MI3 wrote:Hello,

Can someone please assist on how to resolve the below question?

Q: 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 E)199

Thanks,
M

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.

Great explanation, many thanks.