Problem Solving

A test is taken by 100 people and possible scores are the integers between 0 and 50, inclusive. Would the avg. score be greater than 30? (Answer Yes, No or Uncertain)

Condition- No more than 2 test takers scored any given score.

Ans: NO

This one is from MGMAT's Advanced Quant book. I did not understand the explanation to the question. Could someone simplify?

pareekbharat86 wrote:A test is taken by 100 people and possible scores are the integers between 0 and 50, inclusive. Would the avg. score be greater than 30? (Answer Yes, No or Uncertain)

Condition- No more than 2 test takers scored any given score.

Ans: NO

This one is from MGMAT's Advanced Quant book. I did not understand the explanation to the question. Could someone simplify?

Hi Bharat!

Let's start by breaking down the info - there's actually a lot more there than you may think!

Possible scores are the integers 0 to 50, inclusive - that's 51 possible scores.
100 people take the test.
At most, 2 people can get the same score.

Let's assume for a moment that every score gets duplicated. We have 100 people, so that would mean 50 different scores. There are only 51 scores available, so at MOST 1 score gets left out (i.e. either all 51 scores get used or 50 out of 51 get used).

Now let's look at the exact question: is the average score greater than 30?

Well, if 2 people got each of the 51 scores, then the average would be (0+50)/2 = 25

That's below 30, so now let's ask: if we maximize the scores, can we get the average over 30?

Well, to maximize the scores, we have no one get a "0" and 2 people get every other score. So, now our average is (1+50)/2 = 25.5 which is still less than 30.

Accordingly, the average is definitely less than 30, giving us a definite "NO".

The key to this question, like many on the GMAT, is to break down the stem before even starting any calculations. Once you realize that almost every score gets used twice and that the overall average is going to be very close to 25, actually answering the question is intuitive.

I hope that helps!

Stuart

pareekbharat86 wrote:A test is taken by 100 people and possible scores are the integers between 0 and 50, inclusive. Would the avg. score be greater than 30? (Answer Yes, No or Uncertain)

Condition- No more than 2 test takers scored any given score.

Ans: NO

Alternate approach:

Step 1: Plug in the THRESHOLD.
Here, the threshold = an average of 30.
If the average of the 100 scores = 30, then the SUM of the 100 scores = (number)(average) = 100*30 = 3000.

Step 2: Determine the MAXIMUM POSSIBLE SUM
Since each score can be included at most 2 times, the maximum possible sum will be yielded if the 100 scores are as follows:
50, 50, 49, 49, 48, 48....3, 3, 2, 2, 1, 1.
In the list above, each integer between 1 and 50, inclusive, appears TWICE.
Implication:
The maximum possible sum is equal to 2 times the sum of the integers between 1 and 50, inclusive.
Sum of consecutive integers = (number of integers)(average of biggest and smallest).
Thus, the maximum possible sum = 2 * (number of integers)(average of biggest and smallest) = 2 * 50 * (50+1)/2 = 2550.

Result:
Since the maximum possible sum is LESS than 3000, the average of the 100 scores must be LESS than 30.