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

We have 40 employees in a specific order, and we need to find their average bonus in 2023 after certain changes.

What we know for 2022:

First 30 employees: average bonus = Rs. 40000

Last 30 employees: average bonus = Rs. 60000

First 10 + Last 10 employees: average bonus = Rs. 50000

What changes in 2023:

First 10 employees: bonus increases by 100% (doubles)

Last 10 employees: bonus increases by 200% (triples)

Middle 20 employees: bonus unchanged

---

We think of the 40 employees as three overlapping groups:

Group A: Employees 1-10 (first 10)

Group B: Employees 11-30 (middle 20)

Group C: Employees 31-40 (last 10)

The given information talks about "first 30" (A+B), "last 30" (B+C), and "first 10 + last 10" (A+C). This grouping helps us use all the given information systematically.

Let's call their total bonuses in 2022: $a$, $b$, and $c$ respectively.

---

Total = Average × Number of people

From the given information:

First 30 employees (A + B): $a + b = 30 × 40000 = 1200000$

Last 30 employees (B + C): $b + c = 30 × 60000 = 1800000$

First 10 + Last 10 (A + C): $a + c = 20 × 50000 = 1000000$

---

We have three equations:

1. $a + b = 1200000$
2. $b + c = 1800000$
3. $a + c = 1000000$

Instead of solving one by one, let's add all three equations:

$(a + b) + (b + c) + (a + c) = 1200000 + 1800000 + 1000000$

$2a + 2b + 2c = 4000000$

$a + b + c = 2000000$

When we add all three equations, each variable appears exactly twice, so we get $2(a + b + c)$.

Now we can find each group's total:

From equation 2: $a = (a + b + c) - (b + c) = 2000000 - 1800000 = 200000$

From equation 1: $c = (a + b + c) - (a + b) = 2000000 - 1200000 = 800000$

From equation 3: $b = (a + b + c) - (a + c) = 2000000 - 1000000 = 1000000$

---

100% increase means bonus becomes double (original + 100% of original = 2 × original)

200% increase means bonus becomes triple (original + 200% of original = 3 × original)

Group A (employees 1-10):

New sum = $200000 × 2 = 400000$

Group B (employees 11-30):

Sum remains = $1000000$

Group C (employees 31-40):

New sum = $800000 × 3 = 2400000$

---

Total sum in 2023 = $400000 + 1000000 + 2400000 = 3800000$

Average bonus in 2023 = $\dfrac{3800000}{40} = 95000$

Answer: Rs. 95000