A person X started 3 hours earlier at 40 km/h from a place P then another person Y followed him at 60 km/h started his journey at 3 o'clock afternoon.what is the diffrence in time when X was 30 km ahead of Y and when Y was 30 km ahead of X?

Options

A. 2
B. 3
C. 3.5
D. 4.25

Answer

To solve the problem, we need to analyze the movements of both persons, X and Y, and determine the time difference when X is 30 km ahead of Y and when Y is 30 km ahead of X.

### Step-by-Step Solution:

1. **Determine the initial distance covered by X**:
- Person X starts 3 hours earlier than Y at a speed of 40 km/h.
- Distance covered by X in 3 hours = Speed × Time = 40 km/h × 3 h = 120 km.

2. **Establish the starting time for Y**:
- Person Y starts his journey at 3:00 PM.
- By this time, X has already traveled 120 km and is at point Q.

3. **Calculate the relative speed**:
- Since both X and Y are moving in the same direction, we calculate the relative speed.
- Speed of Y = 60 km/h, Speed of X = 40 km/h.
- Relative speed = Speed of Y - Speed of X = 60 km/h - 40 km/h = 20 km/h.

4. **Case 1: When X is 30 km ahead of Y**:
- The distance between X and Y when X is 30 km ahead = 120 km - 30 km = 90 km.
- Using the formula: Distance = Speed × Time, we find the time taken for Y to catch up with X.
- Time (T1) = Distance / Relative Speed = 90 km / 20 km/h = 4.5 hours.

5. **Case 2: When Y is 30 km ahead of X**:
- The distance between X and Y when Y is 30 km ahead = 120 km + 30 km = 150 km.
- Again using the formula: Distance = Speed × Time, we find the time taken for Y to be 30 km ahead of X.
- Time (T2) = Distance / Relative Speed = 150 km / 20 km/h = 7.5 hours.

6. **Calculate the difference in time**:
- Difference in time = T2 - T1 = 7.5 hours - 4.5 hours = 3 hours.

### Final Answer:
The difference in time when X was 30 km ahead of Y and when Y was 30 km ahead of X is **3 hours**.