Lecture Notes: Adding and Subtracting Fractions

Key Concepts

• Common Denominator: Essential for adding or subtracting fractions. Fractions must have the same denominator to be combined.
• Least Common Denominator (LCD): Ideally, use the smallest common multiple of the denominators.
• Equivalent Fractions: Fractions with different numerators and denominators but represent the same value.

Process for Adding/Subtracting Fractions

1. Find a Common Denominator: Ensure fractions have the same denominator.
   • Multiply fractions to achieve a common denominator.
   • Preferably use the least common denominator.
2. Perform the Operation: Add or subtract the numerators.
   • Keep the denominator the same.
3. Simplify the Result: Check if the resulting fraction can be simplified.

Example Problems

Example 1: Adding Fractions

• Problem: (\frac{1}{2} + \frac{1}{4})
  • Least Common Denominator: 4.
  • Convert (\frac{1}{2}) to (\frac{2}{4}).
  • Add: (\frac{2}{4} + \frac{1}{4} = \frac{3}{4}).

Example 2: Common Error

• Error: Adding numerators and denominators directly.
  • Incorrect: (\frac{1}{2} + \frac{1}{4} = \frac{2}{6}).
  • Correct: Use common denominator.

Example 3: Adding with Different Denominators

• Problem: (\frac{1}{4} + \frac{5}{8})
  • LCD: 8.
  • Convert (\frac{1}{4}) to (\frac{2}{8}).
  • Add: (\frac{2}{8} + \frac{5}{8} = \frac{7}{8}).

Example 4: Subtracting Fractions

• Problem: (\frac{5}{6} - \frac{5}{12})
  • LCD: 12.
  • Convert (\frac{5}{6}) to (\frac{10}{12}).
  • Subtract: (\frac{10}{12} - \frac{5}{12} = \frac{5}{12}).

Example 5: Improper Fractions

• Problem: (\frac{5}{7} + \frac{3}{5})
  • LCD: 35.
  • Convert (\frac{5}{7}) to (\frac{25}{35}) and (\frac{3}{5}) to (\frac{21}{35}).
  • Add: (\frac{25}{35} + \frac{21}{35} = \frac{46}{35}).
  • Convert to Mixed Number: 1 (\frac{11}{35}).

Example 6: Subtracting with Different Denominators

• Problem: (\frac{1}{3} - \frac{1}{4})
  • LCD: 12.
  • Convert (\frac{1}{3}) to (\frac{4}{12}) and (\frac{1}{4}) to (\frac{3}{12}).
  • Subtract: (\frac{4}{12} - \frac{3}{12} = \frac{1}{12}).

Conclusion

• Always use a common denominator to add or subtract fractions.
• Simplify answers where possible.
• Convert improper fractions to mixed numbers if required.

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These notes aim to provide a comprehensive overview of the process for adding and subtracting fractions while highlighting common errors and best practices.