The average salary of the entire teaching staff in a college is ₹2,000 per day. The average salary of the male teachers is ₹2,500 and that of the female teachers is ₹1,200. If the number of male teachers is 16, then find the number of female teachers in the college.
10
Solving the Average Salary Problem for Teachers
This problem involves finding the number of female teachers given the average salaries of male teachers, female teachers, and the entire teaching staff, along with the number of male teachers. This is a classic application of the weighted average concept.
Understanding Weighted Average Salary
The average salary of the entire staff is the total salary of all teachers divided by the total number of teachers. When the staff is divided into groups (male and female teachers), the total salary is the sum of the total salaries of each group. The total salary for a group is the average salary of that group multiplied by the number in that group.
Let:
- \(N_m\) be the number of male teachers.
- \(N_f\) be the number of female teachers.
- \(A_m\) be the average salary of male teachers.
- \(A_f\) be the average salary of female teachers.
- \(A_{total}\) be the average salary of the entire teaching staff.
The total salary of male teachers is \(N_m \times A_m\).
The total salary of female teachers is \(N_f \times A_f\).
The total number of teachers is \(N_m + N_f\).
The total salary of all teachers is \((N_m + N_f) \times A_{total}\).
Since the total salary of all teachers is the sum of the total salaries of male and female teachers, we can write the equation:
\(N_m A_m + N_f A_f = (N_m + N_f) A_{total}\)
Applying Given Values to Find Number of Female Teachers
We are given the following information:
- Average salary of the entire teaching staff, \(A_{total}\) = ₹2,000 per day.
- Average salary of the male teachers, \(A_m\) = ₹2,500 per day.
- Average salary of the female teachers, \(A_f\) = ₹1,200 per day.
- Number of male teachers, \(N_m\) = 16.
- We need to find the number of female teachers, \(N_f\).
Substitute these values into the equation:
\(16 \times 2500 + N_f \times 1200 = (16 + N_f) \times 2000\)
Step-by-Step Calculation
Now, let's solve this equation for \(N_f\):
- Calculate the total salary of male teachers:
\(16 \times 2500 = 40000\)
So, the equation becomes:
\(40000 + 1200 N_f = (16 + N_f) \times 2000\)
- Distribute the 2000 on the right side:
\((16 + N_f) \times 2000 = 16 \times 2000 + N_f \times 2000 = 32000 + 2000 N_f\)
So, the equation is:
\(40000 + 1200 N_f = 32000 + 2000 N_f\)
- Rearrange the equation to gather \(N_f\) terms on one side and constant terms on the other side:
\(40000 - 32000 = 2000 N_f - 1200 N_f\)
- Perform the subtractions:
\(8000 = 800 N_f\)
- Solve for \(N_f\) by dividing both sides by 800:
\(N_f = \frac{8000}{800}\)
\(N_f = 10\)
Therefore, the number of female teachers in the college is 10.
| Item | Number | Average Salary (₹) | Total Salary (₹) |
|---|---|---|---|
| Male Teachers | 16 | 2,500 | \(16 \times 2500 = 40,000\) |
| Female Teachers | \(N_f\) | 1,200 | \(N_f \times 1200\) |
| Entire Staff | \(16 + N_f\) | 2,000 | \((16 + N_f) \times 2000\) |
From the table, the total salary of the entire staff must equal the sum of total salaries of male and female teachers:
\(40,000 + 1200 N_f = (16 + N_f) \times 2000\)
\(40,000 + 1200 N_f = 32,000 + 2000 N_f\)
\(40,000 - 32,000 = 2000 N_f - 1200 N_f\)
\(8,000 = 800 N_f\)
\(N_f = \frac{8000}{800} = 10\)
Revision Table: Average Salary Problem
| Concept | Formula/Method | Application |
|---|---|---|
| Average | Total Sum / Number of items | Used for each group (male, female, total) |
| Total Sum (Group) | Number in group × Average of group | Calculated for male teachers and female teachers |
| Weighted Average Relation | Sum of (Number in group × Average of group) = Total Number × Overall Average | Key equation used to solve for the unknown number |
Additional Information: Weighted Averages
A weighted average is an average where some values contribute more than others. In this problem, the average salary of the entire staff is a weighted average of the male and female teachers' average salaries. The "weights" are the number of teachers in each group. The group with more teachers has a greater influence on the overall average.
If the numbers of male and female teachers were equal, the overall average would be the simple average of ₹2,500 and ₹1,200, which is \(\frac{2500+1200}{2} = 1850\). However, the overall average is ₹2,000. Since ₹2,000 is closer to the male average (₹2,500) than the female average (₹1,200), this implies that there must be more male teachers than female teachers, which aligns with the given information ($N_m=16$). Our calculation shows that \(N_f=10\), confirming this observation.
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