Data Sufficiency

This is a classic OVERLAPPING SETS problem. It is usually best to set these up with a DOUBLE-SET MATRIX:



From the question stem, we know that 180 guests have houses in both locations, and we're looking for the ratio of the two circled boxes:



There's nothing else we can infer from the question stem, so we move on to statement 1:

(1) One-half of the guests had a house in Palm Beach.
From this we can also infer that half the guests do NOT have a house in Palm Beach:



This gives us no information about the two circled boxes, though. Insufficient.

Make sure that you always erase what you learned from statement 1 before tackling statement 2 in DS overlapping sets!

(2) Two-thirds of the guests had a house in the Hamptons
From this we can also infer that 1/3 of guests did NOT, but it tell us nothing about our circled boxes:



(1) and (2) together:
Intuitively, you might be able to tell that we don't have enough information to fill in our circled boxes. But if we wanted to prove it, here's what we know algebraically:



If we simplify the equation in our second row, we get:
((2/3)x + 180) + ((1/3)x - ((1/2)x - 180)) = (1/2)x
(2/3)x + 180 + (1/3)x - (1/2)x + 180 = (1/2)x
(2/3)x + (1/3)x - (1/2)x = (1/2)x
x - (1/2)x = (1/2)x

This is a tautology; we cannot solve for x. We can't infer anything about the ratios of the circled boxes.

The answer is E.

Overlapping Sets are a common question type on the GMAT, and the double-set matrix is often the easiest way to solve. For more on how to use the matrix, see:
https://www.beatthegmat.com/among-the-m ... tml#827756
https://www.beatthegmat.com/in-a-class- ... tml#802582
https://www.beatthegmat.com/jefferson-s ... tml#708717
https://www.beatthegmat.com/sandwich-t2 ... tml#730500
https://www.beatthegmat.com/ds-question ... tml#677080
https://www.beatthegmat.com/survery-res ... tml#576105
https://www.beatthegmat.com/rainbow-tro ... tml#555603


You also may notice that with overlapping sets in DS, the answer often ends up being E. That certainly doesn't mean that the answer will always be E, but E seems to be more likely to show up in this topic than in others:
https://www.beatthegmat.com/for-what-pe ... tml#818652
https://www.beatthegmat.com/in-a-produc ... tml#818274
https://www.beatthegmat.com/retailers-q ... tml#809549