Manhattan GMAT Challenge Problem of the Week - 18 Feb 10

by , Feb 19, 2010

Welcome back to this week's Challenge Problem! As always, the problem and solution below were written by one of our fantastic instructors. Each challenge problem represents a 700+ level question. If you are up for the challenge, however, set your timer for 2 minutes and go!

Question

Set S consists of 5 values, not necessarily in ascending order: {4, 8, 12, 16, x}. For how many values of x does the mean of set S equal the median of set S?

(A) Zero

(B) One

(C) Two

(D) Three

(E) More than three

Solution

To solve this problem quickly, you might try to come up with likely values for x that would make the mean equal the median. One sort of set for which the mean equals the median is a set with values symmetrically spaced around its mean/median. The values do not have to be evenly spaced.

Three values that would make the set symmetrical are 0, 10, and 20:

{0, 4, 8, 12, 16}

{4, 8, 10, 12, 16}

{4, 8, 12, 16, 20}

We are down to choices (D) and (E). Now, can we prove that no other values of x make the mean equal the median? After all, some non-symmetrical sets have their mean equal to their median: for instance, {1, 1, 2, 2.5, 3.5}. All you need to do is make the "residuals," or differences, around the middle value cancel out (in the case above, the values to the left of 2 are 1 & 1, leaving a total residual of -2, while the values to the right of 2 are 2.5 and 3.5, leaving a total residual of +2).

Well, we can set up three scenarios, each with a relevant equation.

(1) If x is less than or equal to 8, then the median is equal to 8. We now set the mean equal to the median:

(40 + x)/5 = 8

40 + x = 40

x = 0

(2) If x is between 8 and 12, then the median is equal to x. Again, we set the mean equal to the median:

(40 + x)/5 = x

40 + x = 5x

40 = 4x

x = 10

(3) If x is greater than 12, then the median is equal to 12. Again, we set the mean equal to the median:

(40 + x)/5 = 12

40 + x = 60

x = 20

We have now exhausted all the possibilities for x. In fact, we did not have to actually compute the values of x in each case; rather, we could have simply realized that each equation is linear in x and so would have exactly one solution. Since there are three scenarios, there are exactly three values of x that satisfy the constraint of making the mean and the median equal. Indeed, if we had started with this approach, we might have gotten to the answer more quickly.

The correct answer is (D) Three.

To view the current Challenge Problem, simply visit the Challenge Problem page on Manhattan GMAT's website.