Question

In an office, 20 clerks get a salary of Rs. 2500 each and 10 officers get a salary of Rs. 4000 each. The average salary of the employees in the office is
(a) Rs. 2800
(b) Rs. 3000
(c) Rs. 3200
(d) Rs. 3600

Answer 1

Hint: Here, we have to find the average salary of the employees in the office. We will find the sum of salaries of all employees, and the total number of employees. Then, using the formula for mean, we will find the average salary of the employees.

Formula Used: We will use the formula of the mean of a series, \[\overline x = \dfrac{{\sum x }}{N}\], where \[N\] is the number of terms and \[\sum x \] is the sum of all the terms.

Complete step-by-step answer:
We will find the sum of the salaries of all employees.
First, we will find the sum of salaries of the clerks.
We know that the 20 clerks get Rs. 2500 each.
The total/sum of the salaries earned by 20 clerks is equal to the salary earned by each clerk multiplied by the number of clerks.
Therefore, we can find the sum of the salaries of 20 clerks as
\[{\rm{Rs}}{\rm{. }}2500 \times 20 = {\rm{Rs}}{\rm{. 5}}0000\]
Next, we will find the sum of salaries of the officers.
We know that the 10 officers get Rs. 4000 each.
The total/sum of the salaries earned by 10 officers is equal to the salary earned by each officer multiplied by the number of officers.
Therefore, we can find the sum of the salaries of 10 officers as
\[{\rm{Rs}}{\rm{. }}4000 \times 10 = {\rm{Rs}}{\rm{. }}40000\]
Now, we will find the sum of the salaries of all the employees.
The sum of the salaries of all employees is the sum of salaries of 20 clerks and the sum of salaries of 10 clerks.
Therefore, we get
Sum of salaries of all employees \[ = {\rm{Rs}}{\rm{. 5}}0000 + {\rm{Rs}}{\rm{. }}40000 = {\rm{Rs}}{\rm{. 9}}0000\]
The total number of employees is the sum of the number of clerks and number of officers.
Therefore, we get
Total number of employees \[ = 20 + 10 = 30\]
Now, substituting \[\sum x = 90000\] and \[N = 30\] in the formula for average/mean, we get
Average salary of the employees in the office \[ = \dfrac{{90000}}{{30}}\]
Simplifying the expression, we get
Average salary of the employees in the office \[ = {\rm{Rs}}{\rm{. }}3000\]
Therefore, the average salary of the employees in the office is Rs. 3000.
Thus, the correct option is option (b).

Note: We can also solve this question by using the formula for combined mean. The combined mean of two series is given by \[\overline {{x_{12}}} = \dfrac{{{N_1} \times \overline {{x_1}} + {N_2} \times \overline {{x_2}} }}{{{N_1} + {N_2}}}\], where \[{N_1}\] is the number of terms in the first series, \[{N_2}\] is the number of terms in the second series, \[\overline {{x_1}} \] is the mean of the first series, and \[\overline {{x_2}} \] is the mean of the second series. However, we need to calculate the means of the two series as well. Since \[{N_1} \times \overline {{x_1}} \] is equal to \[\sum {{x_1}} \], the combined mean formula follows the same logic as the solution provided, but includes more irrelevant steps.