$\text{The value of}\mathrm{ 3}\frac{2}{3}÷2\frac{1}{5}\times \frac{1}{4}\times (1+\frac{3}{1-\frac{2}{5}})\text{is:}$

The correct answer is:
$2\frac{1}{2}$

We are asked to find the value of the given expression:
$3\frac{2}{3}÷2\frac{1}{5}\times \frac{1}{4}\times (1+\frac{3}{1-\frac{2}{5}})$

Step 1: Convert all mixed fractions into improper fractions.
For the first fraction:
$3\frac{2}{3}=\frac{3\times 3+2}{3}=\frac{11}{3}$

For the second fraction:
$2\frac{1}{5}=\frac{2\times 5+1}{5}=\frac{11}{5}$

Step 2: Simplify the term inside the parenthesis.
First, evaluate the denominator inside the parenthesis:
$1-\frac{2}{5}=\frac{5-2}{5}=\frac{3}{5}$

Next, evaluate the fraction within the parenthesis:
$\frac{3}{1-\frac{2}{5}}=\frac{3}{\frac{3}{5}}=3\times \frac{5}{3}=5$

Now, add 1 to complete the simplification of the parenthesis:
$1+5=6$

Step 3: Substitute the simplified values back into the expression.
$\frac{11}{3}÷\frac{11}{5}\times \frac{1}{4}\times 6$

Step 4: Perform division and multiplication operations.
Convert division into multiplication by multiplying by the reciprocal of $\frac{11}{5}$:
$=\frac{11}{3}\times \frac{5}{11}\times \frac{1}{4}\times 6$

Cancel out the common factor of $11$ from numerator and denominator:
$=\frac{1}{3}\times 5\times \frac{1}{4}\times 6$

Multiply the terms:
$=\frac{5\times 6}{3\times 4}=\frac{30}{12}$

Simplify the resulting fraction:
$\frac{30}{12}=\frac{5}{2}$

Step 5: Convert the improper fraction back into a mixed fraction.
$\frac{5}{2}=2\frac{1}{2}$

Therefore, the simplified value of the given expression is:
$2\frac{1}{2}$