Two trains starting from cities 300 miles apart head in opposite directions at rates of 70 mph and 50 mph, respectively. How long does it take the trains to cross paths?

This is a classic problem that sends chills up students’ collective spines. I’m now going to add another bone rattling element: The Empty Box.

That’s right—the GRE will have a fill-in-the blank/empty box math problem. There won’t be too many, judging from the ETS Revised GRE book, but even a few should be enough to discomfit most.

Solution

Let’s go back, and attack the above problem the following way. When you have any two entities (trains, bicyclists, cars, etc.) headed towards each other you must add their rates to find the total rates. The logic behind this is the two trains (as is the case here) are coming from opposite directions straight into each other.

This yields 120 mph, a very fast rate (which accounts for the severity of head-on collisions…don’t worry the trains in the problem won’t collide).

To find the final answer, we want to employ our nifty old formula: D = RT, where D stands for distance, R stands for rate, and T stands for time.

We’ve already found R, which is their combined rate of 120 mph. They are 300 miles apart so that is D. Plugging those values in, we get 300 = 120T. Dividing 120 by both sides, we get T = 2.5 hrs.

Now we can confidently fill that box in, and let the trains continue on their ways.