$A$ and $B$ are two stations $10$ km apart. A man, $P$ starts from $A$ and travels towards $B$ at the rate of $3$ km/hr, whereas another man $Q$ starts from $B$ and travels to wards A at the rate of $2$ km/hr. When and where do they meet?

### Concept & Meeting Point Calculation When two objects travel towards each other simultaneously, their relative speed is the sum of their speeds. The meeting time is the total initial distance divided by this relative speed. Once time is known, distance from a specific point is found using the respective traveler's speed. $$ \text{Meeting Time} = \frac{\text{Total Distance}}{\text{Relative Speed}} $$ ### Step-by-Step Solution * **Given:** Distance between $A$ and $B$ = $10$ km. Speed of $P$ (from $A$) = $3$ km/hr. Speed of $Q$ (from $B$) = $2$ km/hr. * **Calculation:** Relative speed of $P$ and $Q$ = $3 + 2 = 5$ km/hr. * Time taken to meet = $\frac{\text{Total Distance}}{\text{Relative Speed}} = \frac{10}{5} = 2$ hours. * The distance from station $A$ where they meet is the distance traveled by $P$ in those $2$ hours. * Distance from $A$ = Speed of $P \times \text{Meeting Time} = 3 \times 2 = 6$ km. ### Exam Strategy & Shortcut Find relative speed ($5$ km/hr). Total distance ($10$ km) divided by $5$ gives $2$ hours. Since the question asks for distance from $A$, multiply the meeting time ($2$ hrs) by the speed of the person leaving from $A$ ($3$ km/hr) to get $6$ km. Match these two values to the options. ### Common Pitfall Calculating the correct meeting time ($2$ hours) but mistakenly calculating the meeting distance from station $B$ ($2 \times 2 = 4$ km) and selecting option (d). ### Final Answer Therefore, the correct answer is **After $2$ hours, $6$ km from A**.