Find the value of

$\frac{12+3\frac{2}{3}÷[1\frac{3}{4}÷\{5\frac{1}{4}\times (3\frac{3}{5}÷4\frac{1}{2})\left\}\right]}{1\frac{1}{3}\times 5\frac{1}{5}}.$

The correct answer is 3.

To evaluate the given complex fraction expression, we follow the standard order of operations (BODMAS/PEMDAS). We will simplify the inner grouping symbols first (parentheses, curly braces, and square brackets), then complete the operations in the numerator, and finally divide by the denominator.

The given expression is:

$\frac{12+3\frac{2}{3}÷[1\frac{3}{4}÷\{5\frac{1}{4}\times (3\frac{3}{5}÷4\frac{1}{2})\left\}\right]}{1\frac{1}{3}\times 5\frac{1}{5}}$

Step 1: Convert all mixed fractions into improper fractions
Let us convert each mixed fraction involved in the expression:
• $3\frac{2}{3}=\frac{3\times 3+2}{3}=\frac{11}{3}$
• $1\frac{3}{4}=\frac{1\times 4+3}{4}=\frac{7}{4}$
• $5\frac{1}{4}=\frac{5\times 4+1}{4}=\frac{21}{4}$
• $3\frac{3}{5}=\frac{3\times 5+3}{5}=\frac{18}{5}$
• $4\frac{1}{2}=\frac{4\times 2+1}{2}=\frac{9}{2}$
• $1\frac{1}{3}=\frac{1\times 3+1}{3}=\frac{4}{3}$
• $5\frac{1}{5}=\frac{5\times 5+1}{5}=\frac{26}{5}$

Step 2: Simplify the innermost parentheses in the numerator
We solve the expression inside the round brackets $(3\frac{3}{5}÷4\frac{1}{2})$:

$\frac{18}{5}÷\frac{9}{2}=\frac{18}{5}\times \frac{2}{9}=\frac{2\times 2}{5}=\frac{4}{5}$

Step 3: Simplify the curly braces in the numerator
Next, we evaluate the curly braces $\{5\frac{1}{4}\times \frac{4}{5}\}$:

$\frac{21}{4}\times \frac{4}{5}=\frac{21}{5}$

Step 4: Simplify the square brackets in the numerator
Now we solve inside the square brackets $[1\frac{3}{4}÷\frac{21}{5}]$:

$\frac{7}{4}÷\frac{21}{5}=\frac{7}{4}\times \frac{5}{21}=\frac{1}{4}\times \frac{5}{3}=\frac{5}{12}$

Step 5: Perform division and addition to find the Numerator
We now simplify the entire numerator: $12+3\frac{2}{3}÷\frac{5}{12}$.
By order of operations, perform division first:

$\frac{11}{3}÷\frac{5}{12}=\frac{11}{3}\times \frac{12}{5}=11\times \frac{4}{5}=\frac{44}{5}$

Now add 12 to $\frac{44}{5}$:

$\text{Numerator}=12+\frac{44}{5}=\frac{60}{5}+\frac{44}{5}=\frac{104}{5}$

Step 6: Evaluate the Denominator
Multiply the two improper fractions in the denominator:

$\text{Denominator}=1\frac{1}{3}\times 5\frac{1}{5}=\frac{4}{3}\times \frac{26}{5}=\frac{104}{15}$

Step 7: Divide the Numerator by the Denominator
Finally, we compute the total value:

$\text{Value}=\frac{104/5}{104/15}=\frac{104}{5}\times \frac{15}{104}=\frac{15}{5}=3$

Thus, the final value of the expression is 3.