6 5/6+2 3/5

To add the mixed numbers \( 6 \frac{5}{6} \) and \( 2 \frac{3}{5} \), follow these steps:

1. Convert the mixed numbers to improper fractions.

   • For \( 6 \frac{5}{6} \): \[ 6 \times 6 + 5 = 36 + 5 = 41 \quad \text{so } 6 \frac{5}{6} = \frac{41}{6} \]
   • For \( 2 \frac{3}{5} \): \[ 2 \times 5 + 3 = 10 + 3 = 13 \quad \text{so } 2 \frac{3}{5} = \frac{13}{5} \]

2. Now you have \( \frac{41}{6} + \frac{13}{5} \).

3. Find a common denominator. The least common multiple of 6 and 5 is 30.

   • Convert \( \frac{41}{6} \) to a fraction with a denominator of 30: \[ \frac{41}{6} = \frac{41 \times 5}{6 \times 5} = \frac{205}{30} \]
   • Convert \( \frac{13}{5} \) to a fraction with a denominator of 30: \[ \frac{13}{5} = \frac{13 \times 6}{5 \times 6} = \frac{78}{30} \]

4. Now add the two fractions: \[ \frac{205}{30} + \frac{78}{30} = \frac{205 + 78}{30} = \frac{283}{30} \]

5. Convert \( \frac{283}{30} \) back to a mixed number:

   • Divide \( 283 \) by \( 30 \): \[ 283 \div 30 \approx 9 \quad \text{(Since } 30 \times 9 = 270\text{)} \]
   • The remainder is \( 283 - 270 = 13 \).

So, \[ \frac{283}{30} = 9 \frac{13}{30} \]

Therefore, the final answer is \[ 6 \frac{5}{6} + 2 \frac{3}{5} = 9 \frac{13}{30}. \]