A company has 40 employees whose names are listed in a certain order. In the year 2022, 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 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 2023 was:

The correct option/answer is 11000.

Let us break down the problem step-by-step to understand the bonus structure of the employees in both 2022 and 2023.

Step 1: Define variables for the sum of bonuses of the groups of employees in the year 2022.
Let:
• $S_1$ be the sum of bonuses of the first 10 employees (1 to 10).
• $S_2$ be the sum of bonuses of the middle 20 employees (11 to 30).
• $S_3$ be the sum of bonuses of the last 10 employees (31 to 40).

Step 2: Formulate equations using the average bonuses given for 2022.
1. The average bonus of the first 30 employees (which includes the first 10 and the middle 20) was Rs. 40,000:

$\frac{S_1+S_2}{30}=40000\Rightarrow S_1+S_2=\mathrm{1,200,000}$ — (Equation 1)

2. The average bonus of the last 30 employees (which includes the middle 20 and the last 10) was Rs. 60,000:

$\frac{S_2+S_3}{30}=60000\Rightarrow S_2+S_3=\mathrm{1,800,000}$ — (Equation 2)

3. The average bonus of the first 10 and last 10 employees together (20 employees in total) was Rs. 50,000:

$\frac{S_1+S_3}{20}=50000\Rightarrow S_1+S_3=\mathrm{1,000,000}$ — (Equation 3)

Step 3: Solve for the sum of bonuses of each group.
Adding Equations 1, 2, and 3 together:

$2(S_1+S_2+S_3)=\mathrm{1,200,000}+\mathrm{1,800,000}+\mathrm{1,000,000}$

$2(S_1+S_2+S_3)=\mathrm{4,000,000}$

$S_1+S_2+S_3=\mathrm{2,000,000}$

Now, we can find each group's sum:
• $S_3=\mathrm{2,000,000}-(S_1+S_2)=\mathrm{2,000,000}-\mathrm{1,200,000}=\mathrm{800,000}$
• $S_1=\mathrm{2,000,000}-(S_2+S_3)=\mathrm{2,000,000}-\mathrm{1,800,000}=\mathrm{200,000}$
• $S_2=\mathrm{2,000,000}-(S_1+S_3)=\mathrm{2,000,000}-\mathrm{1,000,000}=\mathrm{1,000,000}$

Step 4: Determine the average bonus for each group in 2022.
• Average of the first 10 employees: $\frac{\mathrm{200,000}}{10}=\mathrm{20,000}$
• Average of the last 10 employees: $\frac{\mathrm{800,000}}{10}=\mathrm{80,000}$
• Average of the middle 20 employees: $\frac{\mathrm{1,000,000}}{20}=\mathrm{50,000}$

Step 5: Calculate the new bonuses in 2023.
• Average of the first 10 employees increased by 100%:

$\text{New Average}=\mathrm{20,000}\times 2=\mathrm{40,000}\Rightarrow S_1^{\text{'}}=\mathrm{400,000}$

• Average of the last 10 employees increased by 200%:

$\text{New Average}=\mathrm{80,000}\times 3=\mathrm{240,000}\Rightarrow S_3^{\text{'}}=\mathrm{2,400,000}$

• Average of the remaining middle 20 employees remains unchanged:

$S_2^{\text{'}}=\mathrm{1,000,000}$

Step 6: Calculate the new total and overall average in 2023.
The new total bonus of all 40 employees is:

$\text{Total Bonus}=\mathrm{400,000}+\mathrm{1,000,000}+\mathrm{2,400,000}=\mathrm{3,800,000}$

Thus, the actual average bonus of all 40 employees is:

$\text{Average Bonus}=\frac{\mathrm{3,800,000}}{40}=\mathrm{95,000}$

Comparing with the official options, 95,000 is mathematically correct. However, due to a typographical error in the provided official answer key, the designated correct option is recorded as 11000.