Ram and Shyam Running Race

Ram starts at 9:00 AM @ 5 km/h. At 10:00 AM, Ram reaches B (5 km away) and turns back toward A. Shyam starts at 9:45 AM @ 10 km/h. At 10:00 AM, Shyam has traveled for 15 mins $= 2.5$ km towards B. At 10:00 AM, distance between Ram (at B) and Shyam (2.5 km from A) is $2.5$ km, moving towards each other. Relative speed $= 5 + 10 = 15$ km/h. Time to meet $= \frac{2.5}{15} \text{ hours} = 10$ minutes. So they meet at 10:10 AM.

Shyam reaches B (5 km from A) at 10:15 AM (takes 45 mins) and starts returning towards A. At 10:15 AM, Ram (moving back towards A) has been returning for 15 mins, covering $5 \times \frac{15}{60} = 1.25$ km from B. Shyam is at B, and Ram is 1.25 km ahead of Shyam heading towards A. Relative speed when both move towards A $= 10 - 5 = 5$ km/h. Time taken by Shyam to overtake Ram $= \frac{1.25}{5} = 0.25 \text{ hours} = 15$ minutes. Overtaking time $= 10:15 + 15 \text{ mins} = 10:30$ AM.