One hour out of the station, the locomotive of a freight train develops trouble that slows its speed to of its average speed up to the time of the failure. Continuing at this reduced speed it reaches its destination two hours late. Had the trouble occurred 50 miles beyond, the delay would have been reduced by 40 minutes. Find the distance from the station to the destination.

Question

One hour out of the station, the locomotive of a freight train develops trouble that slows its speed to of its average speed up to the time of the failure. Continuing at this reduced speed it reaches its destination two hours late. Had the trouble occurred 50 miles beyond, the delay would have been reduced by 40 minutes. Find the distance from the station to the destination.

Answer 1

"that slows its speed to of its average speed up to the time of the failure"

something missing here.

Answer 2

its speed to 2/5 of its average speed up to the time of the failure.

Answer 3

To solve this problem, we first need to understand the given information and identify the unknowns. Let's break it down step by step:

1. Let's denote the average speed of the train as 'S' (in miles per hour).
2. The train develops trouble one hour after leaving the station. After the trouble, its speed reduces to (2/3)S, which is two-thirds of its average speed.
3. With this reduced speed, the train arrives at its destination two hours late. Let's denote the total time taken to reach the destination as 'T' (in hours). So, without the delay caused by the trouble, the total time to the destination would be T - 2 hours.
4. If the trouble had occurred 50 miles beyond the original point, the delay would have been reduced by 40 minutes. So, without the delay, the total time taken to reach the destination would be T - 2 hours - 40 minutes.

Now, let's express the given information in terms of equations:

1. Distance of the journey (D) = (S - (2/3)S)(T + 2) -- The distance traveled at the reduced speed (2/3S) for (T + 2) hours.
2. Distance of the journey (D) = (S)(T - 2) -- The distance traveled at the average speed (S) for (T - 2) hours.
3. Distance of the journey (D + 50) = (S)(T - 2 - 40/60) -- The distance traveled at the average speed (S) for (T - 2) hours - 40 minutes converted into hours.

Next, we can equate these equations and solve for the unknowns.

Solving equation 1 and 2:

(S - (2/3)S)(T + 2) = (S)(T - 2)

Simplifying:

S(T + 2) - (2/3)S(T + 2) = S(T - 2)

ST + 2S - (2/3)ST - (4/3)S = ST - 2S

Multiplying through by 3:

3ST + 6S - 2ST - 4S = 3ST - 6S

ST + 2S = 3ST - 6S

Bringing the similar terms together:

ST - 4S = 0 ---- (Equation 3)

Now, let's solve equations 2 and 3 simultaneously:

(S)(T - 2) = 0

ST - 2S = 0 ---- (Equation 4)

Subtracting equation 4 from equation 3:

(ST - 4S) - (ST - 2S) = 0 - 0

2S = 2S

The variable 'S' cancels out, which means there are infinitely many possible values for 'S'. This implies that we cannot determine the average speed of the train from the given information.

Therefore, we cannot find the distance from the station to the destination since it depends on the unknown average speed of the train ('S').