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Set Q contains 14 distinct numbers. Which of the following operations would decrease the average of set Q while leaving the median unchanged?

A. Decreasing all 14 numbers by 2 each

B. Increasing the 3 smallest numbers by 4 each and decreasing the 4 largest numbers by 3 each

C. Increasing the 2 smallest numbers by 2 each and decreasing the 3 largest numbers by 1 each

D. Decreasing the 7 smallest numbers by 2 each and increasing the 7 largest numbers by 1 each

E. Decreasing the 2 smallest numbers by 5 each and increasing the 3 largest numbers by 3 each

I could let set the first number as 1 and make my way to 14 and do the operations but that would be tedious and because this is a standardized test question I'm afraid I'd mess up if I were to do that on the test. Is there a better way to solve this problem? can you notice any patterns? take advantage of the multiple choice format of the test

1 Answers1

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First, we can know that only A, D, and E will decrease the sum, and therefore decrease the average.

All of these three operations leave the order of numbers unchanged, and the median of 14 numbers is the average of the 7th and 8th numbers. That means operation A will decrease the median by 2, operation D will decrease the median by 0.5 (the 7th number is decreased by 2 and the 8th number is increased by 1), and operation E leave the median unchanged.

So only operation E will decrease the average and leave the median unchanged.

infmagic2047
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  • How do you know that the median is the average of the 7th and 8th numbers? How do you know the median is the 7th and 8th instead of the 7th and 6th without writing the numbers out? Is there a formula to find which number the median would be in a set with a even number of numbers? –  Sep 17 '14 at 21:17
  • @Ben I mean the 7th and 8th numbers after they are sorted from least to greatest. Since operations A, D, and E don't change the relative order, the 7th and 8th numbers in the new sorted list are those in the original sorted list, plus or minus the number in the operation. – infmagic2047 Sep 18 '14 at 11:11