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 initial number of employees, the percentage of employees hired, and the percentage of employees that left.

### Step-by-Step Solution:

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

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

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 some 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 simplifies to:
\[ \text{Total employees remaining} = n \left(1 + \frac{p}{100}\right) \left(1 - \frac{q}{100}\right) \]

4. **Setting the Equation**:
According to the problem, the total number of employees remaining is equal to \( n \):
\[ n \left(1 + \frac{p}{100}\right) \left(1 - \frac{q}{100}\right) = n \]
Dividing both sides by \( n \) (assuming \( n \neq 0 \)):
\[ \left(1 + \frac{p}{100}\right) \left(1 - \frac{q}{100}\right) = 1 \]

5. **Expanding the Equation**:
Expanding the left-hand side gives:
\[ 1 + \frac{p}{100} - \frac{q}{100} - \frac{pq}{10000} = 1 \]
Subtracting 1 from both sides:
\[ \frac{p}{100} - \frac{q}{100} - \frac{pq}{10000} = 0 \]

6. **Rearranging the Terms**:
Rearranging the equation gives:
\[ \frac{p}{100} - \frac{q}{100} = \frac{pq}{10000} \]
Multiplying through by 10000 to eliminate the denominators:
\[ 10000 \left(\frac{p}{100} - \frac{q}{100}\right) = pq \]
Simplifying this gives:
\[ 100(p - q) = pq \]

7. **Finding \( p - q \)**:
Dividing both sides by 100:
\[ p - q = \frac{pq}{100} \]

### Final Result:
Thus, the value of \( p - q \) is:
\[ p - q = \frac{pq}{100} \]