Imagine you are driving from Chicago to Los Angeles, and you want to know what your average speed needs to be to reach Los Angeles in a certain number of hours. You would probably start by determining the speed you will be able travel during certain parts of your journey. Since most of the distance will be covered by highway, you might plan to travel most of the distance at 70 miles per hour. However, you will also want to plan for some traffic when you are still in or near Chicago and when you get close to Los Angeles. During these parts of your journey let’s say you can plan to travel at 30 miles per hour.
When calculating the average speed at which you will be traveling, you need to avoid the trap of just averaging these speeds together and planning on an average speed of 50 miles per hour. Because the vast majority of your journey will take place at 70 miles per hour and only a relatively small portion will take place at 30 miles per hour, simply averaging the speeds is not sufficient. You need to account for the difference in the amount of time you will be driving at each speed. Once you do so, your average speed will be much closer to 70 miles per hour than 30 miles per hour.
The same principle will apply when you see average speed questions on the GMAT. Average speed is defined as total distance divided by total time, rather than the average of the speeds. Additionally, the average of the speeds will almost always be offered as an answer choice, so be sure to avoid it.
This can be especially tricky when a problem gives little information other than the two speeds. On test day, you should think strategically and pick a number for the distance, calculate the times using this number, and then plug into the average speed formula described above. Give it a try on the following 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?
Step 1: Analyze the Question
A canoeist goes one rate in one direction, turns around, and goes back at a different rate. Whenever you deal with one entity that has different rates at different times, set up a chart to track the data. Otherwise, you’ll find yourself in a six-variable, six-equation system that will take a long time to work through. Also, notice that although the distance and time are never mentioned, there are no variables in the answer choices. Whenever variables will cancel out, consider Picking Numbers.
Step 2: State the Task
Our task is to calculate her average speed for the whole journey.
Step 3: Approach Strategically
The formula is this:
Average Speed = Total Distance / Total Time
But we’re seemingly told nothing about time, and the only thing we know about distance is that it is the same in both directions. So what to do? As with almost every problem involving a multistage journey, set up this chart:
Now plug in the data we’re given:
Now we see clearly that we’ll be able to know the time if we know something about the distance. Since we know whatever variable we put in place will cancel out by the time we get to the answer choices, let’s just pick a number for distance—one that will fit neatly with a rate of 10 and a rate of 15. A distance of 30 should work well:
At this point, we can fill in the rest of the chart very straightforwardly. The entire distance is 60. The time taken upstream must be 3, and the time taken downstream must be 2. That makes the entire time 5.
Step 4: Confirm Your Answer
The speed for the entire trip, then, is 60 / 5 = 12. Answer (B).
Reread the question stem, making sure that you didn’t miss anything about the problem.
To all of you strategic thinkers out there, can you spot a way to quickly eliminate 3 of the answer choices without doing any calculations?