`3(1)/(12)-[1(3)/(4)+{2(1)/(2)-(1(1)/(2)-(1)/(3))}]` का मान है -

Options

A. `(1)/(2)`
B. 2
C. 1
D. 0

Answer

To solve the expression \( 3\left(\frac{1}{12}\right) - \left[1\left(\frac{3}{4}\right) + \left\{2\left(\frac{1}{2}\right) - \left(1\left(\frac{1}{2}\right) - \left(\frac{1}{3}\right)\right)\right\}\right] \), we will follow these steps:

### Step 1: Simplify the first term
The first term is \( 3\left(\frac{1}{12}\right) \).
\[ 3 \times \frac{1}{12} = \frac{3}{12} = \frac{1}{4} \]

### Step 2: Simplify the second term inside the brackets
The second term is \( 1\left(\frac{3}{4}\right) \).
\[ 1 \times \frac{3}{4} = \frac{3}{4} \]

### Step 3: Simplify the expression inside the curly braces
We need to simplify \( 2\left(\frac{1}{2}\right) - \left(1\left(\frac{1}{2}\right) - \left(\frac{1}{3}\right)\right) \).

First, calculate \( 2\left(\frac{1}{2}\right) \):
\[ 2 \times \frac{1}{2} = 1 \]

Next, calculate \( 1\left(\frac{1}{2}\right) \):
\[ 1 \times \frac{1}{2} = \frac{1}{2} \]

Now, substitute these values into the expression:
\[ 1 - \left(\frac{1}{2} - \frac{1}{3}\right) \]

### Step 4: Simplify \( \frac{1}{2} - \frac{1}{3} \)
To subtract these fractions, we need a common denominator, which is 6:
\[ \frac{1}{2} = \frac{3}{6}, \quad \frac{1}{3} = \frac{2}{6} \]
Thus,
\[ \frac{1}{2} - \frac{1}{3} = \frac{3}{6} - \frac{2}{6} = \frac{1}{6} \]

### Step 5: Substitute back into the expression
Now we have:
\[ 1 - \frac{1}{6} \]
Convert 1 into a fraction with a denominator of 6:
\[ 1 = \frac{6}{6} \]
So,
\[ \frac{6}{6} - \frac{1}{6} = \frac{5}{6} \]

### Step 6: Combine the two parts
Now substitute back into the original expression:
\[ \frac{1}{4} - \left(\frac{3}{4} + \frac{5}{6}\right) \]

### Step 7: Simplify \( \frac{3}{4} + \frac{5}{6} \)
To add these fractions, we need a common denominator, which is 12:
\[ \frac{3}{4} = \frac{9}{12}, \quad \frac{5}{6} = \frac{10}{12} \]
So,
\[ \frac{3}{4} + \frac{5}{6} = \frac{9}{12} + \frac{10}{12} = \frac{19}{12} \]

### Step 8: Final calculation
Now we have:
\[ \frac{1}{4} - \frac{19}{12} \]
Convert \( \frac{1}{4} \) into a fraction with a denominator of 12:
\[ \frac{1}{4} = \frac{3}{12} \]
Thus,
\[ \frac{3}{12} - \frac{19}{12} = \frac{3 - 19}{12} = \frac{-16}{12} = -\frac{4}{3} \]

### Final Answer
The value of the expression is:
\[ -\frac{4}{3} \]