Relative Motion – Meet after turnaround on a straight route: A and B start from point P to Q (84 km) at 12 km/h and 16 km/h, respectively. B reaches Q, immediately turns back, and meets A at point R. What is the distance PR?

Introduction / Context:
This is a classic chase-and-return problem. After B reaches Q and turns back, they meet when A's position from P equals B's position from P (computed as 84 km minus B's return distance). We compute the meeting time and then multiply by A's speed.

Given Data / Assumptions:

• Distance P to Q = 84 km.
• Speeds: A = 12 km/h, B = 16 km/h.
• B turns back immediately on reaching Q.

Concept / Approach:
Let t be the time from the start to the meeting. For t ≥ 84/16, B has already turned. After turning, B's position from P is 84 − 16(t − 84/16). Set this equal to A's position 12t and solve for t.

Step-by-Step Solution:
B reaches Q at 84/16 = 5.25 h.Meeting condition: 12t = 84 − 16(t − 5.25) = 168 − 16t.28t = 168 ⇒ t = 6 h ⇒ PR = 12 * 6 = 72 km.

Verification / Alternative check:
At t = 6 h, B has been returning for 0.75 h and covered 12 km back from Q, so B's position is 72 km from P, matching A.

Why Other Options Are Wrong:
76, 78, 70, and 68 km do not satisfy the meeting equality under the given speeds.

Common Pitfalls:
Assuming they meet before B turns or forgetting to subtract the return distance from 84 km.

Final Answer:
72 km