Of a certain group of 100 people, 40 graduated from High Sch

Question

Of a certain group of 100 people, 40 graduated from High School X, 65 graduated from College Y, and 30 live in City Z. What is the greatest possible number of people in this group who did not graduate from High School X, did not graduate from College Y, and do not live in City Z ?

(A) 5

(B) 15

(C) 35

(D) 65

(E) 85

(A) 5
(B) 15
(C) 35
(D) 65
(E) 85

Bunuel · Answer

65 graduated from College Y --> 35 did not. It's possible that all of them are from different school and city.

The group cannot be greater than 35 because College Y graduates limit the size to that number.

Answer: C.

Asifpirlo · Answer

if we had 60 and 70 in the answer choices what could be the answer bunuel? still 35?

....................... 2nd reply :

not from x = 100-40 = 60
not from y = 100-65=35
not from z = 100-30=70

So 35 is the only common number among all because of it's minimum value. 
is this the logic bunuel ?

Bunuel · Answer

If Y were 60, the answer would be 40.
If Y were 70, the answer would be 30.

GMAT40 · Answer

Hi Bunnel,

I solved this as

100 = 40 + 65+ 30 - 0 - 0 + Neither ==> 35

Is this right?

Also couldn't understand

If Y = 60 Then 40
If Y = 70 Then 30

SVaidyaraman · Answer

1. x+ y +z+  xy + yz +xz+ xyz + none=100. For "none" to be maximum others have to be minimum
2. We know the underlined portions sum to 65 and that is the minimum value of others
3. Maximum "none" = 100-65=35

KarishmaB · Answer

You can take the venn diagram perspective here. If you were to draw a venn diagram, you will make one circle for school X, another for college Y and a third for city Z. To find people who do not belong to any of these three sets, you would want to maximize the people outside these three circles. So you would want these three circles to have maximum overlap. Ques3.jpg The circle will be inside one another for maximum overlap and that will leave 35 people outside. That is the greatest number of people who did not graduate from school X, did not graduate from college Y and do not live in city Z.

Asifpirlo · Answer

Thanks. very useful diagram.

only minimum value we can arrange for each = 30 =z 10=x ,25=Y . So somehow common = 65(minimum), so max = 100 - 35=65 So you would want these three circles to have maximum overlap. - this is the gist. i got it in a new way

paddyboy · Answer

I may be reading the question wrongly, but my take on it is as follows.

My reading of the question is - how many people out of the 100 do not fall under any of the classifications mentioned in the question?

Out of the 100% population:
40% graduated from High School X  
65% graduated from College Y  
30% live in City Z

This constitutes 85% of the population.

Therefore the portion of the population which " did not graduate from High School X, did not graduate from College Y,  and  do not live in City Z " is the remaining population, which is 15% or 15 individuals out of the population of 100.

Imho the answer is B.

DavidTutorexamPAL · Answer

Since we're asked about a maximal value ('greatest possible'), we'll look at the extremes.
This is a Logical approach.

In our case we'd like to maximise the people NOT in the sets so we'd like to minimize the people IN the sets.
Since our largest set has 65 people then this is the minimum number of people we need
(As everyone in X or Z can be in Y but not the other way around).
Therefore the maximum number of people NOT in the sets is 100 - 65 = 35.
(C) is our answer.

Shri15kumar · Answer

Hello, is there any other approach to solve this question?
thanks in advance!

GMATinsight · Answer

For outside part to be greatest the intersection of all 3 categories should be minimized

For minimum intersection of three categories, 40 and 30 may be subset of 65 i.e. minimum sum of all three categories = 65

i.e. Maximum outside part = 100-65 = 35

Answer: option C

fskilnik · Answer

This solution is associated with the image attached.

?\,\, = \,\,{\left( {{	ext{none}}} ight)_{\,\max }}

\,\,{\left( {{	ext{none}}} ight)_{\,\max }}\,\,\,\,\, \Leftrightarrow \,\,\,{\left( {X \cup Y \cup Z} ight)_{\min }}\,\,\,\,\,\,\,\,\,\,\,\,\,\left \,\,

{\left( {X \cup Y \cup Z} ight)_{\min }}\,\,\,\, \Leftrightarrow \,\,\,\,\max \,\,{	ext{intersections}}!\,\,\,\,\,\,\, \Rightarrow \,\,\,\,{\left( {X \cup Y \cup Z} ight)_{\min }} = 30 + 10 + 25 = 65

?\,\,\mathop = \limits^{\left( * ight)} \,\,100 - \,\,65 = 35

This solution follows the notations and rationale taught in the GMATH method.

Regards,
fskilnik.

Shri15kumar · Answer

Thank You for your response. it's been helpful!

fskilnik · Answer

Hi Shridhyey!

I hope your kind words were related to my answer (last post above), or at least ALSO to it. 
The "test drive" of my method covers more than 10 videos, 80 exercises and 1 mock... try it!

Regards,
fskilnik.

Shri15kumar · Answer

Yes fkskilnik. I'll check the test drive surely. Thank you.

bumpbot · Answer

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Of a certain group of 100 people, 40 graduated from High Sch

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