A car overtakes a bus travelling from Delhi to Jaipur at 4 : 30 p.m. The car reaches Jaipur at 6 : 00 p.m. After stopping there for 1 hour, it starts back towards Delhi and meets the same bus at 7 : 30 p.m. which was moving towards Jaipur at that time. If both the bus and the car were travelling with uniform speeds on the same route, at what time would the bus reach Jaipur?

### Concept & Logic This problem requires tracking the positions and times of two vehicles to determine their speeds relative to the total distance. $$ \text{Speed} = \frac{\text{Distance}}{\text{Time}} $$ ### Step-by-Step Solution 1. Let the point where the car overtakes the bus (at 4:30 p.m.) be point $A$, and Jaipur be point $B$. Let the distance $AB$ be $D$. 2. **Car's Journey to Jaipur:** The car travels from $A$ to $B$ between 4:30 p.m. and 6:00 p.m. Time taken by car = $1.5$ hours. Speed of car ($v_c$) = $\frac{D}{1.5} = \frac{2D}{3}$. 3. **Car's Return Journey:** The car stops at Jaipur for 1 hour (leaves at 7:00 p.m.) and meets the bus at 7:30 p.m. Let this meeting point be $C$. Time traveled by car from $B$ to $C$ = $0.5$ hours. Distance $BC$ = $v_c \times 0.5 = \frac{2D}{3} \times \frac{1}{2} = \frac{D}{3}$. 4. **Bus's Journey:** The bus has been traveling from $A$ since 4:30 p.m. At 7:30 p.m., it is at point $C$. Distance traveled by bus ($AC$) = $AB - BC = D - \frac{D}{3} = \frac{2D}{3}$. Time taken by bus = 7:30 p.m. - 4:30 p.m. = $3$ hours. Speed of bus ($v_b$) = $\frac{\text{Distance}}{\text{Time}} = \frac{\frac{2D}{3}}{3} = \frac{2D}{9}$. 5. **Bus's Total Time:** Total time for the bus to travel distance $D$ ($AB$) = $\frac{D}{v_b} = \frac{D}{\frac{2D}{9}} = \frac{9}{2} = 4.5$ hours. 6. **Arrival Time:** The bus was at point $A$ at 4:30 p.m. Arrival time at Jaipur = 4:30 p.m. + 4.5 hours = 9:00 p.m. ### Exam Strategy & Shortcut Visualize the timeline. The car travels the whole distance in 1.5 hours. On the way back, it travels for 0.5 hrs, covering $1/3$ of the distance. This means the bus has covered $2/3$ of the distance in 3 hours. To cover the full distance (3/3), the bus will take $3 \times \frac{3}{2} = 4.5$ hours. 4:30 pm + 4.5 hrs = 9:00 pm. ### Common Pitfall A common error is forgetting to account for the 1-hour stop the car makes in Jaipur when calculating the time it takes to travel back and meet the bus. ### Final Answer Therefore, the correct answer is **9 : 00 p.m.**