A bus and a truck are available to cross a jungle. The speed of the truck is thriee that of the bus. The capacity of the truck is 50 persons and that of bus is 30 persons. The average occupancy of the ous is twice that of the truck. The tickets for the bus and the truck cost Re 1 and Re `1.50` respectively. What is the ratio of the average rupee collection of the truck to that of the bus in a day? Assume there is no watage time between trips and the occupancy of the bus/truck is defined as the ratio of the actual number of persons boarding it and its capacity.

Options

A. `15 : 4`
B. `15 : 2`
C. `15 : 8`
D. `15 : 7`

Answer

To solve the problem, we need to find the ratio of the average rupee collection of the truck to that of the bus in a day. Let's break this down step by step.

### Step 1: Define Variables
Let:
- Speed of the bus = \( x \)
- Speed of the truck = \( 3x \) (since the speed of the truck is three times that of the bus)
- Capacity of the bus = 30 persons
- Capacity of the truck = 50 persons

### Step 2: Determine Average Occupancy
Let the average occupancy of the truck be \( y \). According to the problem, the average occupancy of the bus is twice that of the truck, so:
- Average occupancy of the bus = \( 2y \)

### Step 3: Calculate Actual Number of Passengers
The actual number of passengers boarding the bus and truck can be calculated as follows:
- Actual number of passengers in the bus = \( 2y \)
- Actual number of passengers in the truck = \( y \)

### Step 4: Calculate Revenue per Trip
The ticket prices are:
- Ticket price for the bus = Rs 1
- Ticket price for the truck = Rs 1.50

Revenue per trip can be calculated as:
- Revenue from the bus = \( 2y \times 1 = 2y \)
- Revenue from the truck = \( y \times 1.50 = 1.5y \)

### Step 5: Determine Number of Trips in a Day
To find the number of trips each vehicle can make in a day, we need to consider the time taken for each trip. The time taken for a trip is inversely proportional to speed.

Let’s assume they both travel the same distance \( d \):
- Time taken by bus for one trip = \( \frac{d}{x} \)
- Time taken by truck for one trip = \( \frac{d}{3x} \)

Thus, the number of trips made by each vehicle in a day (assuming a day has \( T \) hours) is:
- Number of trips by bus = \( \frac{T}{\frac{d}{x}} = \frac{Tx}{d} \)
- Number of trips by truck = \( \frac{T}{\frac{d}{3x}} = \frac{3Tx}{d} \)

### Step 6: Calculate Total Revenue per Day
Now, we can calculate the total revenue for both vehicles in a day:
- Total revenue from the bus in a day = Number of trips by bus × Revenue per trip
\[ = \frac{Tx}{d} \times 2y = \frac{2Txy}{d} \]
- Total revenue from the truck in a day = Number of trips by truck × Revenue per trip
\[ = \frac{3Tx}{d} \times 1.5y = \frac{4.5Txy}{d} \]

### Step 7: Calculate the Ratio of Revenues
Now, we can find the ratio of the average rupee collection of the truck to that of the bus:
\[ \text{Ratio} = \frac{\text{Total revenue from truck}}{\text{Total revenue from bus}} = \frac{\frac{4.5Txy}{d}}{\frac{2Txy}{d}} = \frac{4.5}{2} = 2.25 \]

### Step 8: Express as a Ratio
To express this as a ratio:
\[ \text{Ratio} = \frac{9}{4} \]

### Final Answer
Thus, the ratio of the average rupee collection of the truck to that of the bus in a day is \( 9:4 \).

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