Answering the CORRECT Question

Many of us fear the multiple choice time limited tests – the kinds of GRE, GMAT, SAT and such. The reason for this fear is quite understandable – those tests have a serious effect over our “final score” that serves us in attending colleges and universities. One single test is almost as important as the entire high-school grades collection! Well, this is true and this is scary.
However, another, more important reason for being terrified by GRE is that this test is DIFFERENT – unlike any tests we have done throughout our formal education. The questions posed seems to be puzzling, many times we don’t know how to start. What do they want from me in this problem? Many times we start our calculations without completely understanding the question, hoping that something, on the way, will help us to uncover the truth. Sometimes this works, sometimes not.

One should know that one of the keys to succeeding in this kind of test is not just to choose the correct answer, but also to answer the right QUESTION! As many times we make a completely correct calculation – and yet get the wrong answer, because we are answering the WRONG question!

Pose your friends the following question:
“The car travels half of the way with the speed of 60 mph and the other half – with the speed of 90 mph. What is the average speed of a car?”
Seems easy, isn’t it? Let me make it even easier for you. I will give you four answers to choose from:

The average speed of a car:

  1. …is more than 75 mph
  2. Equals 75 mph
  3. …is less than 75 mph
  4. Depends on the length of the way

Are you getting suspicious now? It can’t be that easy, right? But… the average speed is (60 + 90) / 2 = 75 mph. Well, if your answer is (2) your consolation is that over 90% of people do the same mistake… So, what’s the catch? I will answer you by stating another question:

“The car travels half of the time with the speed of 60 mph and the other half – with the speed of 90 mph. What is the average speed of a car?”

Did you notice the difference between the two situations? All of the sudden, travelling TIME is the same, not the travelling distance. In case of the second question – the answer 75 mph is correct. But we are trying to answer ANOTHER question.

In the first episode, the TIME is not the same. It is obvious that the car spends more time “travelling half of the way at 60 mph” than on the second part of the way. The average speed then should be less than 75 – answer number (3). But wait! What about number (4)? Doesn’t it depend on the distance travelled? Well, let’s check it:

Say the distance is 360 miles. As speed * time = distance, the cars spends 3 hours on the first half of the way (360 / 2 / 60 = 3) and then 2 hours on the second half (360/2/90). 5 hours in total. The average speed (let’s avoid any tricks here to be sure) is the total distance divided by total time spent: 360 / 5 = 72 mph – less then 75, indeed (in case you were skeptic…)

Let’s now suppose the distance is 1080 miles (the number is provisional, you can choose another if you want, but this one produces easy calculations)

First half of the way will take 540/60 = 9 hours; the second half 540/90 = 6 hours. 15 hours total. 1080/15 = 72 (Did you know that 1080/15 = (1080/10 + 1080/20) / 2 – and this is yet NOT the fastest way to calculate it!). 72 mph once again! A coincidence, maybe? Well, it is not (try any other number and you will get the same result).

Let’s do some algebra here to prove it:
Say the overall distance is W.
Then, time required to travel the first half of the way is W / 2 / 60 = W/120.
And the time required to travel the second half of the way is W/2/90 = W/180.
In total, we have spent W/120 + W/180, which is 5W/360.
Average speed is total distance (W) divided by total time (5W/360). Simple calculation will give us the elimination of W and the answer 360/5 = 72. No magic, no trick.

But if you want a trick, I will tell you, that we could get 72 by an even easier way – proportioning the difference of 90-60 (which is, of course 30) by a fraction of 6/9 (which is, of course 2/3) and getting the numbers 12 and 18. And 60+12 is 72 (as is 90-18). But that is another story… A more straightforward hint is that, as the average speed is independent on the distance, you can choose an arbitrary distance and do all the calculations – much easier than solving equations with X, isn’t it?

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