Of the z students at a certain college, x are studying

Question

Of the z students at a certain college, x are studying French and y are studying German. If w are studying both French and German, which of the following expresses the number of students at the college not studying either French or German ?

(A) z + w- x- y
(B) z - w- x- y
(C) z- w - x + y
(D) w + x + y- z
(E) w - x - y - z

(A) z + w- x- y 
(B) z - w- x- y 
(C) z- w - x + y
(D) w + x + y- z 
(E) w - x - y - z

BrentGMATPrepNow · Accepted Answer

We can also use the Double Matrix Method here. 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  z students, and the two characteristics are: 
- studying French or not studying French
- studying German or not studying German

So, we can set up our diagram as follows:
 https://i.imgur.com/U7GKSEO.png

Note : I placed a  star  in the bottom right box to remind me that this is the value we are trying to determine.

Now, if there are z students ALTOGETHER, and x of them are studying French, then the number of students NOT studying French =  z - x . 
Similarly, if there are z students ALTOGETHER, and y of them are studying German, then the number of students NOT studying German =  z - y . 
So, we can add that information to the diagram. 
 https://i.imgur.com/F12xBIU.png

w are studying both French and German   
When we add this information to our diagram, we get the following:
 https://i.imgur.com/7I3Ni61.png

When we examine the TOP 2 BOXES, we see that they add to x. So, the TOP-RIGHT box must be  x - w 
 https://i.imgur.com/BU8WcZy.png

Finally, we know that the two HIGHLIGHTED boxes below must add to z - y.
 https://i.imgur.com/HU2x4Mv.png

So, the BOTTOM-RIGHT box must equal  (z - y) - (x - w) 
 https://i.imgur.com/ngEtaCv.png

(z - y) - (x - w)  = z - y - x + w

Answer: A

RELATED VIDEO
 https://www.youtube.com/watch?v=jK-tiBrrf04

SimaQ · Answer

Should be A....

Group+Group+Neither-Both=Total

x+y+(unknown)-w=z
(unknown)=z+w-x-y

kook44 · Answer

Hard to answer b/c there are no parenthesis in the answer choices. Looks to as they are written, none of the answer choices are correct. I would say that the answer should be :
z-(w-(x+y))

jlgdr · Answer

OA please? I got A.
Used smart numbers but not sure if it is the best approach here

Bunuel · Answer

{Total} = {French} + {German} - {Both} + {Neither}

z = x + y - w + {Neither}

{Neither} = z + w - x - y.

Answer: A.

Yash12345 · Answer

EMPOWERgmatRichC · Answer

Hi All,

It looks like most of the posters in this thread recognized that this question is a variation on an Overlapping Sets question. It can be solved in a variety of ways (mostly algebraic), but there is an opportunity to TEST VALUES. You just have to be careful with your notes:

We're given 4 variables to work with:
Z = Total number of students
X = Total who study French
Y = Total who study German
W = Total who study French AND German

We're asked for the number that study NEITHER French NOR German.

Let's TEST VALUES. I'm going to keep things simple, but the note-taking here is crucial to getting the correct answer.

IF...
1 studies just French
1 studies just German
1 studies BOTH French and German

We have
X = 2 (since 1 speaks just french and another speaks both)
Y = 2 (since 1 speaks just German and another speaks both
W = 1

Now we can set the "neither" group to any positive value we want; I'm going to choose a larger number to set it apart from the others.
Neither = 5

That makes the TOTAL number of students: 1 + 1 + 1 + 5 = 8
Z = 8

So, using the values....
X = 2
Y = 2
W = 1
Z = 8

We're looking for an answer that equals 5.

Answer A: 8+1-2-2 = 5 This IS a match
Answer B: 8-1-2-2 = 3 This is NOT a match
Answer C: 8-1-2+2 = 7 This is NOT a match
Answer D: 1+2+2-5 = 0 This is NOT a match
Answer E: 1-2-2-5 = -8 This is NOT a match

Final Answer: A

GMAT assassins aren't born, they're made,
Rich

Skywalker18 · Answer

Total = (French) + (German) - (French n German) + (Neither)
=> Neither = z + w - x - y

Answer A

CounterSniper · Answer

total students =z
studying french =x
Studying german=y
both=w
z=x+y-w+neither
neither=z-x-y+w

DJ1986 · Answer

Z(total) = X(french)+Y(German)-W(Both)+N(Neither)
N = z+w-x-y

Answer A

rakshithbabu · Answer

Total Students = French Students + German Students - (French and German) students + Neither
Z = X +Y -W + Neither
Neither = z + w - x - y

ans: A

ScottTargetTestPrep · Answer

Since we have an overlapping set problem, we can use the following formula:

number of students studying French + number of students studying German + number of students studying neither subject - number of students studying both subjects = total number of students

We are given that:

number of students studying French = x

number of students studying German = y

number of students studying both subjects = w

total number of students = z.

If we let number of students studying neither subject = n, we have:
 
x + y + n - w = z

n = z + w - x - y

Answer: A

bumpbot · Answer

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Of the z students at a certain college, x are studying

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Of the z students at a certain college, x are studying

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