Person X starts from place P three hours earlier at 40 km/h. Person Y starts from the same place P at 3:00 pm at 60 km/h in the same direction. What is the difference in time (in hours) between the moment when X is 30 km ahead of Y and the moment when Y is 30 km ahead of X?

Introduction / Context:
This question is about relative motion in the same direction. One person starts earlier at a lower speed, and another starts later at a higher speed. We are asked to find how much time separates two events: first when the earlier person leads by 30 km, and later when the faster person leads by 30 km. The key tool is relative speed between the two travellers.

Given Data / Assumptions:

- Person X starts from P three hours before Y and travels at 40 km/h.
- Person Y starts from P at 3:00 pm and travels at 60 km/h in the same direction.
- At some time after 3:00 pm, X is 30 km ahead of Y.
- At a later time, Y is 30 km ahead of X.
- We need the difference between these two times, measured from the 3:00 pm start of Y.

Concept / Approach:
Take Y's starting time as the reference time t = 0. Express the positions of X and Y as functions of time t (in hours) after 3:00 pm. We then set up equations for the two situations: X leading by 30 km and Y leading by 30 km. Solving for the two times and taking the difference gives the required answer.

Step-by-Step Solution:
Let t be the time in hours after 3:00 pm (when Y starts). X has already been travelling for 3 hours at 40 km/h. So at t = 0, X is 40 * 3 = 120 km ahead of P. Position of X at time t: xX(t) = 120 + 40 t. Position of Y at time t: xY(t) = 60 t. First condition: X is 30 km ahead of Y. So xX(t1) - xY(t1) = 30. That is, 120 + 40 t1 - 60 t1 = 30. Simplify: 120 - 20 t1 = 30, so 20 t1 = 90, giving t1 = 4.5 hours. Second condition: Y is 30 km ahead of X. So xY(t2) - xX(t2) = 30. That is, 60 t2 - (120 + 40 t2) = 30. Simplify: 60 t2 - 120 - 40 t2 = 30, so 20 t2 - 120 = 30. Thus 20 t2 = 150 and t2 = 7.5 hours. Required difference in time = t2 - t1 = 7.5 - 4.5 = 3 hours.

Verification / Alternative check:
At t1 = 4.5 hours after 3:00 pm, Y has travelled 60 * 4.5 = 270 km, and X has travelled 120 + 40 * 4.5 = 300 km, a difference of 30 km. At t2 = 7.5 hours, Y has travelled 60 * 7.5 = 450 km, while X has travelled 120 + 40 * 7.5 = 420 km, so now Y leads by 30 km. The two conditions are satisfied perfectly, confirming the calculations.

Why Other Options Are Wrong:
2 h, 3.5 h, and 4.25 h: These values do not match the actual difference t2 - t1 obtained from correctly solving the position equations. Any other value would imply incorrect positions for X and Y at one or both of the described events.

Common Pitfalls:
Some learners mistakenly ignore the 3 hour head start of X or choose the wrong reference time origin, which leads to incorrect equations. Another common mistake is to misinterpret the phrase “ahead of” and reverse the sign in the equations. Always define positions clearly and be consistent with the direction and the reference time.

Final Answer:
The difference in time between the two events is 3 hours.