In an office there were initially n employees. The HR manager first hired `P%` employees then after a month`q%`employees left the office, then there were finally n employees remained in the office , the value of p-q is :

Options

A. pq
B. `(pq)/100`
C. `p/q`
D. none of these

Answer

To solve the problem step by step, we will analyze the situation involving the employees in the office.

### Step 1: Initial Setup
Let the initial number of employees be \( n \).

### Step 2: Hiring New Employees
The HR manager hires \( P\% \) of the initial employees. The number of employees hired can be calculated as:
\[ \text{Employees hired} = \frac{P}{100} \times n \]
Thus, the total number of employees after hiring becomes:
\[ \text{Total Employees after hiring} = n + \frac{P}{100} \times n = n \left(1 + \frac{P}{100}\right) \]

### Step 3: Employees Leaving
After a month, \( q\% \) of the total employees leave. The number of employees leaving is:
\[ \text{Employees leaving} = \frac{q}{100} \times \left(n \left(1 + \frac{P}{100}\right)\right) \]
Thus, the total number of employees remaining after \( q\% \) leave is:
\[ \text{Total Employees remaining} = n \left(1 + \frac{P}{100}\right) - \frac{q}{100} \times \left(n \left(1 + \frac{P}{100}\right)\right) \]
This can be simplified to:
\[ \text{Total Employees remaining} = n \left(1 + \frac{P}{100}\right) \left(1 - \frac{q}{100}\right) \]

### Step 4: Setting Up the Equation
According to the problem, the total number of employees remaining is still \( n \):
\[ n \left(1 + \frac{P}{100}\right) \left(1 - \frac{q}{100}\right) = n \]

### Step 5: Simplifying the Equation
We can divide both sides by \( n \) (assuming \( n \neq 0 \)):
\[ \left(1 + \frac{P}{100}\right) \left(1 - \frac{q}{100}\right) = 1 \]
Expanding the left side gives:
\[ 1 + \frac{P}{100} - \frac{q}{100} - \frac{Pq}{10000} = 1 \]
Subtracting 1 from both sides results in:
\[ \frac{P}{100} - \frac{q}{100} - \frac{Pq}{10000} = 0 \]

### Step 6: Rearranging the Equation
Rearranging gives:
\[ \frac{P}{100} - \frac{q}{100} = \frac{Pq}{10000} \]
Multiplying through by 10000 to eliminate the fractions:
\[ 100P - 100q = Pq \]

### Step 7: Final Expression
Rearranging this gives:
\[ 100P - Pq = 100q \]
Factoring out \( P \) from the left side:
\[ P(100 - q) = 100q \]
Thus, we have:
\[ P = \frac{100q}{100 - q} \]

### Step 8: Finding \( p - q \)
From the earlier steps, we can deduce that:
\[ p - q = \frac{100q}{100 - q} - q \]
This can be simplified further, but the key point is that \( p - q \) can be expressed in terms of \( q \).

### Conclusion
The value of \( p - q \) can be expressed as:
\[ p - q = \frac{100q}{100 - q} - q \]