2. \( 1 \frac{5}{6}+3 \frac{7}{13}= \)

To solve the addition of the mixed numbers \( 1 \frac{5}{6} + 3 \frac{7}{13} \), follow these steps: ### Step 1: Convert Mixed Numbers to Improper Fractions 1. **Convert \( 1 \frac{5}{6} \):** \[ 1 \frac{5}{6} = \frac{1 \times 6 + 5}{6} = \frac{11}{6} \] 2. **Convert \( 3 \frac{7}{13} \):** \[ 3 \frac{7}{13} = \frac{3 \times 13 + 7}{13} = \frac{46}{13} \] ### Step 2: Find a Common Denominator The denominators are 6 and 13. The least common denominator (LCD) is the least common multiple (LCM) of 6 and 13, which is \( 78 \). ### Step 3: Convert Fractions to Have the Common Denominator 1. **Convert \( \frac{11}{6} \) to have a denominator of 78:** \[ \frac{11}{6} = \frac{11 \times 13}{6 \times 13} = \frac{143}{78} \] 2. **Convert \( \frac{46}{13} \) to have a denominator of 78:** \[ \frac{46}{13} = \frac{46 \times 6}{13 \times 6} = \frac{276}{78} \] ### Step 4: Add the Fractions \[ \frac{143}{78} + \frac{276}{78} = \frac{143 + 276}{78} = \frac{419}{78} \] ### Step 5: Simplify the Result Convert \( \frac{419}{78} \) back to a mixed number: 1. **Divide 419 by 78:** \[ 419 \div 78 = 5 \text{ with a remainder of } 29 \] 2. **Write as a mixed number:** \[ 5 \frac{29}{78} \] Since 29 and 78 have no common factors other than 1, the fraction \( \frac{29}{78} \) is already in its simplest form. ### Final Answer \[ 1 \frac{5}{6} + 3 \frac{7}{13} = 5 \frac{29}{78} \]