Let's solve the expression step by step. The expression is:

$\frac{(9 \frac{1}{3} - 2 \frac{5}{6} + 1 \frac{1}{4})}{1 \frac{1}{6} + \left\{2 \frac{1}{3} + \left(6 + 5 - 4 \frac{1}{2}\right)\right\}}$

Step 1: Simplify the numerators and denominators

1. Numerator: $9 \frac{1}{3} - 2 \frac{5}{6} + 1 \frac{1}{4}$:

   • Convert to improper fractions: $9 \frac{1}{3} = \frac{28}{3}, \quad 2 \frac{5}{6} = \frac{17}{6}, \quad 1 \frac{1}{4} = \frac{5}{4}$
   • Find a common denominator: $\text{Common denominator is } 12: \quad \frac{28}{3} = \frac{112}{12}, \quad \frac{17}{6} = \frac{34}{12}, \quad \frac{5}{4} = \frac{15}{12}$
   • Perform the operation: $\frac{112}{12} - \frac{34}{12} + \frac{15}{12} = \frac{93}{12} = \frac{31}{4}$

2. Denominator: $1 \frac{1}{6} + \left\{2 \frac{1}{3} + (6 + 5 - 4 \frac{1}{2})\right\}$:

   • Convert to improper fractions: $1 \frac{1}{6} = \frac{7}{6}, \quad 2 \frac{1}{3} = \frac{7}{3}, \quad 4 \frac{1}{2} = \frac{9}{2}$
   • Calculate inside the parentheses: [ 6 + 5 = 11, \quad 11 - \frac{9}{2} = \frac{22}{2}