Sample Math Problem: Two trains leave the station…

If a nineteenth-century Russian adulteress goes to a train station…

“It’s not GREek!” will present you with question types you are likely to see on the GRE, as well as a brief explanation on how to arrive at the answer for each question. This week we will turn our attention toward a sample GRE Math problem.

Ah, the dreaded train problem. Surely these kinds of questions must be the the most infamous of all inane word problems. They can haunt the mathematically disinclined for years after leaving school, causing people undue anxiety waiting in traffic for a locomotive to pass. You probably thought you left these behind long ago, but they’re back. Who cares about some stupid trains, you ask? The GRE, that’s who.

Never fear though – all GRE math questions are written so that they they can be solved in less than two minutes, if you know what to do. This means that they aren’t going to require going through a lot of complicated steps to solve, and remember, the GRE doesn’t test anything beyond high school math. It just asks questions in unfamiliar ways that may require you to read carefully, and if you’re more of a verbal person than a math person, that shouldn’t be so bad, right? With some practice, the test makers’ tricks become familiar and recognizable, and problems that once seemed confusing become plain as day. Today, we’ll banish your siderodromophobia (fear of trains) for good.

Consider the following GRE math problem:

“At 10:00 AM train A left the station and an hour later train B left the same station on a parallel track. If train A traveled at a constant speed of 60 miles per hour and train B at 80 miles per hour, then at what time did train B pass train A?”

Monet’s “The Arival of the Train” at the Gare St. Lazare in Paris.

The first step to solving this problem is understanding what the question is really asking. What is this question asking? Well, “at what time did train B pass train A.” Yes, but what does that mean? When will train B pass train A? When they have traveled the same distance.

This is key to understanding how to solve the problem. We are going to need to know how to use the information we have been given, the speeds of the trains and the times at which they left the station, to calculate the distance they have traveled. As you know, the distance formula is usually written as:

speed = distance/time

If we want to find distance, we rearrange this familiar equation like this:

distance = speed(time)

So, if we want to calculate train A’s distance after a given length of time, we would multiply train A’s speed times the length of time it has been traveling. We know train A’s speed is 60 mph, so if we let the variable t represent the number of hours it has been since 10:00, we could write this as:

If you like the Monet painting, consider visiting the Musee d’Orsay. It’s full of them.

train A’s distance = 60(t)

Now, for train B, it’s slightly more complicated. How far has train A gone by 11:00, one hour after leaving the station? 60 miles, of course. But how long has train B gone? Zero, because train B doesn’t start traveling until 11:00. If we were to write this mathematically, we would have to express the distance traveled by train B as:

train B’s distance = 80(t – 1)

We have to write (t – 1) because train B starts an hour later than train A. This makes sense, because if we let t be one, that is, one hour after 10:00, then train B has gone zero miles:

80(1 – 1) = 80(0) = 0

Now, what were we trying to find again? The time when train A and train B have traveled the same distance.In other words, we want to know when:

train A’s distance = train B’s distance

If train A’s distance is equal to 60(t) and train B’s distance is equal to 80(t – 1), then we can just set those termsd equal to each other and solve for t:

60(t) = 80(t – 1)

60t = 80t – 80

80 = 20t

4 = t

So, four hours after 10:00 is when train A and train B have traveled the same distance. So that’s 2:00 PM. Was that so bad?

All you need to do is break it down step by step and practice. Try this one on your own and post the answer as a comment if you think you got it right:

“Train A leaves Paris at noon and travels at a constant speed of 75 mph toward Berlin. At the same time, train B leaves Berlin headed toward Paris at a constant speed of 50 mph. If Paris and Berlin are 500 miles apart, then at what time will the two trains pass each other?”

Remember, if you want, you can always get extra help studying for the GRE from the experts at Test Masters. Good luck!

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6 Responses to Sample Math Problem: Two trains leave the station…

  1. Niko says:


    Took a while, but it is easy to see that the correct distance from Paris will be 3/2 the distance from Berlin.

    • Calvin says:

      Perhaps, but the question asked you to find the time when the two trains would pass each other! Be careful about stopping work on a problem before you have found the solution the question asks for. Very often, numbers that are intermediate solutions rather than final solutions will appear in the answer choices as traps you should avoid. If you give me a time of day, I will be able to confirm if your answer is correct.

  2. Omair says:


    12 noon + 20
    2 am

    am i wrong sir, kindly let me knw the solution.

    • Calvin says:

      Sorry, that is incorrect. An alternative way to solve the problem would be to make a chart with 4 columns: one for time, one for the distance traveled by one train, one for the distance traveled by the other train, and one for the sum of the distances. Since Paris and Berlin are 500 miles apart (in this problem, at least), then the time when the sum of the distances traveled by each train equals 500 miles is the time when the two trains pass each other.

  3. mrittunjoy says:

    after 4 hours train a passes 300 miles and at the same time train B passes 200 miles. 300+200=500, so two trains pass each other at ( 12pm+4h) that means 4 pm. if I wrong plz tell.

  4. Rick Matthews says:

    125 = 500/t (where 125 is the relative speed of both trains heading towards each other)

    125 t = 500

    t = 500/125

    t = 4

    So 4 hours after noon is 4pm

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