A person P is at X and another person Q is atY. 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 they continue to move to and fro between X and Y then what is the distance covered by P when they meet second time?

Options

A. 105 km
B. 100 km
C. 80 km
D. 75 km

Answer

To solve the problem, we need to determine the distance covered by person P when he meets person Q for the second time. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Problem
- Person P starts at point X and moves towards point Y.
- Person Q starts at point Y and moves towards point X.
- 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.

### Step 2: Calculate the Time Until They Meet for the First Time
- The combined speed of P and Q when they are moving towards each other is:
\[ \text{Combined Speed} = \text{Speed of P} + \text{Speed of Q} = 20 \text{ km/h} + 60 \text{ km/h} = 80 \text{ km/h} \]
- The time taken to meet for the first time can be calculated using the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{100 \text{ km}}{80 \text{ km/h}} = \frac{5}{4} \text{ hours} \]

### Step 3: Calculate the Distance Covered by P Until the First Meeting
- The distance covered by P until they meet for the first time is:
\[ \text{Distance} = \text{Speed of P} \times \text{Time} = 20 \text{ km/h} \times \frac{5}{4} \text{ hours} = 25 \text{ km} \]

### Step 4: Calculate the Time Until They Meet for the Second Time
- After meeting for the first time, P continues to Y and then returns to X, while Q continues to X and returns to Y.
- The total distance covered by both P and Q to meet again is 200 km (100 km to Y and 100 km back to X).
- The time taken for this round trip is:
\[ \text{Time for P} = \frac{100 \text{ km}}{20 \text{ km/h}} + \frac{100 \text{ km}}{20 \text{ km/h}} = 5 \text{ hours} \]
\[ \text{Time for Q} = \frac{100 \text{ km}}{60 \text{ km/h}} + \frac{100 \text{ km}}{60 \text{ km/h}} = \frac{10}{3} \text{ hours} \]

### Step 5: Calculate the Total Time Until the Second Meeting
- The total time until they meet for the second time is:
\[ \text{Total Time} = \frac{5}{4} \text{ hours (first meeting)} + 5 \text{ hours} = \frac{5}{4} + \frac{20}{4} = \frac{25}{4} \text{ hours} \]

### Step 6: Calculate the Distance Covered by P Until the Second Meeting
- The distance covered by P until the second meeting is:
\[ \text{Distance} = \text{Speed of P} \times \text{Total Time} = 20 \text{ km/h} \times \frac{25}{4} \text{ hours} = 125 \text{ km} \]

### Final Answer
Thus, the distance covered by P when they meet for the second time is **125 km**.

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