Class Notes: Adding and Subtracting Fractions with Unlike Denominators

Objective

Learn how to add and subtract fractions with different denominators.

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Key Concepts

• Common Denominator: To add or subtract fractions with unlike denominators, the denominators must be the same.
• Steps:
  1. Find a Common Denominator:
     • A quick method: Multiply the two denominators together.
     • For least simplifying later, find the Least Common Denominator (LCD).
  2. Rewrite the Fractions:
     • Multiply each fraction’s numerator and denominator so that both fractions have the common denominator.
     • For the first fraction, multiply numerator and denominator by the denominator of the second fraction (and vice versa).
  3. Add or Subtract the Numerators:
     • Once denominators are the same, add or subtract the numerators as needed.
     • Keep the denominator the same.
  4. Simplify the Result:
     • If possible, simplify the final fraction.

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Example 1: Addition

• Problem: (\frac{2}{7} + \frac{1}{3})
  • Find common denominator: (7 \times 3 = 21)
  • Adjust fractions:
    • (\frac{2}{7} \times \frac{3}{3} = \frac{6}{21})
    • (\frac{1}{3} \times \frac{7}{7} = \frac{7}{21})
  • Add numerators: (6 + 7 = 13)
  • Answer: (\frac{13}{21})

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Example 2: Addition

• Problem: (\frac{1}{6} + \frac{3}{5})
  • Find common denominator: (6 \times 5 = 30)
  • Adjust fractions:
    • (\frac{1}{6} \times \frac{5}{5} = \frac{5}{30})
    • (\frac{3}{5} \times \frac{6}{6} = \frac{18}{30})
  • Add numerators: (5 + 18 = 23)
  • Answer: (\frac{23}{30})

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Example 3: Subtraction

• Problem: (\frac{5}{8} - \frac{3}{7})
  • Find common denominator: (8 \times 7 = 56)
  • Adjust fractions:
    • (\frac{5}{8} \times \frac{7}{7} = \frac{35}{56})
    • (\frac{3}{7} \times \frac{8}{8} = \frac{24}{56})
  • Subtract numerators: (35 - 24 = 11)
  • Answer: (\frac{11}{56}) (already simplified)

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Example 4: Subtraction

• Problem: (\frac{8}{9} - \frac{2}{3})
  • Find common denominator:
    • (9 \times 3 = 27) is a common denominator, but (9) is the LCD.
  • Using 27 as the common denominator:
    • (\frac{8}{9} \times \frac{3}{3} = \frac{24}{27})
    • (\frac{2}{3} \times \frac{9}{9} = \frac{18}{27})
  • Subtract numerators: (24 - 18 = 6)
  • Fraction: (\frac{6}{27})
  • Simplify: (\frac{6}{27} = \frac{2}{9}) (divide numerator and denominator by 3)

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Important Tips

• Always check if the result can be simplified.
• Using the least common denominator (LCD) reduces the need for simplification at the end.
• Multiply each numerator and denominator carefully to avoid mistakes in the calculation.

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Summary

To add or subtract fractions with different denominators:

1. Find a common denominator (preferably the LCD).
2. Rewrite each fraction with the common denominator.
3. Add or subtract numerators.
4. Simplify the final fraction if needed.