At a charity fundraiser, 180 of the guests had a house both

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by BTGmoderatorDC » Thu Mar 21, 2019 9:59 pm

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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

OA E

Source: Manhattan Prep

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Source: Beat The GMAT — Data Sufficiency | Thu Mar 21, 2019 9:59 pm

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ceilidh.erickson: GMAT Instructor; Posts: 2095; Joined: Tue Dec 04, 2012 3:22 pmFollowed by: 247 members; Thanked: 1443 times

by ceilidh.erickson » Fri Mar 22, 2019 10:33 am

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.

Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education

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ceilidh.erickson: GMAT Instructor; Posts: 2095; Joined: Tue Dec 04, 2012 3:22 pmFollowed by: 247 members; Thanked: 1443 times

by ceilidh.erickson » Fri Mar 22, 2019 10:39 am

Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education

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