From GMAT Prep 1:

Last month 15 homes were sold in Town X. The avg. 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

answer is A. I was only able to think about this abstractly and wasn't very sure of my answer after spending 2 min. so, question is, how would you recommend solving?

thanks

the main deal with 'at least one' problems - which come up more often on probability than on other problem types - is that they're very difficult to treat directly. instead, when you see 'at least one', you should treat the OPPOSITE situation - i.e., none.

so, because 'at least one' and 'none' are opposites, the following statements are exactly equivalent:
* there must be at least one
* it's IMPOSSIBLE to have NONE
the second is the easier way to think about it.

so:
in this problem, you should consider the case in which NONE of the homes was sold for whatever price is mentioned in the problem, and see whether it's IMPOSSIBLE.

NOTE: I AM NOT GOING TO WRITE THE THOUSANDS. so, '130' means $130,000. you'll thank me; this problem will be much easier to read.

(preface)
the median of 15 values is the value that comes 8th in the list. therefore, the first seven values are 130 or less, the 8th value is 130, and the 9th-15th values are 130 or more.
also, Sum = Average x Number of data points, so the sum of all the prices is 15 x 150 = 2250.

(i)
let's consider the case in which NONE of the homes was sold for more than 165.
the MAXIMUM sum of prices in this case would be 8(130) + 7(165), which is the case if all of the first 8 values are 130 each (the biggest they can be) and values 9-15 are 165 each.
that's a total of 1040 + 1155 = 2195.
not high enough.
therefore, it's IMPOSSIBLE to have NO prices over 165, so this statement must be true.

(ii)
let's try to create a list with NO such house prices.
let the first 7 prices be, say, 100 each.
the 8th is 130.
so the first 8 have a sum of 830, meaning that the highest 7 have a sum of 2250 - 830 = 1420.
there are all kinds of ways to do that with no values between 130 and 150, but the simplest is to make all seven of them equal to 1420 / 7, which is greater than 200.
so (ii) doesn't have to be true.

(iii)
let's try to create a list with NO such house prices.
this would mean that the first 8 prices are all 130.
so, the last 7 prices sum to 2250 - 8(130) = 1210.
that's an average of 1210/7, which is a shade over 170. you could let all 7 of the high prices equal that value, and it would work.
therefore, (iii) doesn't have to be true.

--

i agree with you that this problem is a lot of work.

very clearly explained
thanks ron

Here's a solution without much calculating that proves that (1) must be true.

It makes sense that if the median price is below the average price, the the average of seven highest prices should be further from 150 than the average of the seven lowest prices, which we know is no more than 130. Since the average of the seven highest prices must therefore be at least 170, at least one of them must be more than 165.

i like it!

well played.

Here's a solution without much calculating that proves that (1) must be true.

It makes sense that if the median price is below the average price, the the average of seven highest prices should be further from 150 than the average of the seven lowest prices, which we know is no more than 130. Since the average of the seven highest prices must therefore be at least 170, at least one of them must be more than


can somebody please explain this further... I am unable to get this.

[quote="kevin"][quote="abovethehead"]From GMAT Prep 1:

Last month 15 homes were sold in Town X. The avg. 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


thanks[/quote]
Here's a solution without much calculating that proves that (1) must be true.

It makes sense that if the median price is [b]below[/b] the average price, the the average of seven highest prices should be [b]further from 150[/b] than the average of the seven lowest prices, which we know is no more than 130. Since the average of the seven highest prices must therefore be at least 170, at least one of them must be more than 165.[/quote]

Why does the average of the seven highest have to be at least 170,000, how was 170 calculated?

the median, which is the eighth value (in order) out of fifteen, is 130,000. this means that you have 8 values that are 130,000 or less.

now think about the OTHER seven values. the smallest these values can get is if the first eight are all exactly 130,000. (if you lower any of those eight, then you have to raise at least one of the other seven to maintain the same average.)

even if the first eight values are all exactly 130,000, the average of the last seven must be GREATER than 170,000 (which is the same distance on the other side of the mean, 150,000). think about it this way: if you average eight 130,000's and seven 170,000's, you'll still get an overall average below 150,000.
so the average of the seven highest numbers (the 9th through 15th numbers) must be even higher than 170,000.

Similar to Kevin's approach:

Total = 15 * 15 = 225
Lets distribute 13 evenly: 15 * 13 = 195

Left out = 225 - 195 = 30
This can only be distributed from 9th to 15th term (totally 7 terms).

Even if you distribute it evenly = 30/7 = 4.xx
So 9th - 15h term will be greater 13 + 4.x = 17.x : I is TRUE

Thanks

lots of different ways to solve this thing.

Another method that I used was leveraging the options. I realized if I can prove statement 2 and 3 as YES/NO, I would be left with only option 1.

Good point!

_________________
Jamie Nelson
ManhattanGMAT Instructor