# GMAT Problem Solving: There Can Be Only One

On the GMAT, there is only one correct answer to each question (How many caught the Highlander reference in the title?  Be honest!).

I know, big surprise, right?

But that simple, obvious statement leads us to a powerful deduction. Some Problem Solving questions on the Quantitative section will have terms, variables, or unknowns that are unsolvable—they could take multiple values on the basis of the information in the stem. And we’re not talking Data Sufficiency here. “Not sufficient” isn’t a choice (Occasionally, “Cannot be determined” is a choice on problem solving questions. This answer is usually a trap, but you can use Data Sufficiency solving techniques to see if multiple answers are possible). So if the answer choices are numbers or proportions, and some term in the question stem is unsolvable, that undetermined x-factor can’t affect the outcome. Some ratio or mathematical step in the solution has to result in that variable “canceling out,” because otherwise the problem would have multiple correct solutions and therefore could not appear on the GMAT!

This is one of the ways that the Kaplan strategy of Picking Numbers works. Once you’ve identified an unknown that cancels out, you can plug in any value for that unknown, and be confident that your result is the right one. For instance, consider the following problem:

A runner runs downhill from point A to point B at 15 kilometers per hour, then runs uphill along the same path from point B to point A at 10 kilometers per hour. Assuming the time spent turning was negligible, what was the runner’s average speed during the round trip?

11.5

12

12.5

13

13.5

Here, the distance the runner travels is unknown and unknowable. But we’re asked for average speed. That’s a ratio; if we double the distance, we’ll double the time, and get the same speed. The answer to this question won’t change if he’s running one meter or one million!

Of course, neither one nor one million makes this problem particular easy to solve. Since every distance will give us the same average speed, we should pick the distance that takes the least effort when we plug it into the question. What’s the right distance to choose to make the arithmetic work? And, how can we use it to find the average speed?

Eli Meyer has been a Kaplan teacher since 2003. He has spent the past four years focused almost exclusively on the GMAT, and also has prior experience helping students ranging from middle-schoolers taking the ISEE to professors retaking the GRE for their second PhD. During his Kaplan career, Elis has also written and revised Kaplan course materials and acted as a community liaison on several popular GMAT message boards, all the while helping his students succeed both in and out of the classroom.

What’s the right distance to choose to make the arithmetic work? And, how can we use it to find the average speed?
The best distance to choose would be the least common multiple of 2 speeds: 15 and 10, so that both the time periods are integers. In this case distance can be taken as 30 km.
Then time1 = 30/15 = 2hours
time2 = 30/10 = 3 hours
Total time = 5 hours
Average speed = total distance/total time = 60/5 = 12
I believe a very common mistake people make in such questions is to directly take the average of both the speeds. This is wrong since in such questions the weights for calculating speed is time and not the distance!!

• Anonymous

Bingo, well done!