The average salary of employees of a factory is ₹15,000. The average salary of 250 of the employees is ₹16,000 and that of the remaining employees is ₹13,750. Total number of employees in the factory is:
450
Understanding the Average Salary Problem
This problem involves calculating the total number of employees in a factory based on their average salaries in different groups. We are given the overall average salary, the average salary for a specific number of employees, and the average salary for the remaining employees.
Key Information Provided
- Overall average salary of all employees: ₹15,000
- Number of employees in the first group: 250
- Average salary of the first group: ₹16,000
- Average salary of the remaining employees: ₹13,750
Our goal is to find the total number of employees in the factory.
Approach to Solving the Average Salary Question
The average salary is calculated as the total salary divided by the number of employees. We can use this relationship to set up equations. Let's denote:
- Total number of employees = \(N\)
- Number of the first group of employees = \(n_1 = 250\)
- Number of the remaining employees = \(n_2\)
- Total salary of all employees = \(S\)
- Total salary of the first group = \(S_1\)
- Total salary of the remaining employees = \(S_2\)
We know that the total number of employees is the sum of the employees in the two groups:
\[N = n_1 + n_2\]
Also, the total salary of all employees is the sum of the total salaries of the two groups:
\[S = S_1 + S_2\]
Calculations Using Average Salary Formula
The average salary is given by:
\[\text{Average Salary} = \frac{\text{Total Salary}}{\text{Number of Employees}}\]
From this, we can express the total salary as:
\[\text{Total Salary} = \text{Average Salary} \times \text{Number of Employees}\]
Using the given information, we can write the total salaries for each group and the total factory:
- For all employees: \(S = 15000 \times N\)
- For the first group: \(S_1 = 16000 \times n_1 = 16000 \times 250\)
- For the remaining group: \(S_2 = 13750 \times n_2\)
Setting up the Equation
Substitute the expressions for \(S\), \(S_1\), and \(S_2\) into the equation \(S = S_1 + S_2\):
\[15000 \times N = (16000 \times 250) + (13750 \times n_2)\]
We also know that \(N = 250 + n_2\). Substitute this into the equation:
\[15000 \times (250 + n_2) = (16000 \times 250) + (13750 \times n_2)\]
Solving for the Number of Remaining Employees (\(n_2\))
Now, we solve the equation for \(n_2\):
First, calculate the product \(16000 \times 250\):
\[16000 \times 250 = 4000000\]
Distribute the 15000 on the left side:
\[15000 \times 250 + 15000 \times n_2 = 4000000 + 13750 \times n_2\]
\[3750000 + 15000 n_2 = 4000000 + 13750 n_2\]
Rearrange the terms to group \(n_2\) on one side and constants on the other:
\[15000 n_2 - 13750 n_2 = 4000000 - 3750000\]
\[1250 n_2 = 250000\]
Now, solve for \(n_2\):
\[n_2 = \frac{250000}{1250}\]
\[n_2 = \frac{25000}{125}\]
\[n_2 = 200\]
So, the number of remaining employees is 200.
Calculating the Total Number of Employees (\(N\))
The total number of employees is the sum of the first group and the remaining group:
\[N = n_1 + n_2\]
\[N = 250 + 200\]
\[N = 450\]
The total number of employees in the factory is 450.
Verification
Let's check if the overall average is correct with 450 employees, 250 at ₹16,000, and 200 at ₹13,750.
- Total salary of first group = \(250 \times 16000 = 4000000\)
- Total salary of remaining group = \(200 \times 13750 = 2750000\)
- Total salary of all employees = \(4000000 + 2750000 = 6750000\)
- Total number of employees = \(250 + 200 = 450\)
Overall average salary = \(\frac{\text{Total Salary}}{\text{Total Number of Employees}} = \frac{6750000}{450}\)
\[\frac{6750000}{450} = \frac{675000}{45}\]
\[\frac{675000}{45} = \frac{135000}{9} = 15000\]
The calculated overall average salary of ₹15,000 matches the given information, confirming our total number of employees is correct.
| Group | Number of Employees | Average Salary (₹) | Total Salary (₹) |
|---|---|---|---|
| Group 1 | 250 | 16,000 | \(250 \times 16000 = 4,000,000\) |
| Remaining Group | \(n_2 = 200\) | 13,750 | \(200 \times 13750 = 2,750,000\) |
| Total Factory | \(N = 250 + 200 = 450\) | 15,000 (Given) | \(4,000,000 + 2,750,000 = 6,750,000\) |
Revision Table: Average Calculations
Understanding average calculations is fundamental. Here's a quick review:
| Concept | Formula | Explanation |
|---|---|---|
| Average | \(\frac{\text{Sum of quantities}}{\text{Number of quantities}}\) | Represents the central value of a set of numbers. |
| Total Sum | \(\text{Average} \times \text{Number of quantities}\) | If you know the average and the number of items, you can find the sum. |
| Weighted Average | \(\frac{\sum (w_i \times x_i)}{\sum w_i}\) | Used when different items have different importance or "weights", like calculating the average of groups with different sizes. In our problem, the number of employees in each group acts as the weight. |
Additional Information: Applications of Averages
Averages are used widely in various fields:
- Statistics: Mean, median, and mode are types of averages used to summarize data sets.
- Economics: Calculating average income, average cost, average consumption.
- Finance: Average returns on investment, average prices.
- Science: Calculating average speed, average temperature.
- Everyday Life: Average score, average height, average expenses.
Understanding how to calculate and interpret averages, especially weighted averages as shown in this salary problem, is a key skill in quantitative reasoning.
Important Questions from Average
Find the mean of prime numbers between 1 and 30.
If mode and mean of a data are 18 and 21 respectively, then median of the data is
The average of the numbers 5, 3, 9, 11, 29 and (p+1) is 12. The average increases by 2 when the numbers (q – 5) and (q +11) are also included. Find the value of q.
In a particular week, the average earning per day of a plumber from Monday to Wednesday remained ₹580 and from Thursday to Saturday it was ₹612. If the average earning for the whole week was ₹675, then how much did he earn on Sunday?
If the mean of the numbers 2, (p + 1), 8, 19, 16, 3 and p is 9, then find the mode of the numbers.
