45})

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Summary

• Always adjust each fraction so they share a common denominator.
• Multiply numerators and denominators appropriately for conversion.
• Subtract only the numerators, keep the common denominator.
• Simplify your final fraction if possible.

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Visual Shortcut

Subtract:     a     b     c
            --- - --- - ---
             d     e     f

Common denominator: d × e × f (or least common multiple)

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Practice

Try subtracting:
[ \frac{5}{8} - \frac{1}{4} - \frac{3}{10} ]

• Find LCD, convert, subtract, and simplify!