Simplify :
`8 (1)/(2)-[3 (1)/(4)-:{1(1)/(4) - (1)/(2) ( 1 (1)/(2)-(1)/(3)-(1)/(6))}]`

Options

A. `4 (1)/(2)`
B. `4(1)/(6)`
C. `9 (1)/(2)`
D. `(2)/(9)`

Answer

To simplify the expression \( 8 \left( \frac{1}{2} \right) - \left[ 3 \left( \frac{1}{4} \right) - \left\{ 1 \left( \frac{1}{4} \right) - \left( \frac{1}{2} \left( 1 \left( \frac{1}{2} \right) - \frac{1}{3} - \frac{1}{6} \right) \right) \right\} \right] \), we will follow the order of operations (BODMAS/BIDMAS).

### Step-by-Step Solution:

1. **Convert Mixed Numbers to Improper Fractions**:
- \( 1 \left( \frac{1}{2} \right) = \frac{3}{2} \)
- \( 1 \left( \frac{1}{4} \right) = \frac{5}{4} \)

2. **Simplify Inside the Innermost Bracket**:
- Calculate \( \frac{1}{2} \left( \frac{3}{2} - \frac{1}{3} - \frac{1}{6} \right) \):
- First, find a common denominator for \( \frac{1}{3} \) and \( \frac{1}{6} \):
- The common denominator is 6.
- \( \frac{1}{3} = \frac{2}{6} \)
- \( \frac{1}{6} = \frac{1}{6} \)
- Now combine:
\[ \frac{3}{2} - \frac{2}{6} - \frac{1}{6} = \frac{3}{2} - \frac{3}{6} = \frac{3}{2} - \frac{1}{2} = \frac{2}{2} = 1 \]

3. **Substituting Back**:
- Now substitute back into the expression:
\[ 8 \left( \frac{1}{2} \right) - \left[ 3 \left( \frac{1}{4} \right) - 1 \right] \]

4. **Calculate Each Part**:
- Calculate \( 8 \left( \frac{1}{2} \right) = 4 \)
- Calculate \( 3 \left( \frac{1}{4} \right) = \frac{3}{4} \)
- Now substitute:
\[ 4 - \left[ \frac{3}{4} - 1 \right] \]

5. **Simplify the Bracket**:
- \( \frac{3}{4} - 1 = \frac{3}{4} - \frac{4}{4} = -\frac{1}{4} \)
- Now substitute back:
\[ 4 - \left( -\frac{1}{4} \right) = 4 + \frac{1}{4} = \frac{16}{4} + \frac{1}{4} = \frac{17}{4} \]

6. **Final Result**:
- The simplified expression is \( \frac{17}{4} \).