At a charity fundraiser, 180 of the guests had a house both in the Hamptons and in Palm Beach. If not everyone at the fundraiser had a house in either the Hamptons or Palm Beach, what is the ratio of the number of people who had a house in Palm Beach but not in the Hamptons to the number of people who had a house in the Hamptons but not in Palm Beach?

(1) One-half of the guests had a house in Palm Beach.

(2) Two-thirds of the guests had a house in the Hamptons

Was having a look at the MGMAT explanation given. My answer coice is 'C'. Please see attachment for the solution.
I assume that MGMAT choice (Ans: 'E') is correct. But the mathematical derivation tells me otherwise.
It would be nice if you could have a look at the attached explanation.

AugiTh

Hello AugiTh,

Thanks for your post. What your solution assumes (incorrectly) is that everyone at the fundraiser has a house either in the Hamptons or in Palm Beach. When you write that T = H + P + 180 (as you did), you are making this assumption. However, the question stem explicitly states that NOT everyone at the fundraiser has a house in either place (I'd certainly fall into this category!). In other words, there are some (>=1) people in the "NEITHER" category. In other words, you'd have to write the equation as follows: T = H + P + 180 + N. This extra variable N is essentially what causes you not to know the answer, even with both statements taken together.

By the way, this is one of the disadvantages of the Venn diagram approach. Conceptually, it favors the "both" category (the intersection of the circles) & can cause you to overlook the "neither" category. It's also harder to deal with subtotals. This is why we favor a matrix approach for many, if not most, overlapping set problems. (An exception would be a case in which you have three sets of categories: say, having a house in Palm Beach or not, having a house in the Hamptons or not, and having a house in Tahiti or not -- again, I'd fall in the "none of the above" category, unfortunately...)

Hope this is helpful!

Chris

This thread hasn't had much activity in a while, but I just hit this problem in one of my MGMAT CAT tests, and it really put a wrench in my plan.

The question for MGMAT folks is: how do you deal with the "neither" case using the double set matrix. In the answer solution it shows the matrix with "house in the hamptons" and "no house in the hamptons" on the top axis, and "house in palms" and "no house in palms" on the vertical axis.

Thanks!

H = the number of people who had a house in the Hamptons but not in Palm Beach
P = the number of people who had a house in Palm Beach but not in the Hamptons
N = the number of people who had house neither in Palm Beach not n the Hamptons
Union = H + P + 180
Universal = H + P + 180 + N
P/H = ??

Statement 1:
(P+180) / (H + P + 180 + N) = 1/2
Not Sufficient

Statement 2:
(H+180) / (H + P + 180 + N) = 2/3
Not Sufficient

Statement 1 and 2
Solve for H and P and find P/H; However, Not possible to eliminate N
Not Sufficient

Ans: E

Ignore previous one, there was a typo

H = the number of people who had a house in the Hamptons but not in Palm Beach
P = the number of people who had a house in Palm Beach but not in the Hamptons
N = the number of people who had house neither in Palm Beach nor n the Hamptons
Union = H + P + 180
Universal = H + P + 180 + N
P/H = ??

Statement 1:
(P+180) / (H + P + 180 + N) = 1/2
Not Sufficient

Statement 2:
(H+180) / (H + P + 180 + N) = 2/3
Not Sufficient

Statement 1 and 2
Solve for H and P and find P/H; However, Not possible to eliminate N
Not Sufficient

Ans: E

"neither" is the intersection of "no house in hamptons" and "no house in palms". it should be the center square in the matrix.

(this is the meaning of the word "neither" -- "neither x nor y" means the combination of "not x" and "not y".)

akhp -- your equations are correct, but you'll find that these problems will become a LOT easier if you take the time to learn how to use the double set matrix.

i wanted to comment on this (emphasis mine):


unless i'm misunderstanding your writing, it appears that you have an implicit assumption that, if you are to find the ratio P/H, then you must find the INDIVIDUAL values of P and H.

this is a bad mistake -- in many (if not most) data sufficiency problems that call for combinations of variables, you'll be able to find the combination WITHOUT finding the individual variables.
the trademark of the test -- especially on harder problems -- is to give problems on which you can find the combination, even though you CAN'T find the variables themselves!

i.e., if there were a situation in which you can find the ratio P/H, you would most likely be able to find at a ratio without the actual values of P and H -- and, in all probability, you'd be able to find that ratio even if P and H themselves were still undetermined.

the answer to this problem is (e), meaning that you can't find anything in this problem anyway, so this observation is irrelevant to the problem at hand. however, if you walk into every data sufficiency problem thinking that you should always solve for individual variables, you will be in for a rude surprise (if not several rude surprises).

Hi Ron,
I understood your concern. I believe you are referring to this.

..............in P..........................not in P ..............Total
in H .........{ 180 (actual) }........._.........................[4]
not in H ....._...........................{ N }....................[2]
Total ........[3]..........................[3]......................[6] (assumed)

Even after combing statement I and II, not possible to make relation between { N } and { 180 (actual) }

However, I believe that above matrix can work only when we have only two circles of ven-diagram. Otherwise, it would be difficult to apply it.

Thanks for suggestion. I appreciate it.

Hi akhp77,

The set up for the Double-Set Matrix is correct, except you put in "assumed" numbers. While in your mind, you may be able to keep track that they are not "actual," it's often confusing when you're using actual and assumed numbers in the same grid. When you aren't sure of something, you should just use a variable.

If we let x by the total number of guests, then statements (1) and (2) produce a combined matrix:

..............in P............not in P .............Total
in H .........180...........2/3 x - 180..........2/3 x
not in H ...1/2 x-180............................1/3 x
Total ......1/2 x............1/2 x..................x



You are correct. The Double Set Matrix only works for two overlapping sets; you need to draw Venn Diagrams for three overlapping sets. However most GMAT questions that use the Venn Diagram only have two overlapping sets.

Hope that helps.

_________________
Ben Ku
Instructor
ManhattanGMAT

Hi,
Can anyone tell me the significance of the below line as posted in the question stem -

"If not everyone at the fundraiser had a house in either the Hamptons or Palm Beach,"

Is it signifying that neither Hampton nor Palm Beach grid area is 0. Please shed some light onto this.

Thanks
H

That line is actually telling you that there is at least one person who does not have a house in either the Hamptons or Palm Beach.

For example, let's assume that there are 10 people at the party (I am making this up). If "everyone" had a house, then I know that all 10 have houses. But if "not everyone" had a house, then I know that at least one person does not have a house in either the Hamptons or Palm Beach.

As a further example, if I say to you "Not everyone has the opportunity to go to either college or trade school" then you know that some people don't have the opportunity to go to either one.

Hope this helps! :-)

_________________
Jamie Nelson
ManhattanGMAT Instructor