Let's break down the expression step by step:

$6 + 2 \frac{5}{11} + \frac{1}{3}$

1. Convert the mixed fraction $2 \frac{5}{11}$ into an improper fraction: $2 \frac{5}{11} = \frac{2 \times 11 + 5}{11} = \frac{22 + 5}{11} = \frac{27}{11}$

Now, the expression becomes: $6 + \frac{27}{11} + \frac{1}{3}$

2. Convert 6 to a fraction with a denominator of 11 (since 11 is a common denominator for $\frac{27}{11}$ and $\frac{1}{3}$): $6 = \frac{6 \times 11}{11} = \frac{66}{11}$

Now, the expression is: $\frac{66}{11} + \frac{27}{11} + \frac{1}{3}$

3. Add the fractions with the same denominator: $\frac{66 + 27}{11} = \frac{93}{11}$

So, the expression becomes: $\frac{93}{11} + \frac{1}{3}$

4. Find a common denominator between 11 and 3, which is 33: $\frac{93}{11} = \frac{93 \times 3}{33} = \frac{279}{33}$ $\frac{1}{3} = \frac{1 \times 11}{33} = \frac{11}{33}$

5. Now add the two fractions: $\frac{279}{33} + \frac{11}{33} = \frac{279 + 11}{33} = \frac{290}{33}$

So, the final result is: $\frac{290}{33}$

This can also be written as a mixed number: $\frac{290}{33} = 8 \frac{26}{33}$

Final Answer: $8 \frac{26}{33}$

Would you like a more detailed breakdown or clarification on any part? Here are some related questions for further exploration:

1. How do you simplify improper fractions into mixed numbers?
2. What are other methods for finding a common denominator for fractions?
3. Can you explain the process of adding fractions with different denominators in more detail?
4. How do you handle fractions in algebraic equations?
5. How would this process change if the fractions involved were decimals instead?

Tip: When adding fractions, always ensure the denominators are the same. If they're not, find the least common denominator (LCD) first.