x97agarwal wrote:

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

For this question "If not everyone at the fundraiser had a house in either the Hamptons or Palm Beach" does this sentence not mean that everyone has a house either in the Hamptions or Palm Beach. So when you draw your grid the No Hamptons and No Palm Beach should be 0". However, in the explanation the answer is different.

Total number of people in the fundraising program = T

the number of people who had a house both in the Hamptons and in Palm Beach = P&H = 180

the number of people who had a house in Palm Beach = P

the number of people who had a house in Palm Beach but not in the Hamptons = P (not H)

so, P = 180 + P (not H)

the number of people who had a house in the Hamptons = H

the number of people who had a house in the Hamptons but not in Palm Beach = H (not P)

so, H = 180 + H (not P)

the number of people who had no house either in the Hamptons or in Palm Beach = P&H (not)

so, total = T = 180 + P (not H) + H (not P) + P&H (not)

The question is: what is 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?

or, what is P (not H) / H (not P) = ?

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

P = T/2

180 + P (not H) = (180 + P (not H) + H (not P) + P&H (not))/2

180 + P (not H) = H (not P) + P&H (not)

180 = H (not P) + P&H (not) - P (not H) ............................(i)

nsf.........

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

H = 2T/3

3H = 2T

3 (180 + H (not P)) = 2 (180 + P (not H) + H (not P) + P&H (not))

180 + H (not P) = 2 (P (not H)) + 2(P&H (not))

180 = 2 (P (not H)) + 2(P&H (not)) - H (not P) ........................... (ii)

nsf.........

from 1 and 2, i and ii are equal. so

H (not P) + P&H (not) - P (not H) = 2 (P (not H)) + 2(P&H (not)) - H (not P)

2 (H (not P)) = P&H (not) + 3 (P (not H))

since we do not know awhat is P&H (not), still insufficient. E.

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GT