Lesson Notes: Adding Fractions with Unlike Denominators

Objective

Learn how to add two fractions that have different denominators by finding a common denominator.

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Key Steps

1. Identify the Fractions

   • Example 1: ( \frac{2}{3} + \frac{1}{4} )
   • Example 2: ( \frac{1}{5} + \frac{2}{7} )

2. Find a Common Denominator

   • Multiply each fraction by a form of 1 using the denominator of the other fraction.
   • This ensures both fractions have the same denominator (the least common multiple, or simply the product, of both denominators).

3. Multiply to Get Equivalent Fractions

   • For ( \frac{2}{3} + \frac{1}{4} ):
     • Multiply ( \frac{2}{3} ) by ( \frac{4}{4} ):
       ( \frac{2 \times 4}{3 \times 4} = \frac{8}{12} )
     • Multiply ( \frac{1}{4} ) by ( \frac{3}{3} ):
       ( \frac{1 \times 3}{4 \times 3} = \frac{3}{12} )
   • For ( \frac{1}{5} + \frac{2}{7} ):
     • Multiply ( \frac{1}{5} ) by ( \frac{7}{7} ):
       ( \frac{1 \times 7}{5 \times 7} = \frac{7}{35} )
     • Multiply ( \frac{2}{7} ) by ( \frac{5}{5} ):
       ( \frac{2 \times 5}{7 \times 5} = \frac{10}{35} )

4. Add the Numerators

   • Make sure denominators are now the same.
   • ( \frac{8}{12} + \frac{3}{12} = \frac{8 + 3}{12} = \frac{11}{12} )
   • ( \frac{7}{35} + \frac{10}{35} = \frac{7 + 10}{35} = \frac{17}{35} )

5. Simplify the Result (if possible)

   • In these examples:
     • ( \frac{11}{12} ) is already simplified.
     • ( \frac{17}{35} ) is already simplified (no common factors).

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Summary

• Always convert fractions to have a common denominator before adding.
• Multiply each fraction by the other's denominator (on top and bottom) to get equivalent fractions.
• Add the numerators, keep the denominator.
• Simplify the final fraction if possible.

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Example Problems

1. ( \frac{2}{3} + \frac{1}{4} = \frac{11}{12} )
2. ( \frac{1}{5} + \frac{2}{7} = \frac{17}{35} )

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Tip: If denominators are large, use the least common multiple (LCM) for efficiency when possible.