To solve the expression:

$3 + \frac{2}{5} \div 3 - \frac{2}{5}$

Step 1: Handle the division first

The division $\frac{2}{5} \div 3$ is equivalent to multiplying $\frac{2}{5}$ by the reciprocal of 3, which is $\frac{1}{3}$:

$\frac{2}{5} \times \frac{1}{3} = \frac{2 \times 1}{5 \times 3} = \frac{2}{15}$

Step 2: Substitute back into the original expression

Now, the expression becomes:

$3 + \frac{2}{15} - \frac{2}{5}$

Step 3: Convert to a common denominator

To combine the fractions $\frac{2}{15}$ and $\frac{2}{5}$, we need a common denominator. The least common denominator of 15 and 5 is 15. So, convert $\frac{2}{5}$ to a fraction with denominator 15:

$\frac{2}{5} = \frac{6}{15}$

Now the expression is:

$3 + \frac{2}{15} - \frac{6}{15}$

Step 4: Combine the fractions

Now combine the fractions $\frac{2}{15}$ and $\frac{6}{15}$:

$\frac{2}{15} - \frac{6}{15} = \frac{-4}{15}$

Step 5: Add the integer and fraction

Now the expression is:

$3 + \frac{-4}{15} = \frac{45}{15} - \frac{4}{15} = \frac{41}{15}$

Final Answer:

$\frac{41}{15}$

Would you like any further details or clarifications?

Here are some related questions for you:

1. What is the value of $\frac{2}{5} \div 4 - \frac{1}{3}$?
2. Can you simplify the expression $4 + \frac{3}{4} - 2 \times \frac{1}{2}$?
3. What is the least common denominator for $\frac{3}{8}$ and $\frac{5}{12}$?
4. How do you solve $2 + \frac{1}{2} \div 2 - \frac{1}{3}$?
5. What happens when you divide one fraction by another?

Tip: When dividing fractions, always multiply by the reciprocal of the divisor.