Question

A company has 40 employees whose names are listed in a certain order. In the year 1925, the average bonus of the first 30 employees was Rs. 40000, of the last 30 employees was Rs. 60000, and of the first 10 and last 10 employees together it was Rs. 50000. Next year, the average bonus of the first 10 employees increased by 100%, of the last 10 employees increased by 200% and of the remaining employees was unchanged. Then, the average bonus, in Rupees, of all the 40 employees together in the year 1926 would have been:

• A. 90000
• B. 78000
• C. 85000
• D. 80000
• E. 60000

Hint:

Be careful with percentage increases: "increased by 100%" means doubled; "increased by 200%" means tripled. Always identify the base value (the original averag(e) correctly.

Answer 1

The Correct Option is D

Step 1: Define Variables:
Let the bonuses of the 40 employees in 1925 be $B_1, B_2, \dots, B_{40}$.
Step 2: Form Equations from Given Averages:
First 30: $\frac{\sum_{i=1}^{30} B_i}{30} = 40000 \implies \sum_{i=1}^{30} B_i = 12,00,000$. (1) Last 30: $\frac{\sum_{i=11}^{40} B_i}{30} = 60000 \implies \sum_{i=11}^{40} B_i = 18,00,000$. (2) First 10 and last 10 together (20 employees): $\frac{\sum_{i=1}^{10} B_i + \sum_{i=31}^{40} B_i}{20} = 50000 \implies \sum_{i=1}^{10} B_i + \sum_{i=31}^{40} B_i = 10,00,000$. (3)
Step 3: Find Sum of Bonuses for Middle 20 (employees 11-30):
Add (1) and (2): $(\sum_{i=1}^{30} B_i) + (\sum_{i=11}^{40} B_i) = 12,00,000 + 18,00,000 = 30,00,000$. The sum $\sum_{i=1}^{30} B_i + \sum_{i=11}^{40} B_i = \sum_{i=1}^{10} B_i + 2\sum_{i=11}^{30} B_i + \sum_{i=31}^{40} B_i$. So, $(\sum_{i=1}^{10} B_i + \sum_{i=31}^{40} B_i) + 2\sum_{i=11}^{30} B_i = 30,00,000$. Using (3): $10,00,000 + 2\sum_{i=11}^{30} B_i = 30,00,000$. Thus, $2\sum_{i=11}^{30} B_i = 20,00,000 \implies \sum_{i=11}^{30} B_i = 10,00,000$.
Step 4: Find Sums for Groups in 1925:
From (1): $\sum_{i=1}^{10} B_i + \sum_{i=11}^{30} B_i = 12,00,000 \implies \sum_{i=1}^{10} B_i + 10,00,000 = 12,00,000 \implies \sum_{i=1}^{10} B_i = 2,00,000$. From (2): $\sum_{i=11}^{30} B_i + \sum_{i=31}^{40} B_i = 18,00,000 \implies 10,00,000 + \sum_{i=31}^{40} B_i = 18,00,000 \implies \sum_{i=31}^{40} B_i = 8,00,000$.
Step 5: Calculate Bonuses for 1926:
First 10: Average increases by 100% means new average = $40000 \times 2 = 80000$. New sum = $10 \times 80000 = 8,00,000$. Last 10: Average increases by 200% means new average = $60000 \times 3 = 1,80,000$. New sum = $10 \times 1,80,000 = 18,00,000$. Middle 20: Unchanged, sum remains $\sum_{i=11}^{30} B_i = 10,00,000$.
Step 6: Find Overall Average for 1926:
Total sum in 1926 = $8,00,000 + 10,00,000 + 18,00,000 = 36,00,000$. Number of employees = 40. Average = $\frac{36,00,000}{40} = 90,000$? Wait, this is Rs. 90,000. There's a miscalculation. Check last 10 average increase: original average of last 30 was 60,000, but the last 10's average in 1925 is not 60,000. We must use the sum we calculated for last 10 in 1925. In 1925, sum for last 10 = Rs. 8,00,000. Their average in 1925 = 80,000. Increase by 200% means new average = $80,000 \times 3 = 2,40,000$. New sum = $10 \times 2,40,000 = 24,00,000$.
Step 7: Recalculate 1926 Sum and Average:
First 10 (1926 sum) = Rs. 8,00,000. Middle 20 (1926 sum) = Rs. 10,00,000. Last 10 (1926 sum) = Rs. 24,00,000. Total sum = $8,00,000 + 10,00,000 + 24,00,000 = 42,00,000$. Average = $\frac{42,00,000}{40} = 1,05,000$? This is not matching optionss. Let's re-read: "average bonus of the first 10 employees increased by 100%" means the average itself increased by 100%, i.e., doubled. We have the 1925 average of first 10 = 2,00,000/10 = 20,000. Double is 40,000. Sum = 4,00,000. "average bonus of the last 10 employees increased by 200%". 1925 average of last 10 = 8,00,000/10 = 80,000. Increase by 200% means new average = 80,000 + 200% of 80,000 = 80,000 + 1,60,000 = 2,40,000. Sum = 24,00,000. Middle 20 unchanged sum = 10,00,000. Total 1926 sum = 4,00,000 + 10,00,000 + 24,00,000 = 38,00,000. Average = 38,00,000/40 = 95,000. This is options ((e). Let's verify 1925 totals: Total 1925 sum = 2,00,000 + 10,00,000 + 8,00,000 = 20,00,000. Average = 50,000. Now, increases: First 10 sum goes from 2,00,000 to 4,00,000 (+2,00,000). Last 10 sum goes from 8,00,000 to 24,00,000 (+16,00,000). Total increase = +18,00,000. New total = 20,00,000 + 18,00,000 = 38,00,000. Average = 95,000.
Step 8: Final Answer:
The average bonus in 1926 is Rs. 95,000.