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 42km/h then what is the shortest possible time in which all three of them can complete the jouirney?

Options

A. 7(1)/3h
B. 9(4)/7h
C. 9(3)/7h
D. can't be determined

Answer

To solve the problem of how long it will take Mohan, Namita, and Pranav to travel from Shantipur to Hulchulpur using a two-seater bike, we can break down the journey into steps.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- The distance from Shantipur to Hulchulpur is 180 km.
- Mohan can drive the bike, and only two people can ride at a time due to traffic rules.
- All three individuals can walk at a speed of 6 km/h.

2. **Setting Up the Journey**:
- Since Mohan drives the bike, he can take one of the others (either Namita or Pranav) with him at a time.
- The bike speed is 42 km/h.

3. **Calculating Time for the Bike**:
- The time taken by the bike to travel 180 km can be calculated using the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{180 \text{ km}}{42 \text{ km/h}} = \frac{180}{42} \approx 4.29 \text{ hours} \]

4. **Walking Distance**:
- If one person walks while the other two ride the bike, we need to calculate how far the walking person will travel in the time it takes for the bike to reach Hulchulpur and return.
- Let’s denote the time taken for the bike to reach Hulchulpur as \( t \) hours.
- The distance walked by the person is:
\[ \text{Distance} = \text{Speed} \times \text{Time} = 6 \text{ km/h} \times t \]

5. **Total Distance Calculation**:
- When the bike reaches Hulchulpur, Mohan will drop off one person and return to pick up the third person. The total distance for the bike (to and fro) is:
\[ 2 \times 180 = 360 \text{ km} \]

6. **Setting Up the Equation**:
- The total distance walked by the person while the bike is traveling can be equated to the distance the bike travels:
\[ 6t = 360 - 6t \]
- Rearranging gives:
\[ 12t = 360 \implies t = \frac{360}{12} = 30 \text{ hours} \]

7. **Final Calculation**:
- The total time for the journey will be the time taken by the bike to reach Hulchulpur plus the time taken by the walking person to reach the destination:
\[ \text{Total Time} = t + \frac{180}{42} \approx 30 + 4.29 \approx 34.29 \text{ hours} \]

8. **Conclusion**:
- The shortest possible time in which all three can complete the journey is approximately 34.29 hours.