Finding the Number of Non-Officers Based on Average Salary

This problem requires us to find the number of non-officers in an office, given the average salaries of different groups and the total staff, along with the specific number of officers. We can solve this using the concept of weighted averages or by setting up an equation based on the total salary.

Understanding Average Salary

The average salary is calculated by dividing the total salary paid to a group by the number of people in that group. Conversely, the total salary for a group is the average salary multiplied by the number of people in that group.

• Average Salary = \(\frac{\text{Total Salary}}{\text{Number of People}}\)
• Total Salary = Average Salary \(\times\) Number of People

Given Information

Let's list the information provided in the question:

• Average salary of the entire staff = Rs. 3,560 per month
• Average salary of officers = Rs. 5,400 per month
• Average salary of non-officers = Rs. 2,600 per month
• Number of officers = 12

We need to find the number of non-officers.

Step-by-Step Solution (Algebraic Method)

Let \(N_{officers}\) be the number of officers and \(N_{non-officers}\) be the number of non-officers. The total number of staff is \(N_{total} = N_{officers} + N_{non-officers}\).

1. Calculate the total salary of officers:

Total Officer Salary = Average Officer Salary \(\times\) Number of Officers

Total Officer Salary = \(5400 \times 12\)

Total Officer Salary = Rs. \(64800\)

2. Express the total salary of non-officers in terms of \(N_{non-officers}\):

Total Non-Officer Salary = Average Non-Officer Salary \(\times\) Number of Non-Officers

Total Non-Officer Salary = \(2600 \times N_{non-officers}\)

3. Express the total salary of the entire staff:

Total Staff Salary = Average Total Salary \(\times\) Total Number of Staff

Total Staff Salary = \(3560 \times (N_{officers} + N_{non-officers})\)

Since \(N_{officers} = 12\):

Total Staff Salary = \(3560 \times (12 + N_{non-officers})\)

4. Set up an equation: The total salary of the entire staff is the sum of the total salaries of officers and non-officers.

Total Staff Salary = Total Officer Salary + Total Non-Officer Salary

\(3560 \times (12 + N_{non-officers}) = 64800 + 2600 \times N_{non-officers}\)

5. Solve the equation for \(N_{non-officers}\):

First, distribute on the left side:

\(3560 \times 12 + 3560 \times N_{non-officers} = 64800 + 2600 \times N_{non-officers}\)

\(42720 + 3560 N_{non-officers} = 64800 + 2600 N_{non-officers}\)

Now, gather the terms with \(N_{non-officers}\) on one side and constant terms on the other:

\(3560 N_{non-officers} - 2600 N_{non-officers} = 64800 - 42720\)

Subtract the terms:

\(960 N_{non-officers} = 22080\)

Finally, divide to find \(N_{non-officers}\):

\(N_{non-officers} = \frac{22080}{960}\)

\(N_{non-officers} = \frac{2208}{96}\)

\(N_{non-officers} = 23\)

Alternative Method (Alligation or Mixture Rule)

This method is often used for problems involving weighted averages. We can visualize the average salaries on a scale:

Non-officers (2600) <---- Total Average (3560) ----> Officers (5400)

The difference between the total average and the non-officer average is \(3560 - 2600 = 960\).

The difference between the officer average and the total average is \(5400 - 3560 = 1840\).

According to the alligation rule, the ratio of the number of non-officers to the number of officers is inversely proportional to these differences. The ratio of quantities (number of people) is equal to the ratio of the differences in average values, but crossed over:

\(\frac{\text{Number of Non-officers}}{\text{Number of Officers}} = \frac{\text{Difference for Officers}}{\text{Difference for Non-officers}}\)

\(\frac{N_{non-officers}}{N_{officers}} = \frac{5400 - 3560}{3560 - 2600}\)

\(\frac{N_{non-officers}}{N_{officers}} = \frac{1840}{960}\)

\(\frac{N_{non-officers}}{N_{officers}} = \frac{184}{96}\)

Simplifying the fraction \(\frac{184}{96}\):

\(\frac{184 \div 8}{96 \div 8} = \frac{23}{12}\)

So, \(\frac{N_{non-officers}}{N_{officers}} = \frac{23}{12}\).

We are given that the number of officers (\(N_{officers}\)) is 12.

\(\frac{N_{non-officers}}{12} = \frac{23}{12}\)

Multiplying both sides by 12:

\(N_{non-officers} = 23\)

Both methods yield the same result. The number of non-officers is 23.

Equation: \(64800 + 2600 N_{non-officers} = 3560 (12 + N_{non-officers})\)

\(64800 + 2600 N_{non-officers} = 42720 + 3560 N_{non-officers}\)

\(64800 - 42720 = 3560 N_{non-officers} - 2600 N_{non-officers}\)

\(22080 = 960 N_{non-officers}\)

\(N_{non-officers} = \frac{22080}{960} = 23\)

Conclusion

The number of non-officers in the office is 23.

Revision Table: Average Salary Concepts

Additional Information: Weighted Averages

A weighted average is an average in which each observation in the data set does not necessarily contribute equally to the final average. When dealing with averages of subgroups that combine to form a larger group, the overall average is a weighted average of the subgroup averages, with the number of elements in each subgroup acting as the weights.

In this problem:

• The entire staff is the combined group.
• Officers and non-officers are the subgroups.
• The average salary of the entire staff (3560) is the weighted average of the officer average salary (5400) and the non-officer average salary (2600).
• The weights are the number of officers (12) and the number of non-officers (\(N_{non-officers}\)).

The formula for the weighted average in this context is:

Average Total Salary = \(\frac{(\text{Avg Officer Salary} \times \text{No. of Officers}) + (\text{Avg Non-Officer Salary} \times \text{No. of Non-officers})}{\text{No. of Officers} + \text{No. of Non-officers}}\)

\(3560 = \frac{(5400 \times 12) + (2600 \times N_{non-officers})}{12 + N_{non-officers}}\)

Multiplying both sides by \((12 + N_{non-officers})\) gives:

\(3560 \times (12 + N_{non-officers}) = (5400 \times 12) + (2600 \times N_{non-officers})\)

This leads back to the same equation we solved using the algebraic method, confirming the relationship between the concepts.