GMAT Tip: Picking Numbers for Unknown Quantities

If you know this cyclist's average speeds up and down the mountain and how long the trip took, figuring the mileage is easy Photograph by Nick Hal/Aurora

This tip on improving your GMAT score was provided by Vivian Kerr at Veritas Prep.

Let’s take a quick look at how we can apply the classic “Picking Numbers” strategy to an advanced Average Rates question.

A cyclist raced to the top of a mountain at an average pace of 20 miles per hour, then took the same trail down. On the way south, his average speed increased to 24 miles per hour. If the entire trip took exactly two hours to complete non-stop, how many miles is the trail one way?

For the way up, we know that D = 20mph x T. For the way down, we know that D = 24mph x T.  We know that the distance up the mountain was the same as the distance down, so let’s choose a value for D. Let’s choose “120″ since that is a multiple of both 20 and 24.  If 120 = 20mph x T, then we know that T = 6 hours. If 120 = 24mph x T, then we know that T = 5 hours.

Now we can use the Average Rate formula to find the average speed for the entire journey:  Average Rate = Total Distance / Total Time.  Using the number we picked, 120, we know that the total distance traveled would be 120 + 120 = 240 miles. The Total Time is 6 hours + 5 hours = 11 hours.  So the Average Rate = 240 miles / 11 hours = 21.8 mph.

It doesn’t matter that the cyclist didn’t “really” go 120 miles on the way up the mountain, or that we know she didn’t “really” go 240 miles total. We picked a value so that we could find the ratio of the Total Distance to the Total Time in order to calculate the Average Rate of the total trip up the mountain. We can use this average rate for the whole trip to find the real-life distance for the entire journey.

D = R x T

D = 21.8mph x 2 hours

We know that T = 2 hours because the problem gave us this given information. Therefore the actual distance for the entire trip was approximately 43.6 miles. The questions asks us how many miles the trail was one way. Since 43.6 / 2 = 21.8 miles. The answer to the question is 21.8 miles.

It’s possible also to solve this question using a system of equations and substitution, but it’s handy to know that you can pick a number for the distance traveled and use it to find the average rate for the entire bike journey. “There and back” questions, in which an inbound and outbound journey follows the same path, can be good opportunities to pick numbers when no exact distance is given.

Vivian Kerr has been teaching and tutoring in the Los Angeles area since 2005. She graduated from the University of Southern California, studied abroad in London, and has worked for several test-prep giants tutoring, writing content, and blogging about all things SAT, ACT, GRE, and GMAT.

For more GMAT advice from Veritas Prep, read “The Hidden Clue in Every GMAT Problem Solving Question”

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