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}

Learn how to solve the mathematical expression (9⅓-2⅚+1¼)÷[1⅙+{2⅓+(6+5-4½)}] step-by-step. This problem involves operations with fractions and follows the order of operations rule. Suitable for students in Grades 6-8.

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