`(1)/(2) + {4 (3)/(4) - (4 (1)/(6) - 2 (1)/(3))}` is equal to

Options

A. `3 (2)/(3)`
B. `1 (1)/(4)`
C. `4 (5)/(12)`
D. `1 (2)/(3)`

Answer

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

### Step-by-Step Solution:

1. **Calculate the inner parentheses**:
\[ 4 \cdot \frac{1}{6} - 2 \cdot \frac{1}{3} \]
- Calculate \(4 \cdot \frac{1}{6} = \frac{4}{6} = \frac{2}{3}\).
- Calculate \(2 \cdot \frac{1}{3} = \frac{2}{3}\).
- Now, substitute these values:
\[ \frac{2}{3} - \frac{2}{3} = 0 \]

2. **Substitute back into the expression**:
\[ 4 \cdot \frac{3}{4} - 0 \]
- Calculate \(4 \cdot \frac{3}{4} = 3\).
- So, we have:
\[ 3 - 0 = 3 \]

3. **Now substitute this back into the original expression**:
\[ \frac{1}{2} + 3 \]

4. **Convert \(3\) into a fraction**:
\[ 3 = \frac{6}{2} \]

5. **Add the fractions**:
\[ \frac{1}{2} + \frac{6}{2} = \frac{1 + 6}{2} = \frac{7}{2} \]

### Final Answer:
The expression \(\frac{1}{2} + \left\{ 4 \cdot \frac{3}{4} - \left( 4 \cdot \frac{1}{6} - 2 \cdot \frac{1}{3} \right) \right\}\) is equal to \(\frac{7}{2}\).