​ 6 38/45+2 5/9 ​

To add the mixed numbers \(6 \frac{38}{45}\) and \(2 \frac{5}{9}\), we first convert them into improper fractions.

1. Convert \(6 \frac{38}{45}\) into an improper fraction: \[ 6 \frac{38}{45} = \frac{6 \times 45 + 38}{45} = \frac{270 + 38}{45} = \frac{308}{45} \]

2. Convert \(2 \frac{5}{9}\) into an improper fraction: \[ 2 \frac{5}{9} = \frac{2 \times 9 + 5}{9} = \frac{18 + 5}{9} = \frac{23}{9} \]

3. Now, we need to add \(\frac{308}{45}\) and \(\frac{23}{9}\). To do this, we first need a common denominator. The least common multiple (LCM) of 45 and 9 is 45.

4. Convert \(\frac{23}{9}\) to have a denominator of 45: \[ \frac{23}{9} = \frac{23 \times 5}{9 \times 5} = \frac{115}{45} \]

5. Now we can add the two fractions: \[ \frac{308}{45} + \frac{115}{45} = \frac{308 + 115}{45} = \frac{423}{45} \]

6. Next, we convert \(\frac{423}{45}\) back into a mixed number:

   • Divide 423 by 45: \[ 423 \div 45 = 9 \quad \text{(quotient)} \]
   • Multiply \(45 \times 9 = 405\) and subtract from 423: \[ 423 - 405 = 18 \quad \text{(remainder)} \]
   • Thus, we write the mixed number: \[ 423 \div 45 = 9 \frac{18}{45} \]

7. Finally, simplify \(\frac{18}{45}\): \[ \frac{18 \div 9}{45 \div 9} = \frac{2}{5} \]

So, the final answer as a mixed number in simplest form is: \[ \boxed{9 \frac{2}{5}} \]