If In denotes the greatest integer `lt` nand (ru denotes the smallest integer `gt` n, where n is any real number, then `(1(1)/(5))xx[1(1)/(5)]-(1(1)/(5))+[1(1)/(5)]+(1.5)` is

Options

A. `1.5`
B. 2
C. `2.5`
D. `3.5`

Answer

To solve the expression \((1 \frac{1}{5}) \times [1 \frac{1}{5}] - (1 \frac{1}{5}) + [1 \frac{1}{5}] + (1.5)\), we need to break it down step by step.

### Step 1: Convert Mixed Numbers to Improper Fractions
First, we convert \(1 \frac{1}{5}\) into an improper fraction:
\[ 1 \frac{1}{5} = \frac{5 \times 1 + 1}{5} = \frac{6}{5} \]

### Step 2: Substitute the Improper Fraction into the Expression
Now, we substitute \(\frac{6}{5}\) into the expression:
\[ \left(\frac{6}{5}\right) \times \left[\frac{6}{5}\right] - \left(\frac{6}{5}\right) + \left[\frac{6}{5}\right] + (1.5) \]

### Step 3: Calculate the Greatest Integer Function
Next, we calculate the greatest integer less than or equal to \(\frac{6}{5}\):
\[ \lfloor \frac{6}{5} \rfloor = \lfloor 1.2 \rfloor = 1 \]
So, we can replace \(\left[\frac{6}{5}\right]\) with \(1\) in the expression:
\[ \left(\frac{6}{5}\right) \times 1 - \left(\frac{6}{5}\right) + 1 + (1.5) \]

### Step 4: Simplify the Expression
Now we simplify the expression:
\[ \frac{6}{5} \times 1 - \frac{6}{5} + 1 + 1.5 \]
This simplifies to:
\[ \frac{6}{5} - \frac{6}{5} + 1 + 1.5 \]
The \(\frac{6}{5} - \frac{6}{5}\) cancels out:
\[ 0 + 1 + 1.5 = 2.5 \]

### Final Answer
Thus, the value of the expression is:
\[ \boxed{2.5} \]