If [x] read as the greatest ingeger less than or equal to `x, {x}` is the least integer greater than interger less than or equal to x, {x} is the least interger greater than or equal to x. Further, `f(x, y) = [x] + {y}` and `g(x, y) = {x} - {y}`
and `P (x,y) = f(x , y) +g(x,y)`
and `Q (x,y) = f(x,y) -g(x,y)`
If `x = 16 and y = 25` then the value of `P(x, y) + Q(x,y)` is:

Options

A. 90
B. 200
C. can't be determined
D. none of these

Answer

To solve the problem step by step, we will follow the definitions provided and calculate the values of \( P(x, y) \) and \( Q(x, y) \) based on the given functions \( f(x, y) \) and \( g(x, y) \).

### Step 1: Identify the values of \( x \) and \( y \)
Given:
- \( x = 16 \)
- \( y = 25 \)

### Step 2: Calculate \( f(x, y) \)
The function \( f(x, y) \) is defined as:
\[ f(x, y) = [x] + \{y\} \]
Where:
- \( [x] \) is the greatest integer less than or equal to \( x \).
- \( \{y\} \) is the least integer greater than or equal to \( y \).

Calculating:
- \( [16] = 16 \) (since 16 is an integer)
- \( \{25\} = 25 \) (since 25 is also an integer)

Thus,
\[ f(16, 25) = 16 + 25 = 41 \]

### Step 3: Calculate \( g(x, y) \)
The function \( g(x, y) \) is defined as:
\[ g(x, y) = \{x\} - \{y\} \]
Where:
- \( \{x\} \) is the least integer greater than or equal to \( x \).
- \( \{y\} \) is the least integer greater than or equal to \( y \).

Calculating:
- \( \{16\} = 16 \) (since 16 is an integer)
- \( \{25\} = 25 \) (since 25 is also an integer)

Thus,
\[ g(16, 25) = 16 - 25 = -9 \]

### Step 4: Calculate \( P(x, y) \)
The function \( P(x, y) \) is defined as:
\[ P(x, y) = f(x, y) + g(x, y) \]

Substituting the values we calculated:
\[ P(16, 25) = 41 + (-9) = 41 - 9 = 32 \]

### Step 5: Calculate \( Q(x, y) \)
The function \( Q(x, y) \) is defined as:
\[ Q(x, y) = f(x, y) - g(x, y) \]

Substituting the values:
\[ Q(16, 25) = 41 - (-9) = 41 + 9 = 50 \]

### Step 6: Calculate \( P(x, y) + Q(x, y) \)
Now, we find:
\[ P(16, 25) + Q(16, 25) = 32 + 50 = 82 \]

### Final Answer
The value of \( P(x, y) + Q(x, y) \) is \( 82 \).

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