55 people live in an apartment complex with three fitness

Question

55 people live in an apartment complex with three fitness clubs (A, B, and C). Of the 55 residents, 40 residents are members of exactly one of the three fitness clubs in the complex. Are any of the 55 residents members of both fitness clubs A and C but not members of fitness club B?

(1) 2 of the 55 residents are members of all three of the fitness clubs in the apartment complex.
(2) 8 of the 55 residents are members of fitness club B and exactly one other fitness club in the apartment complex.

Zarrolou · Accepted Answer

We know that 55-40 people are numbers of 2 or more clubs.
Lets call a the members of only AC, b the members of only AB, c the members of only CB and d the members of all three. And consider n that are the members of none of those clubs!
 n+a+b+c+d=15 
Now the question is: is a >0? there are members in AC only?

1) tells us that d=2. We cannot say anything about a
 m+a+b+2=15 
2)tells us that b+c=8.Again not sufficient
 n+a+8+d=15

1+2) n+a+8+2=15  We cannot say anything about A.
E

Transcendentalist · Answer

The question does not specify if there are people not member of any clubs. So isn't E a possibility as well?

Assume there are 5 people not a part of any club

nt2010 · Answer

Answer is C

I would use the 3 overlapping sets formula to evaluate whether the stat 1 and 2 are sufficient.

Total people = people enrolled in A + people enrolled in B + people enrolled in C - (people enrolled in A & B only + people enrolled in B & C only + people enrolled in A &C only ) - 2 (people enrolled in A, B, and C).

Equation based on question stem :

55 = 40 - (people enrolled in A & B only + people enrolled in B & C only + people enrolled in A &C only ) - 2 (people enrolled in A, B, and C).

Stmt 1 gives only 2 (people enrolled in A, B, and C) - Not Sufficient

Stmt 2 gives only ( people enrolled in A & B only + people enrolled in B & C only  + people enrolled in A &C only ) - Not Sufficient

Considering Stmt 1 and Stmt 2 people enrolled in A &C only can be calculated.

//kudos please, if above explanation is good

Zarrolou · Answer

Indeed my friend, you are right. The question written as it is has E as answer

PratikHP · Answer

Ans - D (since the question does not mention that "every resident belongs to at least one fitness club") Had the question included the above condition the answer would've been an easy C. That's the beauty of GMAT math questions. It not only checks one's ability to solve a question but also his/her ability of analyzing the question. As they say, on good math question on GMAT will also check your verbal skills!

Regards Pratik

raoshahb · Answer

Can anybody please explain the second statement with a venn diagram ? 
What does the bold part means in venn diagram? 8 of the 55 residents are members of fitness club B and  exactly one other fitness club in the apartment complex

Thanks..

Abhishek008 · Answer

This is a problem that involves three overlapping sets. A helpful way to visualize this 
is to draw a Venn diagram as follows:
Each section of the diagram represents a different group of people. Section a
represents those residents who are members of only club a. Section b represents 
those residents who are members of only club b. Section c represents those residents 
who are members of only club c. Section w represents those residents who are 
members of only clubs a and b. Section x represents those residents who are 
members of only clubs a and c. Section y represents those residents who are 
members of only clubs b and c. Section z represents those residents who are 
members of all three clubs.
The information given tells us that a + b + c = 40. One way of rephrasing 
the question is as follows: Is x > 0 ? (Recall that x represents those residents who are 
member of fitness clubs A and C but not B).
Statement (1) tells us that z = 2. Alone, this does not tell us anything about x, which 
could, for example, be 0 or 10, among many other possibilities. This is clearly not 
sufficient to answer the question.
Statement (2) tells us that w + y = 8. This alone does not give us any information 
about x, which, again could be 0 or a number of other values.
In combining both statements, it is tempting to assert the following.
We know from the question stem that a + b + c = 40. We also know 
from statement one that z = 2. Finally, we know from statement two that w + y = 8. 
We can use these three pieces of information to write an equation for all 55 residents 
as follows:
a + b + c + w + x + y + z = 55.
(a + b + c) + x + (w + y) + (z) = 55.
40 + x + 8 + 2 = 55
x = 5
This would suggest that there are 5 residents who are members of both fitness clubs 
A and C but not B.
However, this assumes that all 55 residents belong to at least one fitness club. Yet, 
this fact is not stated in the problem. It is possible then, that 5 of the residents are not 
members of any fitness club. This would mean that 0 residents are members of 
fitness clubs A and C but not B.
Without knowing how many residents are not members of any fitness club, we do 
not have sufficient information to answer this question. 
The correct answer is E: Statements (1) and (2) TOGETHER are NOT sufficient.

KVKool · Answer

When to consider the scenario that neither part could be non 0 too. Because in lot of 2 set questions, we tend to ignore that part. Any thumb rule to be followed?

Posted from my mobile device

MBAB123 · Answer

Depends on the context of the question, but I don't think we should ignore that part. We might consider that part and deem it to be zero.

BodhiChatterjee · Answer

What is the actual formula for 3 overlapping sets?

I am seeing different formulas written in different places.

Posted from my mobile device

KarishmaB · Answer

The data given in the question stem and the two statements independently is shown in the Venn diagram below. Screenshot 2023-11-18 at 12.37.14 PM.png Now when we take both statements together, we are still left with 2 question marks. We do not know how the leftover 5 people are split among the intersection of A and C only area and None. Hence we cannot answer how many people are members of A and C but not B. Answer (E)

KarishmaB · Answer

There are many versions derived from:
Total = A + B + C – (A and B) – (B and C) – (C and A) + (A and B and C) + None
and
Total = None + Exactly one set + Exactly two sets + All three sets

Both these are derived from the Venn diagram. Hence I like to use just the Venn diagram mostly.

aronbhati · Answer

Request you to please elaborate 2nd statement through Video 8 of the 55 residents are members of fitness club B and exactly one other fitness club in the apartment complex

i am not able to understand which portion we have take consider in this

KarishmaB · Answer

Think about which areas represent the fitness club B and exactly one other club. It is the two red regions shown in my 3rd diagram. Check out this post: https://anaprep.com/sets-statistics-thr ... ping-sets/ It will help you map out all the areas of the 3 overlapping sets Venn diagram.

bumpbot · Answer

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55 people live in an apartment complex with three fitness

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