To solve the expression $(9 \frac{1}{3} - 2 \frac{5}{6} + 1 \frac{1}{4}) \div \left[1 \frac{1}{6} + \left\{2 \frac{1}{3} + \left(6 + 5 - 4 \frac{1}{2}\right)\right\}\right]$, we'll proceed step by step.

Step 1: Convert mixed numbers to improper fractions

1. $9 \frac{1}{3} = \frac{28}{3}$
2. $2 \frac{5}{6} = \frac{17}{6}$
3. $1 \frac{1}{4} = \frac{5}{4}$
4. $1 \frac{1}{6} = \frac{7}{6}$
5. $2 \frac{1}{3} = \frac{7}{3}$
6. $4 \frac{1}{2} = \frac{9}{2}$

Step 2: Simplify the innermost expression inside the braces

$6 + 5 - 4 \frac{1}{2} = 11 - \frac{9}{2} = \frac{22}{2} - \frac{9}{2} = \frac{13}{2}$

Step 3: Continue simplifying the expression inside the braces

$2 \frac{1}{3} + \frac{13}{2} = \frac{7}{3} + \frac{13}{2} = \frac{14}{6} + \frac{39}{6} = \frac{53}{6}$

Step 4: Simplify the expression inside the brackets

$1 \frac{1}{6} + \frac{53}{6} = \frac{7}{6} + \frac{53}{6} = \frac{60}{6} = 10$

Step 5: Simplify the numerator

$9 \frac{1}{3} - 2 \frac{5}{6} + 1 \frac{1}{4} = \frac{28}{3} - \frac{17}{6} + \frac{5}{4}$ To subtract and add, find a common denominator: $\frac{28}{3} = \frac{56}{6}, \quad \frac{5}{4} = \frac{15}{12} = \frac{30}{24}$ Now simplify: $\frac{56}{6} - \frac{17}{6} = \frac{39}{6} = \frac{78}{12}, \quad \frac{78}{12} + \frac{30}{24} = \frac{216}{24} + \frac{30}{24} = \frac{246}{24} = \frac{41}{4}$

Step 6: Perform the division

$\frac{41}{4} \div 10 = \frac{41}{4} \times \frac{1}{10} = \frac{41}{40}$

Final Answer:

$\frac{41}{40} \text{ or } 1.025$

Do you want more details or have any questions?

Here are some related questions:

1. How do you convert a mixed number into an improper fraction?
2. What is the process for finding a common denominator?
3. How do you simplify a complex fraction?
4. What is the distributive property, and how is it applied in expressions like this?
5. How does dividing by a fraction work?

Tip: Always simplify expressions inside parentheses, braces, and brackets first, moving from the innermost to the outermost.