Arvind Singh purchased a 40 seater bus. He started his services on route no. 2 (from Terhipuliya to Charbagh with route length of 50 km). His profit (P) from the bus depends upon the no.of passangers over a certain minimum number of passangers n and upon the distance travelled by bus. His profit is Rs. 3600 will 29 passangers in the bus for a journey of 36 km and Rs. 6300 when there are 36 passangers. travelled for 42 km. What is the minimum no. of passangers are required so that he will not suffer any loss?

Options

A. 12
B. 20
C. 18
D. 15

Answer

To solve the problem step by step, we will use the information provided about Arvind Singh's bus service and the profits he earns based on the number of passengers and the distance traveled.

### Step 1: Define Variables
Let:
- \( n \) = minimum number of passengers required to avoid loss
- \( m \) = number of passengers
- \( d \) = distance traveled (in km)
- \( P \) = profit

### Step 2: Establish the Profit Formula
The profit \( P \) is directly proportional to the product of the number of passengers exceeding the minimum number and the distance traveled. We can express this as:
\[ P = k \cdot (m - n) \cdot d \]
where \( k \) is a constant.

### Step 3: Set Up Equations Based on Given Data
From the problem, we have two scenarios:
1. When \( m = 29 \) and \( d = 36 \), the profit \( P = 3600 \):
\[ 3600 = k \cdot (29 - n) \cdot 36 \quad \text{(1)} \]

2. When \( m = 36 \) and \( d = 42 \), the profit \( P = 6300 \):
\[ 6300 = k \cdot (36 - n) \cdot 42 \quad \text{(2)} \]

### Step 4: Rearranging the Equations
From equation (1):
\[ k = \frac{3600}{(29 - n) \cdot 36} \]
From equation (2):
\[ k = \frac{6300}{(36 - n) \cdot 42} \]

### Step 5: Set the Two Expressions for \( k \) Equal
Equating the two expressions for \( k \):
\[ \frac{3600}{(29 - n) \cdot 36} = \frac{6300}{(36 - n) \cdot 42} \]

### Step 6: Cross-Multiply to Solve for \( n \)
Cross-multiplying gives:
\[ 3600 \cdot (36 - n) \cdot 42 = 6300 \cdot (29 - n) \cdot 36 \]

### Step 7: Simplify the Equation
Expanding both sides:
\[ 151200 - 3600n = 226800 - 6300n \]
Rearranging gives:
\[ 6300n - 3600n = 226800 - 151200 \]
\[ 2700n = 75600 \]

### Step 8: Solve for \( n \)
Dividing both sides by 2700:
\[ n = \frac{75600}{2700} = 28 \]

### Step 9: Conclusion
The minimum number of passengers required so that Arvind Singh will not suffer any loss is:
\[ \boxed{28} \]