To solve this problem, let's break it down into smaller parts and analyze each piece of information given.
Let's assume the original speed of the train is "v" miles per hour and the time it takes to cover the distance between the two stations is "t" hours.
1) First, we are given that after traveling for an hour at the original speed, the train goes at 3/5th of the original speed. This means that the new speed of the train is (3/5)v miles per hour. The time taken to cover the remaining distance at this reduced speed is (t + 2) hours.
2) Next, we are told that if the fault had occurred after traveling another 50 miles, the train would have reached the destination 40 minutes earlier. This means that the time taken to cover the remaining distance would be (t - 40/60) hours.
Now, let's use these pieces of information to form equations and solve for the distance between the two stations.
Distance = Speed × Time
1) Using the information given in point 1, we can write the equation as:
(v × 1) + ((3/5)v × (t + 2)) = v × t
2) Using the information given in point 2, we can write the equation as:
(v × 1) + ((3/5)v × (t - 40/60)) = v × (t - 2)
Simplifying these equations will give us the value of "t" and subsequently the distance between the two stations.
Let's solve these equations step by step:
1) v + (3/5)v(t + 2) = vt
5v + 3v(t + 2) = 5vt
5v + 3vt + 6v = 5vt
8v + 3vt = 5vt
8v = 2vt
4v = vt
2) v + (3/5)v(t - 40/60) = vt - 2v
5v + 3v(t - 2/3) = 5vt - 10v
5v + 3vt - 2v = 5vt - 10v
3vt + 3v = 5vt - 5v
3v = 2vt - 5v
5v = 2vt
From equation 1) and equation 2), we can see that 4v = 5v. This implies that v = 0, which is not possible.
Hence, there is no valid solution for this problem.
Therefore, the given question is incorrect, and none of the options provided (A, B, C, D) are correct.