`7(1)/(2)-[2(1)/(4)-:{1(1)/(4)-(1)/(2)((3)/(2)-bar((1)/(3)-(1)/(6)))}]`

Answer

To solve the expression \( 7\frac{1}{2} - \left[ 2\frac{1}{4} \div \left\{ 1\frac{1}{4} - \frac{1}{2} \left( \frac{3}{2} - \left( \frac{1}{3} - \frac{1}{6} \right) \right) \right\} \right] \), we will follow the order of operations, which is to simplify the innermost brackets first and work our way outwards.

### Step 1: Convert Mixed Numbers to Improper Fractions
1. Convert \( 7\frac{1}{2} \) to an improper fraction:
\[ 7\frac{1}{2} = \frac{7 \times 2 + 1}{2} = \frac{15}{2} \]

2. Convert \( 2\frac{1}{4} \) to an improper fraction:
\[ 2\frac{1}{4} = \frac{2 \times 4 + 1}{4} = \frac{9}{4} \]

3. Convert \( 1\frac{1}{4} \) to an improper fraction:
\[ 1\frac{1}{4} = \frac{1 \times 4 + 1}{4} = \frac{5}{4} \]

### Step 2: Simplify the Innermost Bracket
Now, we simplify the innermost expression \( \frac{1}{3} - \frac{1}{6} \):
\[ \text{LCM of } 3 \text{ and } 6 = 6 \]
\[ \frac{1}{3} = \frac{2}{6}, \quad \frac{1}{6} = \frac{1}{6} \]
\[ \frac{1}{3} - \frac{1}{6} = \frac{2}{6} - \frac{1}{6} = \frac{1}{6} \]

### Step 3: Substitute Back into the Expression
Now substitute back into the expression:
\[ \frac{3}{2} - \left( \frac{1}{6} \right) \]
\[ \text{LCM of } 2 \text{ and } 6 = 6 \]
\[ \frac{3}{2} = \frac{9}{6} \]
\[ \frac{9}{6} - \frac{1}{6} = \frac{8}{6} = \frac{4}{3} \]

### Step 4: Substitute into the Larger Expression
Now substitute back into the larger expression:
\[ 1\frac{1}{4} - \frac{1}{2} \left( \frac{4}{3} \right) \]
Convert \( \frac{1}{2} \) to a fraction:
\[ \frac{1}{2} = \frac{3}{6} \]
\[ \frac{1}{2} \left( \frac{4}{3} \right) = \frac{4}{6} = \frac{2}{3} \]
Now substitute:
\[ \frac{5}{4} - \frac{2}{3} \]
\[ \text{LCM of } 4 \text{ and } 3 = 12 \]
\[ \frac{5}{4} = \frac{15}{12}, \quad \frac{2}{3} = \frac{8}{12} \]
\[ \frac{15}{12} - \frac{8}{12} = \frac{7}{12} \]

### Step 5: Final Division
Now we substitute this back into the expression:
\[ \frac{9}{4} \div \frac{7}{12} \]
To divide by a fraction, multiply by its reciprocal:
\[ \frac{9}{4} \times \frac{12}{7} = \frac{9 \times 12}{4 \times 7} = \frac{108}{28} = \frac{27}{7} \]

### Step 6: Final Calculation
Now we have:
\[ \frac{15}{2} - \frac{27}{7} \]
\[ \text{LCM of } 2 \text{ and } 7 = 14 \]
\[ \frac{15}{2} = \frac{105}{14}, \quad \frac{27}{7} = \frac{54}{14} \]
\[ \frac{105}{14} - \frac{54}{14} = \frac{51}{14} \]

### Final Answer
The final answer is:
\[ \frac{51}{14} \]

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