A person travels 285 km in 6 hours in two stages. In the first part of the journey, he travels by bus at the speed of 40 km/hr. In the second part of the journey, he travels by train at the speed of 55 km/hr. How much distance does he travel by train? (M.A.T., 2007)

### Concept & Linear Equations When a total distance and total time are given for a journey split into two parts with different speeds, we can set up a linear equation based on time. Total Time = (Time on Bus) + (Time on Train) ### Step-by-Step Solution * **Given:** Total distance = 285 km. Total time = 6 hours. Speed of bus = 40 km/hr. Speed of train = 55 km/hr. * **Establish Variables:** Let the time traveled by train be $t$ hours. Therefore, the time traveled by bus is $(6 - t)$ hours. * **Formulate Equation:** * Distance by bus + Distance by train = Total Distance * $40(6 - t) + 55t = 285$ * $240 - 40t + 55t = 285$ * $15t = 285 - 240$ * $15t = 45 \Rightarrow t = 3$ hours. * **Calculate Distance:** * Distance traveled by train = Speed of train $\times$ Time by train * Distance = $55 \times 3 = 165$ km. ### Exam Strategy & Shortcut You can also use the method of alligation by finding the average speed of the whole journey ($285/6 = 47.5$ km/hr) and applying it to the two speeds (40 and 55) to find the ratio of times. Ratio of time = $(55 - 47.5) : (47.5 - 40) = 7.5 : 7.5 = 1:1$. Total time is 6 hours, so 3 hours each. Train distance = $3 \times 55 = 165$ km. ### Common Pitfall A common mistake is assigning the variable $x$ to distance rather than time, which creates a more complex fractional equation: $\frac{x}{55} + \frac{285 - x}{40} = 6$. While valid, it takes longer to solve. ### Final Answer Therefore, the correct answer is **165 km**.