A bus left X for point Y. Two hours later a car left point X for Y and arrived at Y at the same time as the bus. If the car and the bus left simultaneously from the opposite ends X and Y towards each other, they would meet $1 \frac{1}{3}$ hours after the start. How much time did it take the bus to travel from X to Y?

### Concept & Equating Relative Speed When dealing with unknown distances, you can often define speed in terms of Distance over Time ($\frac{D}{T}$) and substitute these into the relative speed formula for objects moving towards each other. $$Relative\ Speed = \frac{Total\ Distance}{Meet\ Time}$$ ### Step-by-Step Solution 1. Let the total distance between X and Y be $D$. Let the time taken by the bus be $t$ hours. 2. The car leaves 2 hours later but arrives at the same time, so its time taken is $(t - 2)$ hours. 3. Express their speeds: Speed of bus = $\frac{D}{t}$ Speed of car = $\frac{D}{t - 2}$ 4. They meet in $1 \frac{1}{3}$ hours, which is $\frac{4}{3}$ hours, when moving towards each other. Relative speed = $\frac{D}{t} + \frac{D}{t - 2}$ 5. Apply the formula $\text{Distance} = \text{Speed} \times \text{Time}$: $$D = \left(\frac{D}{t} + \frac{D}{t - 2}\right) \times \frac{4}{3}$$ 6. Cancel $D$ from both sides and solve for $t$: $$1 = \left(\frac{1}{t} + \frac{1}{t - 2}\right) \times \frac{4}{3}$$ $$\frac{3}{4} = \frac{t - 2 + t}{t(t - 2)}$$ $$3t(t - 2) = 4(2t - 2)$$ $$3t^2 - 6t = 8t - 8$$ $$3t^2 - 14t + 8 = 0$$ $$(3t - 2)(t - 4) = 0$$ 7. $t$ can be $\frac{2}{3}$ or $4$. Since the car takes $(t - 2)$ hours, $t$ must be greater than 2. Thus, $t = 4$. ### Exam Strategy & Shortcut Use Option Elimination based on logical constraints. Try Option (b): Bus takes 4 hours. Car takes 4 - 2 = 2 hours. Assume a convenient distance, say 12 km (LCM of 4 and 2, divisible by 3). Bus speed = 3 km/hr. Car speed = 6 km/hr. Relative speed = 9 km/hr. Time to meet = $\frac{12}{9} = \frac{4}{3}$ hours = $1 \frac{1}{3}$ hours. This matches perfectly! ### Common Pitfall Solving the quadratic equation incorrectly or blindly picking a mathematical root that is impossible in physical reality (like $t = \frac{2}{3}$ where car time becomes negative). ### Final Answer Therefore, the correct answer is **4 hours**.