`9(3)/(4)-:[2(1)/(6)+{4(1)/(3)-(1(1)/(2)+1(3)/(4))}]`

Answer

To solve the expression \(9 \frac{3}{4} \div \left[ 2 \frac{1}{6} + \left\{ 4 \frac{1}{3} - \left( 1 \frac{1}{2} + 1 \frac{3}{4} \right) \right\} \right]\), we will follow these steps:

### Step 1: Convert Mixed Numbers to Improper Fractions
First, we convert all mixed numbers to improper fractions.

1. \(9 \frac{3}{4} = \frac{9 \times 4 + 3}{4} = \frac{36 + 3}{4} = \frac{39}{4}\)
2. \(2 \frac{1}{6} = \frac{2 \times 6 + 1}{6} = \frac{12 + 1}{6} = \frac{13}{6}\)
3. \(4 \frac{1}{3} = \frac{4 \times 3 + 1}{3} = \frac{12 + 1}{3} = \frac{13}{3}\)
4. \(1 \frac{1}{2} = \frac{1 \times 2 + 1}{2} = \frac{2 + 1}{2} = \frac{3}{2}\)
5. \(1 \frac{3}{4} = \frac{1 \times 4 + 3}{4} = \frac{4 + 3}{4} = \frac{7}{4}\)

### Step 2: Substitute Back into the Expression
Now substitute these improper fractions back into the expression:
\[ \frac{39}{4} \div \left[ \frac{13}{6} + \left\{ \frac{13}{3} - \left( \frac{3}{2} + \frac{7}{4} \right) \right\} \right] \]

### Step 3: Solve the Innermost Bracket
Calculate \( \frac{3}{2} + \frac{7}{4} \):
- Find a common denominator (which is 4):
\[ \frac{3}{2} = \frac{6}{4} \quad \text{(multiply numerator and denominator by 2)} \]
Now add:
\[ \frac{6}{4} + \frac{7}{4} = \frac{13}{4} \]

### Step 4: Substitute and Simplify the Curly Bracket
Now substitute back:
\[ \frac{39}{4} \div \left[ \frac{13}{6} + \left\{ \frac{13}{3} - \frac{13}{4} \right\} \right] \]

Next, calculate \( \frac{13}{3} - \frac{13}{4} \):
- Find a common denominator (which is 12):
\[ \frac{13}{3} = \frac{52}{12}, \quad \frac{13}{4} = \frac{39}{12} \]
Now subtract:
\[ \frac{52}{12} - \frac{39}{12} = \frac{13}{12} \]

### Step 5: Substitute Back into the Expression
Now substitute back:
\[ \frac{39}{4} \div \left[ \frac{13}{6} + \frac{13}{12} \right] \]

### Step 6: Solve the Square Bracket
Now calculate \( \frac{13}{6} + \frac{13}{12} \):
- Find a common denominator (which is 12):
\[ \frac{13}{6} = \frac{26}{12} \]
Now add:
\[ \frac{26}{12} + \frac{13}{12} = \frac{39}{12} \]

### Step 7: Substitute and Simplify
Now substitute back:
\[ \frac{39}{4} \div \frac{39}{12} \]

### Step 8: Change Division to Multiplication
Change the division to multiplication by taking the reciprocal:
\[ \frac{39}{4} \times \frac{12}{39} \]

### Step 9: Cancel Out Common Factors
The \(39\) in the numerator and denominator cancels out:
\[ = \frac{12}{4} = 3 \]

### Final Answer
The final answer is \(3\).

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