Evaluate:

$8\frac{1}{3}-[\frac{7}{9}÷\{\frac{2}{5}-(\frac{1}{3}\times (\frac{5}{6}-\frac{1}{6}\left)\right)\left\}\right]$

The correct option is $\frac{95}{24}$.

To evaluate the given mathematical expression, we follow the standard order of operations (BODMAS / PEMDAS), which prioritizes simplifying nested brackets from the innermost outward.

Original Expression:

$8\frac{1}{3}-[\frac{7}{9}÷\{\frac{2}{5}-(\frac{1}{3}\times (\frac{5}{6}-\frac{1}{6}\left)\right)\left\}\right]$

Step 1: Simplify the innermost parenthetical term

Subtract the fractions inside the innermost round brackets:

$\frac{5}{6}-\frac{1}{6}=\frac{5-1}{6}=\frac{4}{6}=\frac{2}{3}$

Step 2: Multiply by $\frac{1}{3}$ inside the round brackets

$\frac{1}{3}\times \frac{2}{3}=\frac{2}{9}$

Step 3: Simplify the expression inside the curly braces

Subtract $\frac{2}{9}$ from $\frac{2}{5}$:

$\frac{2}{5}-\frac{2}{9}=\frac{18-10}{45}=\frac{8}{45}$

Step 4: Simplify inside the square brackets

Divide $\frac{7}{9}$ by $\frac{8}{45}$:

$\frac{7}{9}÷\frac{8}{45}=\frac{7}{9}\times \frac{45}{8}=\frac{7\times 5}{8}=\frac{35}{8}$

Step 5: Perform the final subtraction

Convert the mixed fraction $8\frac{1}{3}$ to an improper fraction $\frac{25}{3}$, then subtract:

$\frac{25}{3}-\frac{35}{8}=\frac{(25\times 8)-(35\times 3)}{24}=\frac{200-105}{24}=\frac{95}{24}$