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# Set S has 5 numbers such that its average (arithmetic mean) is greater

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Senior SC Moderator
Joined: 14 Nov 2016
Posts: 1315
Location: Malaysia
Set S has 5 numbers such that its average (arithmetic mean) is greater  [#permalink]

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21 Feb 2017, 19:20
2
00:00

Difficulty:

65% (hard)

Question Stats:

39% (01:48) correct 61% (02:29) wrong based on 36 sessions

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Set S has 5 numbers such that its average (arithmetic mean) is greater than its median. Also, set T has 7 numbers such that its average (arithmetic mean) is greater than its median. If all the elements in set S and set T are different, is the average (arithmetic mean) of set S and set T combined greater than the median of set S and set T?

1) The average (arithmetic mean) of set T is greater than that of set S.

2) The median of set T is greater than that of set S.

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Magoosh GMAT Instructor
Joined: 28 Dec 2011
Posts: 4494
Re: Set S has 5 numbers such that its average (arithmetic mean) is greater  [#permalink]

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23 Feb 2017, 17:30
2
AustinKL wrote:
Set S has 5 numbers such that its average (arithmetic mean) is greater than its median. Also, set T has 7 numbers such that its average (arithmetic mean) is greater than its median. If all the elements in set S and set T are different, is the average (arithmetic mean) of set S and set T combined greater than the median of set S and set T?

1) The average (arithmetic mean) of set T is greater than that of set S.

2) The median of set T is greater than that of set S.

Dear AustinKL,

I'm happy to respond.

This is interesting as a pure math problem. It involves either lengthy calculations or some particularly felicitous skill in picking numbers, but it doesn't really lend itself to any elegance. To that extent, I am not sure it is the best model for the kind of challenge that the GMAT Quant presents. It is as if the question were written by someone who knows a lot of math, who knows the topics on the GMAT, but doesn't necessarily have a keenly honed sense of the quality of the challenge that the MGAT Quant poses.

It is going to be hard to establish sufficiency here, because there are infinite numbers of possibilities for the sets. The individual statements are not sufficient. Let's fast forward to the combined statement case.

We have to satisfy the requirements
a) mean (S) > median (S)
b) mean (T) > median (T)
c) mean (T) > mean (S)
d) median (T) > median(S)

The number picking strategy will pit sets with single extremely large values, vs. set with numbers close together.

Here is one pair of sets, with some extreme values, that easily satisfy all the requirements of the prompt & two statements:
Set S = {1, 2, 3, 4, 100}
Set T = {1, 2, 3, 4, 5, 6, 200}
For these choices the combined set is
Set ST = {1, 1, 2, 2, 3, 3, 4, 4, 5, 6, 100, 200}
Clearly, mean (ST) > median(ST). This indicates a "yes" for the prompt question.

After extensive fiddling with possible sets, here is a counterexample set:
Set S = {4, 4, 4, 5, 5}, median = 4, mean < 5
Set T = {2, 2, 5, 5, 5, 8, 9}, median = 5, mean > 5
For these choices the combined set is
Set ST = {2, 2, 4, 4, 4, 5, 5, 5, 5, 5, 8, 9}
median = 5
mean < 5
This produces a "no" answer to the prompt.

Two different possibilities product two different answers to the prompt. Nothing is sufficient. OA = (E)

I will say that I solve most official GMAT Quant questions in about 30 seconds, and it took me a good 15 minutes of experimenting with numbers to create the second combination. There is really no elegant way to approach this problem--it requires extensive play with number picking. While this is quite enjoyable in terms of recreational mathematics, this is NOT the sort of problem the GMAT Quant would have.

Does all this make sense?
Mike
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Mike McGarry
Magoosh Test Prep

Education is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)

Re: Set S has 5 numbers such that its average (arithmetic mean) is greater &nbs [#permalink] 23 Feb 2017, 17:30
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