There were 25 employees in an office. The mean salary of the employees was Rs. 30,000 per month. A senior officer having a salary of Rs. 60,000 per month retired, and a new clerk was appointed in his place. The mean salary of the employees in the office was thus reduced by Rs. 1,000 per month. The salary of the new clerk (per month) was:
Rs. 35,000
Understanding the Mean Salary Problem
This problem involves understanding how the mean (average) salary of a group of employees changes when one employee leaves and another joins. The mean is calculated as the total sum of salaries divided by the number of employees.
The key is to find the total salary of all employees before the change and after the change. The difference in the total salary, after accounting for the retired officer's salary, will reveal the new clerk's salary.
Initial Situation Analysis
Initially, we have:
- Number of employees: 25
- Mean salary: Rs. 30,000
The total salary of all employees initially can be calculated using the formula:
\(\text{Total Salary} = \text{Number of Employees} \times \text{Mean Salary}\)
Initial Total Salary = \(25 \times 30,000\)
Initial Total Salary = Rs. 750,000
Changes in the Office
A senior officer retired with a salary of Rs. 60,000. This means the total salary of the remaining 24 employees is reduced by this amount.
A new clerk is appointed in his place. The number of employees is back to 25.
The new mean salary is reduced by Rs. 1,000. So, the new mean salary is:
New Mean Salary = Initial Mean Salary - Rs. 1,000
New Mean Salary = \(30,000 - 1,000 = 29,000\)
Calculating the New Total Salary
With the new mean salary and the number of employees back to 25, we can calculate the new total salary of all employees:
New Total Salary = \(25 \times 29,000\)
New Total Salary = Rs. 725,000
Finding the New Clerk's Salary
Let the salary of the new clerk be \(x\).
The initial total salary (Rs. 750,000) was the sum of the salaries of the senior officer (Rs. 60,000) and the salaries of the other 24 employees.
So, Sum of salaries of the other 24 employees = Initial Total Salary - Senior Officer's Salary
Sum of salaries of the other 24 employees = \(750,000 - 60,000 = 690,000\)
The new total salary (Rs. 725,000) is the sum of the salaries of these same 24 employees and the new clerk's salary (\(x\)).
New Total Salary = (Sum of salaries of the other 24 employees) + New Clerk's Salary
\(725,000 = 690,000 + x\)
Now, we can solve for \(x\):
\(x = 725,000 - 690,000\)
\(x = 35,000\)
The salary of the new clerk is Rs. 35,000 per month.
Summary of Calculations
| Initial Number of Employees | 25 |
| Initial Mean Salary | Rs. 30,000 |
| Initial Total Salary | \(25 \times 30,000 = \text{Rs. } 750,000\) |
| Retired Officer's Salary | Rs. 60,000 |
| Number of Employees after appointment | 25 |
| New Mean Salary | \(30,000 - 1,000 = \text{Rs. } 29,000\) |
| New Total Salary | \(25 \times 29,000 = \text{Rs. } 725,000\) |
| Difference in Total Salary | \(750,000 - 725,000 = \text{Rs. } 25,000\) |
| Salary of 24 remaining employees | \(750,000 - 60,000 = \text{Rs. } 690,000\) |
| New Clerk's Salary | \(725,000 - 690,000 = \text{Rs. } 35,000\) |
Conclusion
The salary of the new clerk is Rs. 35,000 per month.
Revision Table: Mean and Salary Calculations
| Concept | Formula/Explanation | Application in Problem |
|---|---|---|
| Mean (Average) | Sum of values / Number of values | Used to calculate initial and final total salaries. |
| Total Sum from Mean | Mean \(\times\) Number of items | Initial Total Salary = \(30000 \times 25\), New Total Salary = \(29000 \times 25\). |
| Change in Total Sum | New Total Sum - Old Total Sum | The difference in total salary reflects the change due to the swap of employees. |
| Finding Unknown Value | Set up an equation based on total sum before and after the change. | Old Total - Retired Salary + New Salary = New Total. |
Additional Information: Impact of Individual Value on Mean
This problem illustrates how a single value (an individual's salary) affects the overall mean of a dataset (the average office salary). When a high value (the senior officer's salary) is replaced by a lower value (the new clerk's salary), the total sum decreases, and consequently, the mean decreases, assuming the number of items remains the same.
If the new employee's salary were higher than the retiring employee's, the mean would increase. If the new salary were exactly the same, the mean would remain unchanged.
The change in the mean is directly related to the difference between the value leaving the group and the value entering the group, divided by the number of items in the group.
Change in Mean = (Value Added - Value Removed) / Number of Items
In this case: \(-1000 = (x - 60000) / 25\)
\(-1000 \times 25 = x - 60000\)
\(-25000 = x - 60000\)
\(x = 60000 - 25000\)
\(x = 35000\)
This confirms the result obtained through the total sum method.
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