Simplify : `7 (1)/(2) - [2 (1)/(4) + {(1)/(4)-(1)/(2) (1 (1)/(2) - (1)/(3) - (1)/(6))}]`

Options

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

Answer

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

### Step 1: Convert Mixed Numbers to Improper Fractions
First, we convert all mixed numbers to improper fractions:
- \( 7 \frac{1}{2} = \frac{15}{2} \)
- \( 2 \frac{1}{4} = \frac{9}{4} \)
- \( 1 \frac{1}{2} = \frac{3}{2} \)

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

### Step 2: Simplify Inside the Bracket
Now we need to simplify the expression inside the curly brackets:
\[ \frac{3}{2} - \frac{1}{3} - \frac{1}{6} \]
To do this, we find the least common multiple (LCM) of the denominators (2, 3, and 6), which is 6.

- Convert \( \frac{3}{2} \) to sixths: \( \frac{3}{2} = \frac{9}{6} \)
- Convert \( \frac{1}{3} \) to sixths: \( \frac{1}{3} = \frac{2}{6} \)
- \( \frac{1}{6} \) remains as is.

Now we can combine:
\[ \frac{9}{6} - \frac{2}{6} - \frac{1}{6} = \frac{9 - 2 - 1}{6} = \frac{6}{6} = 1 \]

### Step 3: Substitute Back into the Expression
Now substitute back into the expression:
\[ \frac{15}{2} - \left[ \frac{9}{4} + \left\{ 1 \right\} \right] \]
This simplifies to:
\[ \frac{15}{2} - \left[ \frac{9}{4} + 1 \right] \]

### Step 4: Convert 1 to a Fraction
Convert 1 to a fraction with a denominator of 4:
\[ 1 = \frac{4}{4} \]
So we have:
\[ \frac{15}{2} - \left[ \frac{9}{4} + \frac{4}{4} \right] = \frac{15}{2} - \frac{13}{4} \]

### Step 5: Find a Common Denominator
The LCM of 2 and 4 is 4. Convert \( \frac{15}{2} \) to fourths:
\[ \frac{15}{2} = \frac{30}{4} \]
Now the expression is:
\[ \frac{30}{4} - \frac{13}{4} = \frac{30 - 13}{4} = \frac{17}{4} \]

### Step 6: Convert to Mixed Number (if needed)
To convert \( \frac{17}{4} \) to a mixed number:
\[ 17 \div 4 = 4 \quad \text{(remainder 1)} \]
So, \( \frac{17}{4} = 4 \frac{1}{4} \).

### Final Answer
The simplified expression is:
\[ \frac{17}{4} \quad \text{or} \quad 4 \frac{1}{4} \]