Points P and Q are 27 km apart along a straight line. Two trains start simultaneously, one from P at 24 km/h and the other from Q at 18 km/h, both moving in the same direction (from P towards beyond Q). They meet at a point R beyond Q. Find the distance QR.

Introduction / Context:
This is a catch-up problem with initial separation. One train starts from P and another from Q, both moving in the same direction (towards and beyond Q). The faster train from P eventually overtakes the train from Q at point R beyond Q.

Given Data / Assumptions:

• Initial gap between trains at t = 0 is PQ = 27 km.
• v_P = 24 km/h (from P), v_Q = 18 km/h (from Q), same direction.
• They meet at R beyond Q ⇒ the P-train catches the Q-train.

Concept / Approach:
For catch-up: relative speed = v_P − v_Q. Catch-up time t = gap / relative speed. Then the distance QR equals distance traveled by the Q-train in time t.

Step-by-Step Solution:

Verification / Alternative check:
Total distance from P to R = 24 * 4.5 = 108 km; hence PR − PQ = 108 − 27 = 81 km = QR ✔

Why Other Options Are Wrong:

• 126, 48, 36 km do not satisfy both relative motion and initial separation conditions.

Common Pitfalls:

• Mistaking opposite-direction closure for same-direction catch-up (sum vs difference of speeds).

Final Answer:
81 km