Of the guests at a charity fundraiser, 180 own both a ...

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Hello,

This question is from MGMAT CAT:

Of the guests at a charity fundraiser, 180 own both a house in the Hamptons and a house in Palm Beach. If not every guest at the fundraiser owns a house in either the Hamptons or Palm Beach, what is the ratio of the number of people who own a house in Palm Beach but not in the Hamptons to the number of people who own a house in the Hamptons but not in Palm Beach?

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

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



I was attempting to solve as follows:

Palm Beach Not in Palm Beach Total
Hampton 180 a -
Not in Hampton b - -
Total - - x


b/a = ?

1) Palm Beach Not in Palm Beach Total
Hampton 180 a -
Not in Hampton b - -
Total x/2 - x


In-sufficient

2) Palm Beach Not in Palm Beach Total
Hampton 180 a 2/3x
Not in Hampton b - -
Total x/2 - x

In-sufficient

1 & 2 - Still Insufficient.

OA: E



Now the question mentions the following:

If not every guest at the fundraiser owns a house in either the Hamptons or Palm Beach,


I was just wondering if this information is really needed since we are not using it anywhere. Thanks a lot - Sri

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by rakeshd347 » Sun Oct 06, 2013 5:26 pm
gmattesttaker2 wrote:Hello,

This question is from MGMAT CAT:

Of the guests at a charity fundraiser, 180 own both a house in the Hamptons and a house in Palm Beach. If not every guest at the fundraiser owns a house in either the Hamptons or Palm Beach, what is the ratio of the number of people who own a house in Palm Beach but not in the Hamptons to the number of people who own a house in the Hamptons but not in Palm Beach?

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

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



I was attempting to solve as follows:

Palm Beach Not in Palm Beach Total
Hampton 180 a -
Not in Hampton b - -
Total - - x


b/a = ?

1) Palm Beach Not in Palm Beach Total
Hampton 180 a -
Not in Hampton b - -
Total x/2 - x


In-sufficient

2) Palm Beach Not in Palm Beach Total
Hampton 180 a 2/3x
Not in Hampton b - -
Total x/2 - x

In-sufficient

1 & 2 - Still Insufficient.

OA: E



Now the question mentions the following:

If not every guest at the fundraiser owns a house in either the Hamptons or Palm Beach,


I was just wondering if this information is really needed since we are not using it anywhere. Thanks a lot - Sri
Hi Sri,

I think answer should be E.

Well it asks about the ratio of two people with Plam only and with hampton only. Now the information that you have highlighted it means that there are some people who don't own either of them. So there are some people who neither own palm nor own hampton.

If you try this problem with double matrix method, its very easy to see and you will quickly get answer E.

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by melguy » Sun Oct 06, 2013 8:40 pm
Hello

As Rakesh has mentioned its best to set up a matrix. It helps you organize the info a lot more efficiently. Make 3 - one for Statement 1, Statement 2 and combining statements.

Hope that helps.
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by Brent@GMATPrepNow » Mon Oct 07, 2013 7:19 am
gmattesttaker2 wrote: Now the question mentions the following:

If not every guest at the fundraiser owns a house in either the Hamptons or Palm Beach,


I was just wondering if this information is really needed since we are not using it anywhere. Thanks a lot - Sri
Hi Sri,

You're right; this information doesn't play a role in the question.

Cheers,
Brent
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by Brent@GMATPrepNow » Mon Oct 07, 2013 7:22 am
I thought I'd point out that Melguy is referring to a technique known as the Double Matrix Method. This technique can be used for most questions featuring a population in which each member has two characteristics associated with it.
Here, we have a population of guests, and the two characteristics are:
- owning or not owning a house in the Hamptons
- owning or not owning a house in Palm Beach

To learn more about this technique, watch our free video: https://www.gmatprepnow.com/module/gmat- ... ems?id=919

Then try these additional practice questions that can be solved using the Double Matrix Method:
- https://www.beatthegmat.com/mba/2011/05/ ... question-1
- https://www.beatthegmat.com/mba/2011/05/ ... question-2
- https://www.beatthegmat.com/mba/2011/05/ ... question-3
- https://www.beatthegmat.com/ds-quest-t187706.html
- https://www.beatthegmat.com/overlapping- ... 83320.html
- https://www.beatthegmat.com/finance-majo ... 67425.html
- https://www.beatthegmat.com/ds-french-ja ... 22297.html
- https://www.beatthegmat.com/sets-t269449.html#692540
- https://www.beatthegmat.com/in-costume-f ... tml#692116

Cheers,
Brent
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by mcdesty » Mon Jul 14, 2014 9:41 pm
Here is what my scratch paper looked like when I solved this one.
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by ashok.tprk » Wed Mar 04, 2015 9:57 pm


MY ANSWER IS C

let total guests=X

house only in the hampton be H

house only in the palm beach be P

house in both hampton and palm beach=180

so total guests X=H+P+180

STATEMENT (1)

ONE-HALF OF THE GUESTS HAD A HOUSE AT PALM BEACH


X/2 =P+180------(A)


NOT SUFFICIENT

STATEMENT (2)


2X/3 OF THE GUESTS HAD A HOUSE IN HAMPTONS


2X/3= H+180----(B)

NOT SUFFICIENT

USE BOTH

by A+B we get

x/2 + 2x/3 = H+P+180+180

7X/6 = X+180

X=1080

FROM EQUATION A & B

P=360 & H=540

SO REQUIRED RATIO IS P:H=360:540
P:H=2:3


ANSWER IS "C"

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by GMATGuruNY » Thu Mar 05, 2015 3:15 am
Of the guests at a charity fundraiser, 180 own both a house in the Hamptons and a house in Palm Beach. If not every guest at the fundraiser owns a house in either the Hamptons or Palm Beach, what is the ratio of the number of people who own a house in Palm Beach but not in the Hamptons to the number of people who own a house in the Hamptons but not in Palm Beach?

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

(2) Two-thirds of the guests own a house in the Hamptons.
Statement 1: One-half of the guests own a house in Palm Beach
No information about the Hamptons.
INSUFFICIENT.

Statement 2: Two-thirds of the guests own a house in the Hamptons.
No information about Palm Beach.
INSUFFICIENT.

Statements combined:
1. One-half of the guests own a house in Palm Beach.
2. Two-thirds of the guests own a house in the Hamptons.


Strategy:
TEST TWO CASES.
Let P = Palm Beach and H = the Hamptons.

Case 1: Guests = 600.
Total P = (1/2)600 = 300.
Thus:
Only P = (total P) - (both H and P) = 300-180 = 120.
Total H = (2/3)600 = 400.
Thus:
Only H = (total H) - (both H and P) = 400-180 = 220.
Resulting ratio:
Only P : Only H = 120:220 = 6:11.

Case 2: Guests = 900.
Total P = (1/2)900 = 450.
Thus:
Only P = (total P) - (both H and P) = 450-180 = 270.
Total H = (2/3)900 = 600.
Thus:
Only H = (total H) - (both H and P) = 600-180 = 420.
Resulting ratio:
Only P : Only H = 270:420 = 9:14.

Since the ratio can be different values, INSUFFICIENT.

The correct answer is E.
Last edited by GMATGuruNY on Wed Mar 11, 2015 2:37 am, edited 1 time in total.
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by MartyMurray » Thu Mar 05, 2015 8:45 am
Of the guests at a charity fundraiser, 180 own both a house in the Hamptons and a house in Palm Beach. If not every guest at the fundraiser owns a house in either the Hamptons or Palm Beach, what is the ratio of the number of people who own a house in Palm Beach but not in the Hamptons to the number of people who own a house in the Hamptons but not in Palm Beach?

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

(2) Two-thirds of the guests own a house in the Hamptons
Ashok,

You can look at this without really doing any math and realize that the number of people invited is never really defined and so the ratio cannot be determined.

When we add up all the information, the clues we have are the following. 180 own both, 1/2 and 2/3 own each, and 1/2 and 2/3 are greater than 1.

We can use those clues to determine the minimum and maximum numbers of guests, but as long as we stay within those constraints, various combinations can work.

1) This means that in addition to the 180 who own both there are at least 180 guests who either own a only house in the Hamptons or do not own a house in either Palm Beach or the Hamptons.

2) This means that in addition to the 180 who own both there are at least 90 guests who either own only a house in Palm Beach or do not own a house in either Palm Beach or the Hamptons.

Even given both those constraints, there could be various different numbers of guests and the 180 who own both could comprise a relatively large section of the guests or a relatively small section of the guests.

As the number of guests becomes larger, the 180 loses significance and the ratio of guests who own only a house in Palm Beach to those who own only a house in the Hamptons gets larger.

As the number of guests get smaller, the 180 who own both becomes more significant and could be the entirety of the number of guests who own a house in Palm Beach, at which point the number of people who own only houses in Palm Beach becomes 0 and the ratio is then 0/(the number who own only a house in the Hamptons) = 0

So seeing the logic of what's going on is another way to get this one, and avoid mistakes.
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by Matt@VeritasPrep » Wed Mar 11, 2015 1:52 am
I might be crazy, but I'm seeing some mistakes here.
Case 2: Guests = 1200.
Total P = (1/2)1200 = 600.
Thus:
Only P = (total P) - (both H and P) = 600-180 = 420.
Total H = (2/3)1200 = 800.
Thus:
Only H = (total H) - (both H and P) = 800-180 = 620.
Resulting ratio:
Only P : Only H = 420:620 = 21:31.


In this case, we have 420 only in the Hamptons, 620 only in Palm Beach, and 180 in both. This gives a total of 1220 guests, forcing the number with a home in neither the Hamptons nor Palm Beach to be -20.

... which is a specific case of what I think is the general error (red font is mine):
Even given those constraints, there could be infinite different numbers of guests and the 180 who own both could comprise a relatively large section of the guests or a relatively small section of the guests.

As the number of guests becomes larger, the 180 loses significance and the ratio of guests who own only a house in Palm Beach to those who own only a house in the Hamptons approaches (1/2)/(2/3) = 3/4


We can't go to infinity here, so limits as we approach infinity are misleading. Let's say that t = the total # of guests and that n = the number who own a home in neither the Hamptons nor Palm Beach. We know n > 0, and obviously t > 0. We're also constrained to integers. Since (1/2)t and (2/3)t are integers, we know that t is divisible by 6.

Taking the two statements together, we have (1/2)t + (2/3)t - 180 + n = t, or n = 180 - t/6.

Since n > 0, we must have 180 > t/6, or 1080 > t. We also know that there must be at least 360 guests, since (1/2)t is at least 180. So t can be any multiple of 6 from 360 to 1074, inclusive, and we have 120 distinct solutions, not ∞.

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by GMATGuruNY » Wed Mar 11, 2015 2:45 am
Good catch, Matt.
In my post above, the total in Case 2 has been lowered to 900, a value that yields 30 guests owning neither type of house.
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by MartyMurray » Wed Mar 11, 2015 4:59 am
Too funny. I guess 1/2 + 2/3 is greater than 1 isn't it. Fixing my answer now.
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by Matt@VeritasPrep » Wed Mar 11, 2015 12:18 pm
GMATGuruNY wrote:Good catch, Matt.
In my post above, the total in Case 2 has been lowered to 900, a value that yields 30 guests owning neither type of house.
No sweat! I'm so used to your explanations being the gold standard that I checked my math three times to be sure I wasn't seeing things. Neat how subtle the implications of these questions can be: we know there are multiple cases, but even then we can blunder into an impossible case. Bleeping GMAT!