Question 1 (c)
Age limits after retrenchment and the average age before and after
9 marks
- Time to write
- About 18 minutes
Question
Question 1 (c) · 9 marksThe age distribution of 2,000 employees employed in a large textile mill is given below:
| Age | Number of Employees | Age | Employees (Number) |
|---|---|---|---|
| 15 but less than 20 | 90 | 45 but less than 50 | 255 |
| 20 but less than 25 | 240 | 50 but less than 55 | 160 |
| 25 but less than 30 | 310 | 55 but less than 60 | 70 |
| 30 but less than 35 | 325 | 60 up to less than 65 | 25 |
| 35 but less than 40 | 275 | 65 but less than 70 | 20 |
| 40 but less than 45 | 230 |
Owing to heavy losses, the management decides to bring the strength down to 40% of the present number as per the scheme below:
To retrench the first 10% of employees from the lower age group,
To absorb the next 40% in other branches,
To make 10% from the highest age group retire prematurely.
What will the age limits be for the employees retained in the mill and for those moved to other branches? Also calculate the average age of the employees before and after retrenchment:
Reworded from the university paper: names and figures differ, the task is the same. The original paper belongs to the University of Delhi. See DU's question bank.
Before you write
Read off the question and its marks
Read from the question
- What…?Calculate…Two asks in one question: answer both.
- calculateFormula first, then the working, then the answer with its unit (₹, %, units).
Write it in this order
- Read the data; note exactly what is asked 1 min
- 1 Cumulative frequencies and the mean age before 2 marks 3 min
- 2 P₁₀, median and P₉₀ by interpolation 3 marks 5 min
- 3 Age limits of the groups retained and moved 1 mark 2 min
- 4 Mean age of the employees retained 2½ marks 4 min
- 5 Comment ½ mark 2 min
- Check: arithmetic, units (₹, %), final answer underlined 1 min
Try it first. About 18 minutes for 9 marks. Write it out, then open the solution.
Solution
Hover or tap a marked figure, like 72,000, to see how it is made and where else it appears.
Reading the scheme. Of the 2,000 employees, the youngest 10% (200) are retrenched, the next 40% (800) move to other branches, and the oldest 10% (200) retire early. The 40% left in the mill (800) are therefore those between the 50th and the 90th percentiles of age. The limits are the 10th, 50th and 90th percentiles (P₁₀, the median and P₉₀), found by interpolation within the class: P = L + [(k − c.f.) / f] × h, where k is the number of employees below the limit. The mean age uses the class mid-points, x̄ = Σfx / N.
Cumulative frequencies and the mean before retrenchment
| Age (years) | Employees (f) | c.f. | Mid-point (x) | fx |
|---|---|---|---|---|
| 15–20 | 90 | 90 | 17.5 | 1,575 |
| 20–25 | 240 | 330 | 22.5 | 5,400 |
| 25–30 | 310 | 640 | 27.5 | 8,525 |
| 30–35 | 325 | 965 | 32.5 | 10,562.5 |
| 35–40 | 275 | 1,240 | 37.5 | 10,312.5 |
| 40–45 | 230 | 1,470 | 42.5 | 9,775 |
| 45–50 | 255 | 1,725 | 47.5 | 12,112.5 |
| 50–55 | 160 | 1,885 | 52.5 | 8,400 |
| 55–60 | 70 | 1,955 | 57.5 | 4,025 |
| 60–65 | 25 | 1,980 | 62.5 | 1,562.5 |
| 65–70 | 20 | 2,000 | 67.5 | 1,350 |
| Total | 2,000 | 73,600 |
Average age before retrenchment = 73,600 / 2,000 = 36.80 years.
- P₁₀ (200th employee) lies in 20–25: 20 + [(200 − 90) / 240] × 5 = 22.29 years.
- Median (1,000th employee) lies in 35–40: 35 + [(1,000 − 965) / 275] × 5 = 35.64 years.
- P₉₀ (1,800th employee) lies in 50–55: 50 + [(1,800 − 1,725) / 160] × 5 = 52.34 years.
| Group | Number | Ages |
|---|---|---|
| Retrenched | 200 | below 22.29 years |
| Moved to other branches | 800 | 22.29 to 35.64 years |
| Retained in the mill | 800 | 35.64 to 52.34 years |
| Retired early | 200 | above 52.34 years |
Average age after retrenchment (the 800 employees retained in the mill; within a class, employees are taken as spread evenly, so a part-class is represented by its own mid-point)
| Age (years) | Employees | Mid-point | Age total |
|---|---|---|---|
| 35.64–40 | 240 | 37.82 | 9,076.36 |
| 40–45 | 230 | 42.5 | 9,775 |
| 45–50 | 255 | 47.5 | 12,112.5 |
| 50–52.34 | 75 | 51.17 | 3,837.89 |
| Total | 800 | 34,801.75 |
The 240 in the first row are 1,240 − 1,000; the 75 in the last are 1,800 − 1,725. Average age after retrenchment = 34,801.75 / 800 = 43.50 years.
Comment. The scheme removes both the youngest and the oldest groups and moves the next-youngest out, so the workforce left in the mill is older on average (43.50 against 36.80 years) and much more uniform in age.
The other reading. If after retrenchment is read as everyone the company still employs (the 1,600 retained or moved, aged 22.29 to 52.34), their average age is 36.37 years; those moved to other branches average 29.25 years. The answer above takes the employees retained in the mill, the group the question names.
Where the 9 marks sit
| # | Component | Marks |
|---|---|---|
| 1 | Cumulative frequencies and the mean age before | 2 |
| 2 | P₁₀, median and P₉₀ by interpolation | 3 |
| 3 | Age limits of the groups retained and moved | 1 |
| 4 | Mean age of the employees retained | 2.5 |
| 5 | Comment | 0.5 |
| Total | 9 |
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