Problem Solving

I am solving these sums half and seem to be missing out the final step. Please help

Last month 15 homes were sold in Town X. The average (arithmetic mean) sale price of the homes was $150,000 and the median sale price was $130,000. Which of the following statements must be true?

I. At least one of the homes was sold for more than $165,000.
II. At least one of the homes was sold for more than $130,0000 and less than $150,000
III. At least one of the homes was sold for less than $130,000.

A. I only
B. II only
C. III only
D. I and II
E. I and III

Ok so the mean is 150,000 and the median is 130,000.

The median is the middle number in the group, so it's in the middle of of the 15 prices, and you could call it the 8th highest.

If we were to have only the median to consider, then anything goes as long as the middle price is 130,000. All of them could cost 130,000. One could cost 130,000 and the rest could be evenly split above and below 130,000. Anything that keeps 130,000 in the middle of the fifteen prices works.

We do have another piece of information though; the mean is 150,000. Now we know something else. At least one of the homes has to have been sold for more than 130,000 in order to create that mean.

So let's go to the statements.

I. Does one have to cost more than 165,000? What this is really asking is if the median is 130,000 does another have to cost over 165,000 to make the mean 150,000?

In other words this is asking how low the highest price can be and still create a mean of 150,000.

Start by maximizing the prices at or below the median. By working with this maximum, we can find the minimum prices above the median that create that 150,000 mean. There will be eight prices at or below the median, and the maximum is 130,000 each. So we have a total of 8 * 130,000 = 1,040,000.

If the mean is 150000, and there are 15 houses, the total of all of their costs is 150,000 * 15 = 2,250,000.

So by taking away from the total of the fifteen the total of the prices at or below the median, 2,250,000 - 1,040,000 = 1,210,000, we get the total value of the other 7 houses. Divide by 7 to get 1,210,000/7 = approx. 173,000 So even if all of prices above the mean were the same and as high priced as possible, the minimum price of any of them would be about 173,000. So statement I has to be true.

II. We already figured out that 8 of them can go for 130,000 and the other 7 for about 173,000. So this does not have to be true.

III. This is maybe a little bit of a trap in case you thought some had to be below the median, but really it is not true either, as was discussed above.

So only I is true.

Choose A.

As for the other problem, how about posting it in a separate post so that we can keep it to one question per post.