The question is from manhattan math challenging problems

11/19/07
Question
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


My method is by plugging in numbers
City X
Leftie: 1
Tall : 1
Both : 1
Neither : X
City Y
Leftie : 3
Tall : 3
Both : 3
Neither : 0

Equation becomes

3+x= 9*4

x=33


But there is no multiple of 33 in the answers? Where am I going wrong?

It should be a multiple of 11 i.e. I think the answer is D.
Can you post the OA?

You are correct

The official answer is given below which makes sense but I wanted to find out where amI going wrong?

OA

The best way to approach this question is to construct a matrix for each town. Let's start with Town X. Since we are not given any values, we will insert unknowns into the matrix:
Left-Handed Not Left-Handed Total
Tall A C A + C
Not Tall B D B + D
Total A + B C + D A + B + C + D


Now let's create a matrix for Town Y, using the information from the question and the unknowns from the matrix for Town X:
Left-Handed Not Left-Handed Total
Tall 3A 3C 3A + 3C
Not Tall 3B 0 3B
Total 3A + 3B 3C 3A + 3B +3C


Since we know that the total number of people in Town X is four times greater than the total number of people in Town Y, we can construct and simplify the following equation:



Since D represents the number of people in Town X who are neither tall nor left-handed, we know that the correct answer must be a multiple of 11. The only answer choice that is a multiple of 11 is 143 (= 11 × 13).

The correct answer is D.

The problem with your method is you started out with too many people in each quadrant, if you had done (0, 0, 1) instead of (1, 1, 1) in Town X and, as a result, (0, 0, 3) in Town Y you would have gotten

x + 1 = 3 *4
x = 11

which will give you the lowest factor, as opposed to 33.

Ratio Order
Left Hand : Tall : Both : Neither

Town X = a : b : c : N
Town Y = 3a : 3b : 3c
N = ??
X = 4Y
X = a+b+c+N, Y = 3a+3b+3c

N = 11(a+b+c)
So, N is a multiple of 11
143 = 11 * 13
Ans: D

The key here is that you were "plugging in numbers." Typically this approach will help especially if you have percents or fractions with no definite values.

There were two main problems with your approach:
(1) One problem was that you're picking numbers for quantities that are unrelated to each other. You decided that there are the same numbers of tall, lefties, and both. However, this assumption is not necessarily true.
(2) The second and probably the more significant error was the way you picked numbers:
My method is by plugging in numbers


I'm guessing by Leftie, you mean "Leftie and NOT tall"; by Tall you mean "Tall and NOT Leftie." However, the problem is not saying "In Town Y, three times as many people are left-handed AND NOT TALL as are left-handed AND NOT TALL in Town X." Instead this comparison is related to ALL Lefties: those tall and those NOT tall. This applies to the numbers of people who are TALL, and also BOTH.

To make sure you don't fall into this error in substitution, just set up a chart for each town organizing the Leftie and Non-Lefties, and Tall and Non-Talls.

Hope that helps.

_________________
Ben Ku
Instructor
ManhattanGMAT