If [x] is read as the greatest integer less than or equal to x, {x} is the least integer 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, 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, we need to evaluate the expressions \( P(x,y) \) and \( Q(x,y) \) using the definitions provided. Let's break it down step by step.

### Step 1: Identify the values of \( x \) and \( y \)
We are 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 \( y \)

For \( x = 16 \):
\[ [x] = 16 \quad (\text{since } 16 \text{ is an integer}) \]

For \( y = 25 \):
\[ \{y\} = 25 \quad (\text{since } 25 \text{ is also an integer}) \]

Now, substituting these values into \( f(x,y) \):
\[ f(16, 25) = 16 + 25 = 41 \]

### Step 3: Calculate \( g(x,y) \)
The function \( g(x,y) \) is defined as:
\[ g(x,y) = [x] - \{y\} \]

Using the same values:
\[ [x] = 16 \quad (\text{as calculated before}) \]
\[ \{y\} = 25 \quad (\text{as calculated before}) \]

Now substituting these values into \( g(x,y) \):
\[ g(16, 25) = 16 - 25 = -9 \]

### 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 found:
\[ P(16, 25) = 41 + (-9) = 41 - 9 = 32 \]

### 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 we found:
\[ Q(16, 25) = 41 - (-9) = 41 + 9 = 50 \]

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

### Final Answer
Thus, the value of \( P(x,y) + Q(x,y) \) is:
\[ \boxed{82} \]