The value of `3 (1)/(2) - [2 (1)/(4) + {1 (1)/(4) - (1)/(2)(1 (1)/(2) - (1)/(3) - (1)/(6))}]` is

Options

A. `(1)/(2)`
B. `2 (1)/(2)`
C. `3 (1)/(2)`
D. `9 (1)/(2)`

Answer

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

### Step-by-Step Solution:

1. **Convert Mixed Numbers to Improper Fractions:**
- \( 3 \frac{1}{2} = \frac{7}{2} \)
- \( 2 \frac{1}{4} = \frac{9}{4} \)
- \( 1 \frac{1}{4} = \frac{5}{4} \)
- \( 1 \frac{1}{2} = \frac{3}{2} \)

So, the expression becomes:
\[ \frac{7}{2} - \left[ \frac{9}{4} + \left\{ \frac{5}{4} - \left( \frac{1}{2} \left( \frac{3}{2} - \frac{1}{3} - \frac{1}{6} \right) \right) \right\} \right] \]

2. **Simplify the Inner Bracket:**
- First, simplify \( \frac{3}{2} - \frac{1}{3} - \frac{1}{6} \).
- Find a common denominator (LCM of 2, 3, and 6 is 6):
- Convert \( \frac{3}{2} = \frac{9}{6} \)
- Convert \( \frac{1}{3} = \frac{2}{6} \)
- Convert \( \frac{1}{6} = \frac{1}{6} \)

Now, substituting these values:
\[ \frac{9}{6} - \frac{2}{6} - \frac{1}{6} = \frac{9 - 2 - 1}{6} = \frac{6}{6} = 1 \]

3. **Substitute Back:**
- Now, substitute \( 1 \) back into the expression:
\[ \frac{7}{2} - \left[ \frac{9}{4} + \left\{ \frac{5}{4} - \frac{1}{2} \cdot 1 \right\} \right] \]

4. **Calculate \( \frac{5}{4} - \frac{1}{2} \):**
- Convert \( \frac{1}{2} = \frac{2}{4} \):
\[ \frac{5}{4} - \frac{2}{4} = \frac{3}{4} \]

5. **Substitute Back Again:**
- Now substitute \( \frac{3}{4} \) back into the expression:
\[ \frac{7}{2} - \left[ \frac{9}{4} + \frac{3}{4} \right] \]

6. **Combine the Terms Inside the Bracket:**
\[ \frac{9}{4} + \frac{3}{4} = \frac{12}{4} = 3 \]

7. **Final Calculation:**
- Now we have:
\[ \frac{7}{2} - 3 \]
- Convert \( 3 \) to a fraction with a denominator of 2:
\[ 3 = \frac{6}{2} \]
- Now perform the subtraction:
\[ \frac{7}{2} - \frac{6}{2} = \frac{1}{2} \]

Thus, the final value of the expression is:
\[ \frac{1}{2} \]