# How to Solve Word Problems About Train Speeds

Word problems involving trains usually go something like this:

A set of parallel train tracks is 400 miles long. On one end of the track, Train X leaves the station at 5PM. On the opposite end of the track, Train Y leaves the station at 6:30PM. If Train X travels 40 mph and Train Y travels 60 mph, what time will they meet?

Math problems involving multiple distances, speeds, and times can be difficult to solve because there are many numbers and variables to work with. The following guide will help you solve them quickly and accurately.

The first step is to draw a diagram of the track on a sheet of paper, and recall the three formulas that relate distance, speed and time:

D = ST
S = D/T
T = D/S

Whenever you know any two of these quantities, you can always solve for the third with one of these equations.

Next, remember the following principle about objects moving in opposite directions toward each other: the sum of their speeds is equal to the speed at which the distance between the objects decreases.

We can apply this principle to the example given in the intro. If Train X and Train Y are moving toward each other at 40 mph and 60 mph respectively, then the distance between them is shrinking at a rate of 100 mph. This fact will be very useful in solving the problem!

Next, if necessary, adjust the problem so that both trains start at the same time.

For example, since Train X starts one and a half hours earlier, we can adjust the problem to make both trains start at 6:30PM. In that 1.5 hour time span between 5PM and 6:30PM, Train X travels (1.5 hours)x(40 mph) = 60 miles. So at 6:30PM, the two trains are only 400 - 60 = 340 miles apart.

Now we can calculate when the trains will meet. At 6:30PM the trains are 340 miles apart and moving toward each other at a combined rate of 100 mph. Thus, the amount of time it takes for the distance between them to shrink from 340 miles to 0 miles is

T = (340 miles)/(100 mph)
= 3.4 hours, or 3 hours and 24 minutes.

So they will meet at 9:54PM.

Use this word problem solving strategy above to solve another word problem:

At 10AM, Train X and Train Y are 480 miles apart on opposite ends of a set of parallel train tracks. At 10AM, Train X starts traveling 50 mph, and Train Y starts traveling 30 mph in the opposite direction. When they meet, how will Train X have traveled? (Answer: 300 miles)