Simplified value of `7 \frac{1}{2}-[2 \frac{1}{4} \div{1 \frac{1}{4}-\frac{1}{2}(1 \frac{1}{2}-\frac{1}{3}-\frac{1}{6})}]` is:

Options

A. `2/9`
B. `4 1/2`
C. `9 1/2`
D. `1 77/228`

Answer

To simplify the expression \( 7 \frac{1}{2} - \left[ 2 \frac{1}{4} \div \left( 1 \frac{1}{4} - \frac{1}{2} \left( 1 \frac{1}{2} - \frac{1}{3} - \frac{1}{6} \right) \right) \right] \), we will follow these steps:

### Step 1: Convert Mixed Numbers to Improper Fractions
Convert all mixed numbers to improper fractions for easier calculations.

- \( 7 \frac{1}{2} = \frac{15}{2} \)
- \( 2 \frac{1}{4} = \frac{9}{4} \)
- \( 1 \frac{1}{4} = \frac{5}{4} \)
- \( 1 \frac{1}{2} = \frac{3}{2} \)

So, the expression becomes:
\[ \frac{15}{2} - \left[ \frac{9}{4} \div \left( \frac{5}{4} - \frac{1}{2} \left( \frac{3}{2} - \frac{1}{3} - \frac{1}{6} \right) \right) \right] \]

### Step 2: Simplify the Inner Expression
Now, we need to calculate the inner expression:
\[ \frac{3}{2} - \frac{1}{3} - \frac{1}{6} \]

To do this, we find a common denominator. The least common multiple of 2, 3, and 6 is 6.

- Convert \( \frac{3}{2} \) to sixths:
\[ \frac{3}{2} = \frac{9}{6} \]
- \( \frac{1}{3} = \frac{2}{6} \)
- \( \frac{1}{6} = \frac{1}{6} \)

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

### Step 3: Substitute Back into the Expression
Now substitute this result back into the expression:
\[ \frac{15}{2} - \left[ \frac{9}{4} \div \left( \frac{5}{4} - \frac{1}{2} \cdot 1 \right) \right] \]

### Step 4: Simplify Further
Calculate \( \frac{5}{4} - \frac{1}{2} \):
- Convert \( \frac{1}{2} \) to fourths:
\[ \frac{1}{2} = \frac{2}{4} \]
So,
\[ \frac{5}{4} - \frac{2}{4} = \frac{3}{4} \]

Now, we have:
\[ \frac{15}{2} - \left[ \frac{9}{4} \div \frac{3}{4} \right] \]

### Step 5: Division of Fractions
Dividing \( \frac{9}{4} \) by \( \frac{3}{4} \):
\[ \frac{9}{4} \div \frac{3}{4} = \frac{9}{4} \cdot \frac{4}{3} = \frac{9 \cdot 4}{4 \cdot 3} = \frac{9}{3} = 3 \]

### Step 6: Final Calculation
Now substitute back:
\[ \frac{15}{2} - 3 \]

Convert 3 to halves:
\[ 3 = \frac{6}{2} \]
So,
\[ \frac{15}{2} - \frac{6}{2} = \frac{15 - 6}{2} = \frac{9}{2} \]

### Step 7: Convert to Mixed Number
Convert \( \frac{9}{2} \) back to a mixed number:
\[ \frac{9}{2} = 4 \frac{1}{2} \]

### Final Answer
The simplified value is:
\[ \boxed{4 \frac{1}{2}} \]