Direction: These questions are based on the following information: Ram and Shyam run a race between points $A$ and $B$, 5 km apart. Ram starts at 9 a.m. from $A$ at a speed of 5 km/hr, reaches $B$ and returns to $A$ at the same speed. Shyam starts at 9 : 45 a.m. from $A$ at a speed of 10 km/hr, reaches $B$ and comes back to $A$ at the same speed. At what time does Shyam overtake Ram?

### Concept & Overtaking Overtaking occurs when two bodies move in the same direction and the faster body catches up to the slower body from behind. The relative speed is the difference between their speeds. $$ \text{Relative Speed} = V_{\text{fast}} - V_{\text{slow}} $$ ### Step-by-Step Solution 1. **Track Ram's Return Journey:** Ram reaches $B$ at 10:00 a.m. and starts heading back to $A$ at 5 km/hr. 2. **Track Shyam's Journey to $B$:** Shyam starts at 9:45 a.m. at 10 km/hr. To cover 5 km, he takes $\frac{5}{10} = 0.5 \text{ hours}$ (30 minutes). He reaches $B$ at 10:15 a.m. and immediately turns back towards $A$. 3. **Locate Ram at 10:15 a.m.:** From 10:00 a.m. to 10:15 a.m., Ram has been walking back towards $A$ for 15 minutes (0.25 hours). Distance covered from $B$ = $5 \text{ km/hr} \times 0.25 \text{ hrs} = 1.25 \text{ km}$. 4. **Set up the Overtake Scenario:** At exactly 10:15 a.m., Shyam is at point $B$, and Ram is 1.25 km ahead of him on the path back to $A$. Both are moving in the same direction (towards $A$). 5. **Calculate Catch-up Time:** The relative speed is $10 \text{ km/hr} - 5 \text{ km/hr} = 5 \text{ km/hr}$. The gap to close is 1.25 km. Time to catch up = $\frac{1.25 \text{ km}}{5 \text{ km/hr}} = 0.25 \text{ hours} = 15 \text{ minutes}$. 6. **Find Exact Overtake Time:** 10:15 a.m. + 15 minutes = 10:30 a.m. ### Exam Strategy & Shortcut Determine exactly when Shyam turns around (10:15 a.m.). At that exact moment, calculate Ram's lead. Ram had a 15-minute head start on the return leg, creating a 1.25 km gap. Since Shyam's relative speed advantage is 5 km/hr, closing a 1.25 km gap takes exactly a quarter of an hour (15 minutes). 10:15 + 15 mins = 10:30. ### Common Pitfall Confusing the initial "meeting" (going in opposite directions) with "overtaking" (going in the same direction). Overtaking requires the relative speed to be the difference, not the sum. ### Final Answer Therefore, the correct answer is **10 : 30 a.m.**