A person P is at X and another person Q is at Y.The distance between X and Y is 100 km.The speed of P is 20 km/h.While the speed of Q is 60 km/h?if P and Q continue to move between X and Y in the given manner and if they meet for the fourth time at a place M somewhere between X and Y then the distance between X and M is:

Options

A. 10 km/h
B. 90 km
C. 75 km
D. 25 km

Answer

To solve the problem step-by-step, we need to find the distance between point X and point M, where persons P and Q meet for the fourth time.

### Step 1: Determine the time taken for the first meeting
- The distance between X and Y is 100 km.
- The speed of P is 20 km/h and the speed of Q is 60 km/h.
- Since they are moving towards each other, their relative speed is:
\[ \text{Relative Speed} = \text{Speed of P} + \text{Speed of Q} = 20 \text{ km/h} + 60 \text{ km/h} = 80 \text{ km/h} \]
- The time taken to meet for the first time is calculated using the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Relative Speed}} = \frac{100 \text{ km}}{80 \text{ km/h}} = 1.25 \text{ hours} \text{ (or } 1 \text{ hour and } 15 \text{ minutes)} \]

### Step 2: Calculate the distance traveled by P and Q until the first meeting
- Distance traveled by P in 1.25 hours:
\[ \text{Distance}_P = \text{Speed of P} \times \text{Time} = 20 \text{ km/h} \times 1.25 \text{ hours} = 25 \text{ km} \]
- Distance traveled by Q in 1.25 hours:
\[ \text{Distance}_Q = \text{Speed of Q} \times \text{Time} = 60 \text{ km/h} \times 1.25 \text{ hours} = 75 \text{ km} \]

### Step 3: Determine the time taken for subsequent meetings
- After the first meeting, they continue to travel towards each other. They will meet again after they have covered a total distance of 200 km (100 km to meet and 100 km back).
- The time taken for the second meeting will be the same as the first meeting, which is 1.25 hours.
- Thus, the time for the third and fourth meetings will also be 1.25 hours each.

### Step 4: Calculate the total time for four meetings
- Total time for four meetings:
\[ \text{Total Time} = \text{Time for first meeting} + 3 \times \text{Time for subsequent meetings} = 1.25 \text{ hours} + 3 \times 1.25 \text{ hours} = 1.25 + 3.75 = 5 \text{ hours} \]

### Step 5: Calculate the distance traveled by Q in 5 hours
- Distance traveled by Q in 5 hours:
\[ \text{Distance}_Q = \text{Speed of Q} \times \text{Total Time} = 60 \text{ km/h} \times 5 \text{ hours} = 300 \text{ km} \]

### Step 6: Determine the number of complete trips and remaining distance to M
- Q travels 300 km, which includes:
- 100 km to reach X (1st trip)
- 100 km to return to Y (2nd trip)
- 100 km to reach X again (3rd trip)
- After 300 km, Q has completed 3 full trips (300 km) and is now at point X.
- The remaining distance to M is:
\[ \text{Remaining Distance} = 300 \text{ km} - 300 \text{ km} = 0 \text{ km} \text{ (at X)} \]
- Q then travels an additional 25 km towards Y to meet P at point M.

### Final Calculation: Distance between X and M
- The distance between X and M is:
\[ \text{Distance}_{X \text{ to } M} = 25 \text{ km} \]

### Conclusion
The distance between X and M is **25 km**.

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