GMAT Math: Avoiding the Most Common Mistakes in Average Speed Questions

Written by Kelly Granson. Posted in GMAT Study Guide

11223512-speedMost people consider motion problems to be among the easiest on the quantitative section of the GMAT. Nevertheless, many test takers fall prey to one particular type of motion problem—the Average Speed question. In this article we will discuss the two most common errors you should avoid when you see an average speed question.

Common mistake #1

One of the most common mistakes in average speed problems occurs when the question asks you to calculate the average speed of a car or other object moving at two different speeds for different parts of the journey. It is tempting in such questions to calculate the arithmetic mean (average) of the two speeds, and select a corresponding answer. (Trust me; test creators will make sure to include that corresponding but wrong answer among the answer choices.) This averaging approach is wrong. The average speed of an object moving at two different speeds during two different periods is NOT calculated by taking the arithmetic mean of the two speeds.

This may not be obvious in GMAT problems that ask for the average speed of a car that first travels 60 miles at 60 miles per hour and then travels another 60 miles at 40 miles per hour. It may be tempting to define the average speed of the car for the entire journey as the arithmetic mean of the two speeds, 60 and 40 miles per hour, or 50 miles per hour, but this is not correct.

To calculate the average speed for the entire journey, you need to use the simple but effective formula:

CodeCogsEqn copy

In this example, the car travelled 60 + 60 = 120 miles total. For the first part of the journey, the time taken would be CodeCogsEqn (1) copy_copy For the second part of the journey, the time taken would be CodeCogsEqn (2) copy_copy The total time would equal CodeCogsEqn (3) copy Now, just plug in these values into the formula CodeCogsEqn (4) copy

CodeCogsEqn (6) copy CodeCogsEqn (7) copy miles per hour, not 50. 

To better understand why simply finding the average of the two speeds regardless of time and distance for each speed is not the way to go, consider a more obvious example. A car travelled at 90 miles per hour for 5 hours and at 10 miles per hour for 4 min. Obviously the average speed of the car will be nearly 90 miles per hour, since the fraction of the distance and the amount of time travelled at 10 miles per hour is almost negligible. But if you simply find the average of the two speeds, you will get CodeCogsEqn (8) copy miles per hour, which is obviously wrong.

Remember: Average speed ≠ Average of the two speeds

Common mistake #2

Quite often GMAT problems use different units of measurement in one question. For example, if a car takes 30 minutes to travel 60 miles, and then takes 2 hours to travel 60 miles, you calculate the average speed of the car for the entire journey expressing 30 minutes as 0.5 hours or by converting 2 hours to 120 minutes. Either way, you never calculate time, distance, or speed using different units of measurement for the different parts of the journey. In this problem, if you had wrongly used two different units of time for calculating average speed, you would obtain the wrong answers even with the correct formula.

Shortcuts for average speed problems

There are two situations in which you can save time when dealing with average speed questions:

-When the distances are equal

-When times are equal

Equal distance shortcut

If the same distance is first travelled at a speed of CodeCogsEqn (9) copy and then travelled at a speed of CodeCogsEqn (10) copy for example, when travelling from City A to City B and then back to City A, the average speed for the entire trip can be calculated quickly by the formula:

CodeCogsEqn (11) copy

Note that this shortcut is applicable to the previous example, since the car travelled 60 miles at each speed. Using this formula can significantly shorten and hasten your calculations:

CodeCogsEqn (12) miles per hour.

Equal time shortcut

There is one special occasion where you can use the arithmetic mean of two speeds to calculate the average speed for the entire journey. If an object moves at a rate of CodeCogsEqn (9) copy first and then moves at a rate of CodeCogsEqn (13) for the same amount of time, then you can calculate the average speed by taking the arithmetic mean of the two speeds, CodeCogsEqn (9) copy and CodeCogsEqn (13).

For example, if a plane travels for two hours at 600 miles per hour and then travels at 1000 miles per hour for another two hours, its average speed for the whole trip would be CodeCogsEqn (14) miles per hour. Make sure you use this shortcut only when the amount of time travelled was the same at each speed.

Otherwise, whenever you see an average speed question on the GMAT, make sure you do not confuse average speed with the average of the two speeds. You may use the shortcut formula CodeCogsEqn (11) copy, if the distances covered at each speed were equal (usually when going to a certain point and returning by the same route but at a different speed), and you may use the arithmetic average of the two speeds if the time travelled at each speed was the same, but these are the only exceptions. Should you have any doubts or uncertainties about whether one of these shortcuts is applicable, use the general formula CodeCogsEqn (4) copy This formula is applicable to any average speed question.

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