Data Sufficiency

At a charity fundraiser, 180 of the guests had a house both in the Hampton's and in Palm Beach. If not everyone at the fundraiser had a house in either the Hampton's 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 Hampton's.

given: H&PM=180, H isn't equal to PM, find PM/H-?
st(1) 1/2(H+PM-H&PM)=PM, H+PM-H&PM=2PM, PM-H=180, PM=180+H, PM/H= (180+H)/H Not Sufficient;
st(2) 2/3(H+PM-H&PM)=H, 2H+2PM-2H&PM=3H, H-2PM=360, 2PM=H-360, Not Sufficient;
Combined st(1&2): PM=180+H AND 2PM=H-360 --> 2(180+H)=H-360, H=0 Not Sufficient, as H=0 this ratio cannot be defined PM/H= PM/0

IOM E

sanju09 wrote:At a charity fundraiser, 180 of the guests had a house both in the Hampton's and in Palm Beach. If not everyone at the fundraiser had a house in either the Hampton's 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 Hampton's.

Conceptually speaking, there's one huge giveaway that the answer will be E, and that's the lack of information about the "neither" group.

We're told explicitly in the prompt that "not everyone at the fundraiser had a house in either the Hampton's or Palm Beach", meaning that some people at the fundraiser didn't have a house at either location.

Both Statements (1) and (2) give us fractional amounts of the TOTAL NUMBER OF GUESTS at the fundraiser. But there's still no information about that "neither" group. Different values of "neither" will produce different values for the ratio we're asked for in the prompt. Thus, there's no way to nail down a definitive value of that ratio.

Hope that helps!

@razz: does your pick "giveaway" make any sense here?
we are looking for the following ratio --> 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
we are not interested in those who have no house(s) ...

raz1024 wrote:Conceptually speaking, there's one huge giveaway that the answer will be E, and that's the lack of information about the "neither" group.

We're told explicitly in the prompt that "not everyone at the fundraiser had a house in either the Hampton's or Palm Beach", meaning that some people at the fundraiser didn't have a house at either location.

Both Statements (1) and (2) give us fractional amounts of the TOTAL NUMBER OF GUESTS at the fundraiser. But there's still no information about that "neither" group. Different values of "neither" will produce different values for the ratio we're asked for in the prompt. Thus, there's no way to nail down a definitive value of that ratio.

Hope that helps!

The "neither" group is not our final concern. But to reiterate what I said in my previous post, it is an essential piece of information that enables us to find the ratio asked for in the problem.

If you want to represent this algebraically, you could do it as follows:

P+180+H+N = Total

where P = Palm Beach only, H = Hamptons only, and N = neither

(1) tells us that 1/2 of the guests have a house in Palm Beach

So P+180 = Total/2 or 2P+360 = Total

(2) similarly gives us H+180 = (2/3)(Total) or 1.5H + 270 = Total

If you wanted to, you could combine the statements to get 2P+360 = 1.5H + 270

and that leads you to 1.5H - 2P = 90

But then with our original equation (P+180+H+N = Total), we still know nothing about that neither group. You could plug in several different values of P and H depending upon the N.

So while the N is not the final concern, it is important to figuring out P/H, which is what the question asks for.

Hope that makes sense.

180 is the pool to which only P and H should contribute (in your interpretation of P and H), and we are not seeking the Total as such. If the question was the ratio of P to all guests then Yes your pick is useful - otherwise it's just a leave for another question.

ac is not only defined by a/(a+b+c, but could be if the question asks) you can always rewrite a/b, a/c and b/c
according to you we cannot perform the above?

If all else fails, use concrete numbers to illustrate:

We know 180 people have houses in both locations.
We know 1/2 of the total have a house in Palm Beach
We know 2/3 of the total have a house in the Hamptons

Let's say 30 people have a house in Palm Beach only. That's 180+30 = 210 people total that have a house in Palm Beach. And thus 210*2 = 420 people total.

If there are 420 people total, then there are (2/3)*(420) = 280 people who own a house in the Hamptons.
Thus, there are 280-180 = 100 people who own a house in the Hamptons only.

The ratio we're being asked for would thus be 30/100 = 3/10.

Now notice that Hampton only = 100, Palm Beach only = 30, Both = 180, and Total = 420. That means Neither = 110.

Now, let's say 60 people have a house in Palm Beach only. That's 240 people total that have a house in Palm Beach. And thus 240*2 = 480 people total.

If there are 480 people total, then there are (2/3)*(480) = 320 people who own a house in the Hamptons.
Thus, there are 320-180 = 140 people who own a house in the Hamptons only.

The ratio we're being asked for would thus be 60/140 = 3/7.

Now notice that Hampton only = 140, Palm Beach only = 60, Both = 180, and Total = 480. That means Neither = 100. (a different number of "neither")


So we end up with completely different ratios in the two examples, which means we've just shown insufficiency. But what allowed us to vary our choices of numbers? It was the fact that no restriction was given regarding the "neither" group, and thus we have no information about the total, and furthermore no information about the asked ratio.

Again, hope this makes sense.

yes I was about to capture this in my last post - but overlooked st(1) ratio of One to Total and the same with st(2) and definitely there's a place for Neither here and we need the Total

the thing is if I correct my previous solution to include N (neither)

given: H&PM=180, N=neither, find PM/H-?
st(1) 1/2(H+PM+N-H&PM)=PM, H+PM+N-H&PM=2PM, PM+N-H=180, PM=180+H-N Not Sufficient;
st(2) 2/3(H+PM+N-H&PM)=H, 2H+2PM+2N-2H&PM=3H, H-2PM+2N=360, 2PM=H+2N-360, Not Sufficient;
Combined st(1&2): PM=180+H-N AND 2PM=H+2N-360 --> 2(180+H-N)=H+2N-360, 720=4N-H; PM/H=(180+H-N)/(4N-720) where H=4N-720 (again) and PM/H=(180+4N-720)/(4N-720) --> 180/(4N-720) +1= 45/(N-180)+1 /// and we stepped into area where N must be an integer, hence N could be (N-180) the factor of 45. Here N could be 181, 183 or 185 various values ... and depending on each value of N we would get various values hence ratios for PM and H