Overlapping Sets - Problem 1

[aditiniyer]

Some of the people in Town X are left handed, some are tall, some are both, some are neither.In Town Y, three times as many people are left handed as are left handed in Town X, three times as many people are tall as are tall in town X, three times as many people are both as are both in Town X, but no one is neither. If the total number of people in Town X is four times greater than the total number of people in Town Y, which of the following could be the number of people in town X who are neither left-handed nor tall ?

a) 23
b) 39
c) 72
d) 143
e) 199

[GMATGuruNY]

Some of the people in Town X are left-handed, some are tall, some are both, and some are neither. In Town Y, three times as many people are left-handed as are left-handed in Town X, three times as many people are tall as are tall in Town X, three times as many people are both as are both in Town X, but no one is neither. If the total number of people in Town X is four times greater than the total number of people in Town Y, which of the following could be the number of people in Town X who are neither left-handed nor tall?


A 23 B 39 C 72 D 143 E199

One formula for overlapping groups:

T = Group1 + Group2 - Both + Neither

The big idea is to subtract the overlap. In the question above, there is an overlap between the left-handed people and the tall people. Thus, when we count all the left-handed people and all the tall people, the overlap -- the number who belong to both groups -- will be counted twice. So that we don't double-count these people, we need to subtract them from the total.

Let L = left-handed people in town X
Let T = tall people in town X
Let B = the people in X who are both left-handed and tall
Let N = the people in X who are neither left-handed nor tall

Town X:
X = L + T - B + N

In Y, there are 3 times as many left-handed people, 3 times as many tall people, and 3 times as many who are both:
Y = 3L + 3T - 3B

The total in X is 4 times the total in Y:
L + T - B + N = 4(3L + 3T - 3B)
L + T - B + N = 12L + 12T - 12B
N = 11L + 11T - 11B
N = 11(L + T - B).

Thus, the number of people in X who are neither left-handed nor tall must be a multiple of 11.

The correct answer is D.

[Mo2men]

If the total number of people in Town X is four times greater than the total number of people in Town Y, which of the following could be the number of people in Town X who are neither left-handed nor tall?

Dear GMAT Guru,

I feel something wrong in the phrasing above, doe not 'Town X is four times greater than the total number of people in Town Y' mean X=5Y. According to your interpretation, it means that Town X is four times as many as in total number of people in Town Y'.

Can you clarify about construction?

Thanks

[Matt@VeritasPrep]

Dear GMAT Guru,

I feel something wrong in the phrasing above, doe not 'Town X is four times greater than the total number of people in Town Y' mean X=5Y. According to your interpretation, it means that Town X is four times as many as in total number of people in Town Y'.

Can you clarify about construction?

Thanks

Guru's interpretation of that phrase is the one commonly used/accepted, even if there is some ambiguity. A good discussion on the topic is here. The GMAT, as far as I know, will never use "five times greater than" to meet "six times", so there shouldn't be anything to worry about.

[ceilidh.erickson]

To Matt's point, the GMAT will almost certainly avoid this ambiguity by not using the phrase "4 times greater than" and instead saying "4 times the number of," etc. I can't think of a real GMAT problem that uses the phrasing "4 times greater than." (Though I can't claim perfect photographic memory of all OG questions. If anyone has evidence of an OG question that uses this phrasing, please let me know).

My guess is that you won't have to worry about this issue on the real GMAT. It was probably just an oversight of whatever prep company wrote the problem.

[GMATGuruNY]

If the total number of people in Town X is four times greater than the total number of people in Town Y, which of the following could be the number of people in Town X who are neither left-handed nor tall?

Dear GMAT Guru,

I feel something wrong in the phrasing above, doe not 'Town X is four times greater than the total number of people in Town Y' mean X=5Y. According to your interpretation, it means that Town X is four times as many as in total number of people in Town Y'.

Can you clarify about construction?

Thanks

The total number of people in Town X is 4 times greater than the total number of people in Town Y.

Given the OA, we know that this sentence is intended to convey that X=4Y.
Do not worry about the correct interpretation of the phrase in red.
While this phrase has appeared in an official Verbal problem -- https://www.beatthegmat.com/the-gyrfalco ... 83853.html -- to my knowledge it has never appeared in an official Quant problem.
As Ceilidh has suggested, the GMAT is likely say one of the following:
The total number of people is Town X is 4 times the total number of people in Town Y.
There are 4 times as many people in Town X as in Town Y.