Ajanta and Taj express start simultaneously from Meerut towards Nagpur (distance 390 km) on parallel tracks. Their initial speed ratio is 6:7. After Taj has covered a distance D km, the trains exchange speeds instantly and continue so that both reach Nagpur at the same time. Find D.

Introduction / Context:
This is a speed-exchange puzzle. Two trains start together over the same origin–destination span. After the faster train has covered D km, both instantly swap speeds and continue, arriving simultaneously. We form total-time expressions before and after the swap and equate them.

Given Data / Assumptions:

• Total distance L = 390 km.
• Initial speeds: Ajanta = 6u, Taj = 7u.
• Speed swap occurs at the same instant for both trains (not necessarily at the same location).

Concept / Approach:
Let t1 = time until swap. Then D = 7u * t1 (distance Taj covers), while Ajanta covers 6u * t1. After swapping, Taj's remaining distance = L − D at speed 6u; Ajanta's remaining = L − 6u t1 at speed 7u. Equal arrival times ⇒ post-swap travel times are equal.

Step-by-Step Solution:

Verification / Alternative check:
Plugging D = 210 satisfies symmetry of post-swap times; any other D breaks equality.

Why Other Options Are Wrong:
150 km and 190 km do not make the post-swap times equal; “can’t be determined” is incorrect because algebra yields a unique D.

Common Pitfalls:
Assuming they must meet at the swap point; the swap is instantaneous and simultaneous, not collocated.

Final Answer:
210 km