Aman and Kapil start from Delhi and Gwalior respectively towards each other at the same time. They meet at Mathura. After meeting, Aman takes 196 minutes to reach Gwalior, while Kapil takes 225 minutes to reach Delhi. If Aman’s speed is 30 km/h, what is the speed of Kapil (in km/h)?

Introduction / Context:
This is a more advanced relative-speed problem involving two people travelling towards each other from different cities and meeting at an intermediate point, Mathura. After meeting, they continue to the opposite cities, and we are given the times they take to complete the remaining parts of their journeys. Using this post-meeting information along with the fact that they met at the same instant, we can deduce the speed of Kapil. This problem requires careful use of distances before and after meeting and a bit of algebra.

Given Data / Assumptions:
- Aman starts from Delhi and Kapil from Gwalior at the same time, moving towards each other.
- They meet at Mathura, an intermediate point on the straight route between Delhi and Gwalior.
- Aman’s speed = 30 km/h (constant).
- After meeting, Aman takes 196 minutes to reach Gwalior.
- After meeting, Kapil takes 225 minutes to reach Delhi.
- 196 minutes = 196/60 hours; 225 minutes = 225/60 hours.
- Let Kapil’s speed be v km/h.
- Distances and speeds remain constant throughout their journeys.

Concept / Approach:
When two travellers meet, the point of meeting splits the total route in two parts. The segment from Mathura to Gwalior is covered by Aman after meeting and by Kapil before meeting. Similarly, the segment from Mathura to Delhi is covered by Kapil after meeting and by Aman before meeting. By using the times and speeds after meeting, we can compute the lengths of these segments. Then, equating the time taken to reach Mathura for each traveller before meeting gives an equation involving v. Solving this equation yields Kapil’s speed.

Step-by-Step Solution:
Step 1: Compute the distance from Mathura to Gwalior.Aman covers this distance after meeting at 30 km/h in 196/60 hours.Distance Mathura–Gwalior = 30 * (196/60) = 30 * (49/15) = 98 km.Step 2: Let the distance from Mathura to Delhi be x km.Kapil covers this distance after meeting at v km/h in 225/60 hours.So x = v * (225/60) = v * (15/4).Step 3: Use the fact that before meeting, Aman covers x km to reach Mathura from Delhi and Kapil covers 98 km to reach Mathura from Gwalior.Time taken by Aman to reach Mathura: x / 30 hours.Time taken by Kapil to reach Mathura: 98 / v hours.These times must be equal because they start together and meet at the same instant: x / 30 = 98 / v.Step 4: Substitute x = (15v) / 4 into x / 30 = 98 / v.( (15v) / 4 ) / 30 = 98 / v ⇒ (15v) / 120 = 98 / v.Simplify: 15v / 120 = v / 8.So v / 8 = 98 / v ⇒ v^2 = 784 ⇒ v = 28 km/h (taking the positive root).

Verification / Alternative check:
With v = 28 km/h, x = (15 * 28) / 4 = 15 * 7 = 105 km from Mathura to Delhi. Aman’s time to reach Mathura originally is 105 / 30 = 3.5 hours. Kapil’s time to reach Mathura originally is 98 / 28 = 3.5 hours as well, so they meet at the same time. After meeting, Aman needs 196/60 ≈ 3.27 hours to reach Gwalior, and Kapil needs 225/60 = 3.75 hours to reach Delhi, consistent with the given values. Everything fits perfectly.

Why Other Options Are Wrong:
- 30 km/h would imply Kapil’s speed equals Aman’s, which destroys the asymmetry created by the different post-meeting times.
- 225/7 km/h and 392/15 km/h, when substituted, do not keep the pre-meeting times equal and violate at least one of the time constraints given for travelling from Mathura to the endpoints.

Common Pitfalls:
Several errors commonly occur: mixing up which segment is 98 km, forgetting to convert minutes into hours, or not equating the pre-meeting travel times correctly. Some students also try to involve the total distance unnecessarily, which complicates the algebra. The cleanest path is to use post-meeting distances first, then rely on equal pre-meeting times to solve for Kapil’s speed.

Final Answer:
Kapil’s speed is 28 km/h.