The given mathematical expression is:

$3 + \frac{3}{8} \div 6 - \frac{2}{4}$

Let's solve this step by step.

1. Division: Start by performing the division of $\frac{3}{8} \div 6$. To divide by 6, multiply by the reciprocal of 6:

   $\frac{3}{8} \div 6 = \frac{3}{8} \times \frac{1}{6} = \frac{3 \times 1}{8 \times 6} = \frac{3}{48} = \frac{1}{16}$

2. Substitution: Now substitute this result back into the equation:

   $3 + \frac{1}{16} - \frac{2}{4}$

3. Simplify the subtraction: Simplify $\frac{2}{4}$, which is equal to $\frac{1}{2}$.

4. Substitute and combine terms: The expression is now:

   $3 + \frac{1}{16} - \frac{1}{2}$

   To combine these terms, we need a common denominator. The common denominator for 16 and 2 is 16. Convert $\frac{1}{2}$ to $\frac{8}{16}$.

   Now, the expression becomes:

   $3 + \frac{1}{16} - \frac{8}{16}$

   Combine the fractions:

   $3 + \frac{1 - 8}{16} = 3 - \frac{7}{16}$

5. Final calculation: Now subtract $\frac{7}{16}$ from 3:

   $3 - \frac{7}{16} = \frac{48}{16} - \frac{7}{16} = \frac{48 - 7}{16} = \frac{41}{16}$

   This is equivalent to:

   $2 \frac{9}{16}$

Thus, the correct answer is A. $2 \frac{9}{16}$.

Would you like more details on any step?

Here are 5 related questions:

1. How do you divide fractions?
2. Why do we need a common denominator when adding or subtracting fractions?
3. How do you convert an improper fraction to a mixed number?
4. What is the reciprocal of a number, and how is it used in division?
5. How can you simplify a fraction like $\frac{2}{4}$?

Tip: Always perform operations in the correct order (division/multiplication before addition/subtraction) to avoid errors.