What should you do when you see a GMAT problem asking you for the average rate over an entire journey? Try your hand at this problem and let’s see.
Problem:
A canoeist paddled upstream at 10 meters per minute, turned around, and drifted downstream at 15 meters per minute. If the distance traveled in each direction was the same, and the time spent turning the canoe around was negligible, what was the canoeist’s average speed over the course of the journey, in meters per minute?
(A) 11.5
(B) 12
(C) 12.5
(D) 13
(E) 13.5
Solution:
In average rate problems many students forget that average rate means total distance divided by total time and not the average of the rates. This is especially true on problems, such as this one, that give the test-taker two rates, but no distances and no times. When this occurs, the most concrete strategy, which will be quickest for moth test-takers, is to pick numbers.
Keep in mind that when picking numbers, the numbers you choose must conform to any constraints in the problem. Here we are told that the distance was the same in both directions, so we should pick a number that is a multiple of both speeds. The lowest common multiple of our speeds, 10 and 15, is 30, so we will set our distance as 30 meters in each direction.
Next, we calculate the time in each direction using this distance. Going upstream we travel 10 meters per minute for 30 meters. Since distance/rate equals time, we know 30/10= 3 minutes. Returning we travel the same distance at 15 meters per minute. Thus, we know that 30/15 = 2 minutes.
Finally, we need total distance/total time to find our average rate. Our total distance is 30 + 30 = 60. Our total time is 2 + 3 = 5. 60/5 = 12, which is answer choice (B).
This can also be solved algebraically, and if you read the question and VERY QUICKLY and ACCURATELY can set up the appropriate equations and solve, that might be your best approach. However for the majority of test-takers, there is some uncertainty and hesitation when setting up the equations, and hence it can be riskier in case your equations or algebra are not perfectly accurate. Plus the equations will include dividing by variables and won’t be extremely straight forward, whereas picking ‘30’ for distance here made the scenario concrete and the test-taker who took that approach will often be done before the algebraic solver as they confidently solve. It’s great to practice both techniques as you study (algebra and picking numbers), so that you have multiple tools you can use on test day, which might serve you better for different problems.
~Bret Ruber
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