Mohan Namita and Pranav travel from Shantipur to Hulchulpur. They have a two seeter bike which can be driven by only mohan. It is known that due to very stringent traffic rules only two persons can ride at a time. Hulchulpur is 180 km away from shantipur. All of them can walk at 6 km/h but reach to hulchulpur simultaneously also they started their journey simultaneously. If the speed of the bike is 30 km/h then what is the total distance that the bike travels ?

Options

A. 400km
B. 380km
C. 200km
D. 420km

Answer

To solve the problem step by step, we need to analyze the situation involving Mohan, Namita, and Pranav traveling from Shantipur to Hulchulpur on a bike.

### Step 1: Understand the Journey
- The distance from Shantipur to Hulchulpur is 180 km.
- The bike can only carry two people at a time, and it can be driven only by Mohan.
- All three of them (Mohan, Namita, and Pranav) start their journey simultaneously.

### Step 2: Calculate the Time Taken by Each Person
- The speed of the bike is 30 km/h.
- The walking speed of all three is 6 km/h.
- Let’s denote the distance the bike travels to pick up Pranav as \( x \) km.

### Step 3: Time Taken for Each Journey
- The time taken by Pranav to walk \( x \) km is given by:
\[ \text{Time}_{\text{Pranav}} = \frac{x}{6} \]
- The bike travels to pick up Pranav and then returns to the point where Namita is waiting. The distance for the bike to travel is \( 180 + (180 - x) \) km (to go to Hulchulpur and return back to pick Pranav).
- The time taken by the bike for this journey is:
\[ \text{Time}_{\text{Bike}} = \frac{180 + (180 - x)}{30} = \frac{360 - x}{30} \]

### Step 4: Set Up the Equation
- Since they all arrive at the same time, we can set the two times equal:
\[ \frac{x}{6} = \frac{360 - x}{30} \]

### Step 5: Solve the Equation
- Cross-multiplying gives:
\[ 30x = 6(360 - x) \]
- Expanding the right side:
\[ 30x = 2160 - 6x \]
- Bringing all terms involving \( x \) to one side:
\[ 30x + 6x = 2160 \]
\[ 36x = 2160 \]
- Dividing both sides by 36:
\[ x = 60 \]

### Step 6: Calculate the Total Distance Traveled by the Bike
- The bike travels to pick up Pranav (60 km) and then returns to the point where Namita is waiting (60 km). After that, it continues to Hulchulpur (180 km).
- Total distance traveled by the bike:
\[ \text{Total Distance} = 60 + 60 + 180 = 300 \text{ km} \]

### Final Answer
The total distance that the bike travels is **300 km**.