Simplify the following expression.
`4-3 "of" (1(1)/2+1/3div1/2"of"4-1/4)`

Options

A. `-8`
B. `17/12`
C. `-1/4`
D. `-(19)/4`

Answer

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

### Step-by-Step Solution:

1. **Convert Mixed Numbers and Fractions**:
- Convert \( 1 \frac{1}{2} \) to an improper fraction:
\[ 1 \frac{1}{2} = \frac{3}{2} \]
- The expression now looks like:
\[ 4 - 3 \text{ of } \left( \frac{3}{2} + \frac{1}{3} \div \frac{1}{2} \text{ of } 4 - \frac{1}{4} \right) \]

2. **Calculate \( \frac{1}{3} \div \frac{1}{2} \text{ of } 4 \)**:
- First, calculate \( \frac{1}{2} \text{ of } 4 \):
\[ \frac{1}{2} \text{ of } 4 = \frac{1}{2} \times 4 = 2 \]
- Now, perform the division:
\[ \frac{1}{3} \div 2 = \frac{1}{3} \times \frac{1}{2} = \frac{1}{6} \]

3. **Substitute Back into the Expression**:
- Now substitute \( \frac{1}{6} \) back into the expression:
\[ 4 - 3 \text{ of } \left( \frac{3}{2} + \frac{1}{6} - \frac{1}{4} \right) \]

4. **Combine the Fractions Inside the Bracket**:
- To combine \( \frac{3}{2} + \frac{1}{6} - \frac{1}{4} \), find a common denominator. The least common multiple of 2, 6, and 4 is 12.
- Convert each fraction:
\[ \frac{3}{2} = \frac{18}{12}, \quad \frac{1}{6} = \frac{2}{12}, \quad \frac{1}{4} = \frac{3}{12} \]
- Now combine:
\[ \frac{18}{12} + \frac{2}{12} - \frac{3}{12} = \frac{18 + 2 - 3}{12} = \frac{17}{12} \]

5. **Substitute Back Again**:
- Substitute \( \frac{17}{12} \) back into the expression:
\[ 4 - 3 \text{ of } \frac{17}{12} \]

6. **Calculate \( 3 \text{ of } \frac{17}{12} \)**:
- Calculate \( 3 \text{ of } \frac{17}{12} \):
\[ 3 \text{ of } \frac{17}{12} = 3 \times \frac{17}{12} = \frac{51}{12} \]

7. **Final Calculation**:
- Now, substitute this back into the expression:
\[ 4 - \frac{51}{12} \]
- Convert 4 to a fraction with a denominator of 12:
\[ 4 = \frac{48}{12} \]
- Now perform the subtraction:
\[ \frac{48}{12} - \frac{51}{12} = \frac{48 - 51}{12} = \frac{-3}{12} = -\frac{1}{4} \]

### Final Answer:
The simplified expression is:
\[ -\frac{1}{4} \]