# Math Help - Train Word Problem

1. ## Train Word Problem

Hi all,

I am curious whether there is a better approach to solving this problem than just guessing and checking. Here is the problem.

One hour out of the station, the locomotive of a freight train develops trouble that slows its speed to 3/5 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.

a) What mathematical questions arise from this story?

I am more interested in the thought process and setup than the answer itself.

Let the distance between the station be $d$ miles the average speed $v$ miles/hour, the average time take by the train is $t=\frac{d}{v}$ hours.
Distance traveled by the train in 1 hour is $v$ miles. Time taken to cover the remaining distance is $\frac{5}{3}\frac{d-v}{v}$. Total time taken is $\frac{5}{3}\frac{d-v}{v} + 1 = t + 2 \implies t=4$ hours.
$\frac{v+50}{v} + \frac{5}{3}\frac{d-v-50}{v} = t+2 -\frac{40}{60} \implies 1+\frac{50}{v} + \frac{5}{3}(4-1-\frac{50}{v}) = 4+2 -\frac{2}{3} \implies v = 50$