A small company pays each of its 5 category ‘C’ workers Rs. 20,000, each of its 3 category ‘B’ workers Rs. 25,000 and a category ‘A’ worker Rs. 65,000. The number of workers earning less than the mean salary is
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Finding Workers Earning Less Than Mean Salary
This problem asks us to calculate the mean salary of workers in a small company and then determine how many workers earn less than this calculated mean salary. We are given the number of workers and their respective salaries for three different categories: Category ‘C’, Category ‘B’, and Category ‘A’.
Step-by-Step Mean Salary Calculation
To find the mean salary, we need to calculate the total salary paid by the company and divide it by the total number of workers.
1. Calculate the Total Number of Workers
The company has workers in three categories:
- Category ‘C’: 5 workers
- Category ‘B’: 3 workers
- Category ‘A’: 1 worker
The total number of workers is the sum of workers in all categories:
\(\text{Total Workers} = \text{Number of C workers} + \text{Number of B workers} + \text{Number of A workers}\)
\(\text{Total Workers} = 5 + 3 + 1 = 9\)
So, there are a total of 9 workers in the company.
2. Calculate the Total Salary Paid
Now, we calculate the total salary paid to all workers. We multiply the number of workers in each category by their respective salary and sum these amounts.
- Total salary for Category ‘C’ workers: \(5 \times \text{Rs. } 20,000 = \text{Rs. } 100,000\)
- Total salary for Category ‘B’ workers: \(3 \times \text{Rs. } 25,000 = \text{Rs. } 75,000\)
- Total salary for Category ‘A’ worker: \(1 \times \text{Rs. } 65,000 = \text{Rs. } 65,000\)
The total salary paid is the sum of salaries for all categories:
\(\text{Total Salary Paid} = \text{Salary C} + \text{Salary B} + \text{Salary A}\)
\(\text{Total Salary Paid} = \text{Rs. } 100,000 + \text{Rs. } 75,000 + \text{Rs. } 65,000 = \text{Rs. } 240,000\)
The total salary paid by the company is Rs. 240,000.
3. Calculate the Mean Salary
The mean salary is calculated by dividing the total salary paid by the total number of workers.
\(\text{Mean Salary} = \frac{\text{Total Salary Paid}}{\text{Total Workers}}\)
\(\text{Mean Salary} = \frac{\text{Rs. } 240,000}{9}\)
\(\text{Mean Salary} \approx \text{Rs. } 26,666.67\)
The mean salary is approximately Rs. 26,666.67.
Identifying Workers Earning Less Than the Mean Salary
Now we need to compare the salary of each worker category with the mean salary (approximately Rs. 26,666.67) to find out which workers earn less than the mean salary.
- Category ‘C’ workers earn Rs. 20,000 each. Is Rs. 20,000 < Rs. 26,666.67? Yes.
- Category ‘B’ workers earn Rs. 25,000 each. Is Rs. 25,000 < Rs. 26,666.67? Yes.
- Category ‘A’ worker earns Rs. 65,000. Is Rs. 65,000 < Rs. 26,666.67? No.
The workers earning less than the mean salary are those in Category ‘C’ and Category ‘B’.
Counting Workers Earning Less Than Mean Salary
The number of workers earning less than the mean salary is the sum of workers in Category ‘C’ and Category ‘B’.
\(\text{Number of workers earning less than mean} = \text{Number of C workers} + \text{Number of B workers}\)
\(\text{Number of workers earning less than mean} = 5 + 3 = 8\)
Therefore, 8 workers earn less than the mean salary.
| Category | Number of Workers | Salary per Worker (Rs.) | Total Salary per Category (Rs.) | Earning < Mean Salary (Rs. 26,666.67)? |
|---|---|---|---|---|
| C | 5 | 20,000 | 100,000 | Yes |
| B | 3 | 25,000 | 75,000 | Yes |
| A | 1 | 65,000 | 65,000 | No |
| Total | 9 | 240,000 |
Number of workers earning less than mean = 5 (from Category C) + 3 (from Category B) = 8 workers.
Conclusion on Workers Earning Less Than Mean
Based on the calculation, 8 workers in the company earn less than the calculated mean salary of approximately Rs. 26,666.67. These are the 5 workers from category ‘C’ and the 3 workers from category ‘B’.
Revision Table: Key Calculations
| Item | Calculation | Result |
|---|---|---|
| Total Workers | 5 + 3 + 1 | 9 |
| Total Salary Paid | (5 × 20,000) + (3 × 25,000) + (1 × 65,000) | Rs. 240,000 |
| Mean Salary | 240,000 ÷ 9 | Rs. 26,666.67 (approx) |
| Workers Earning < Mean | Count of workers with salary < 26,666.67 (approx) | 8 |
Additional Information: Understanding Mean Salary
The mean salary, also known as the average salary, is a measure of central tendency. It is calculated by summing up all the individual salaries and dividing by the total number of employees. While the mean is a useful indicator, it can sometimes be influenced by extreme values (like a very high salary for a Category A worker in this case). Other measures like the median salary might give a different picture of typical earnings, especially if there's a large variation in salaries. The median salary would be the middle value when all salaries are listed in order. In this case, with salaries 20000 (5 times), 25000 (3 times), and 65000 (1 time), the ordered list would be 20000, 20000, 20000, 20000, 20000, 25000, 25000, 25000, 65000. The median is the 5th value, which is Rs. 20,000. This shows how the mean (Rs. 26,666.67) is pulled up by the higher Category A salary compared to the median (Rs. 20,000).
Important Questions from Average
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If the average of 5 consecutive odd integers in increasing order is 11 , then the average of the last 3 of them is:
