The mean of set S does not exceed mean of any subset of...

BREAKING: Target Test Prep releases Brand New 2026 On Demand GMAT prep course

Redeem

This topic has expert replies

Post new topic Post Reply

Previous Topic Next Topic

BTGmoderatorLU: Moderator; Posts: 2505; Joined: Sun Oct 15, 2017 1:50 pm; Followed by:6 members

The mean of set S does not exceed mean of any subset of...

by BTGmoderatorLU » Mon Mar 12, 2018 2:27 pm

Timer

00:00

Your Answer

Global Stats

The mean of set S does not exceed mean of any subset of set S. Which of the following must be true about set S?

I. Set S contains only one element.
II. All elements in set S are equal.
III. The median of set S equals the mean of set S.

A. None of the three qualities is necessary
B. II only
C. III only
D. II and III only
E. I, II, and III

The OA is D.

I'm really confused by this PS question. Experts, any suggestion about how can I solve it? Thanks in advance.

Quote

Source: Beat The GMAT — Problem Solving | Mon Mar 12, 2018 2:27 pm

GMAT/MBA Expert

ErikaPrepScholar: Legendary Member; Posts: 503; Joined: Thu Jul 20, 2017 9:03 am; Thanked: 86 times; Followed by:15 members; GMAT Score:770

by ErikaPrepScholar » Tue Mar 13, 2018 5:32 am

We'll think about this question logically.

If our set is, for example, 2 4 6 8, the mean of the set is 5. If we take subset 2 8 or 4 6, the mean will still be 5. However, if we take subset 2 4 6, the mean will be 4. If we take subset 4 6 8, the mean will be 6. We should note that if our subset only eliminates numbers above the mean, we will get a subset mean that is less than the original mean; if our subset only eliminates numbers below the mean, we will get a subset mean that is greater than the original mean.

We want to find a set where the mean of every possible subset is the same as the mean of the set. If the numbers in our set are different, we will always be able to pick a subset that eliminates more of the higher numbers or more of the lower numbers, yielding a smaller or larger mean. However, if all of the numbers in our set are the same, we can eliminate as many of them as we'd like, and our mean will be the same. For example, in set 4 4 4 4 4, our mean is 4. If we take subset 4 4 4, our mean is still 4.

Knowing that, we'll look at our options.

I. If the original set contains only one element, all subsets will be the same as the original set, which will result in the same mean. However, we established that this doesn't *need* to be true - we can have multiple elements, as long as they have the same value. We can eliminate answer choice E.

II. This is exactly what we stated must be true. We can eliminate A and C.

III. If all numbers in a set or subset are the same, the median will have the same value as the numbers. For instance, in our hypothetical 4 4 4 4 4 set AND our 4 4 4 subset, the median is 4. So since all of the numbers in the set and all of its possible subsets must be the same, the median must also be the same. This must be true, so the correct answer is II and III, or D.

Erika John - Content Manager/Lead Instructor
https://gmat.prepscholar.com/gmat/s/

Get tutoring from me or another PrepScholar GMAT expert: https://gmat.prepscholar.com/gmat/s/tutoring/

Learn about our exclusive savings for BTG members (up to 25% off) and our 5 day free trial

Check out our PrepScholar GMAT YouTube channel, and read our expert guides on the PrepScholar GMAT blog

Quote

GMAT/MBA Expert

Scott@TargetTestPrep: GMAT Instructor; Posts: 8083; Joined: Sat Apr 25, 2015 10:56 am; Location: Los Angeles, CA; Thanked: 43 times; Followed by:29 members

by Scott@TargetTestPrep » Wed Mar 14, 2018 3:31 pm

LUANDATO wrote:The mean of set S does not exceed mean of any subset of set S. Which of the following must be true about set S?

I. Set S contains only one element.
II. All elements in set S are equal.
III. The median of set S equals the mean of set S.

A. None of the three qualities is necessary
B. II only
C. III only
D. II and III only
E. I, II, and III

If the mean of set S does not exceed the mean of any subset of set S, then all the elements in S must be equal. That is because if there is an element that is not equal to the others, then the mean of S will exceed the mean of at least one subset of S. For example, if S = {1, 4, 4}, we see that the mean of S is 3, and the mean of a subset of S, say {1, 4}, is only 2.5.

Thus, we see that Roman numeral II is true. Furthermore, if all the elements in S are equal, then the median of S equals the mean of S, so Roman numeral III is true also.

Answer: D

Scott Woodbury-Stewart
Founder and CEO
[email protected]