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# Of a certain group of 100 people, 40 graduated from High School X, 65

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Math Expert
Joined: 02 Sep 2009
Posts: 50776
Of a certain group of 100 people, 40 graduated from High School X, 65  [#permalink]

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19 Feb 2018, 21:54
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68% (01:52) correct 32% (02:05) wrong based on 86 sessions

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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

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Of a certain group of 100 people, 40 graduated from High School X, 65  [#permalink]

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19 Feb 2018, 23:02
2
Bunuel wrote:
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

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.
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Re: Of a certain group of 100 people, 40 graduated from High School X, 65  [#permalink]

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05 Sep 2018, 22:38
DavidTutorexamPAL wrote:
Bunuel wrote:
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

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.

Hello, is there any other approach to solve this question?
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Re: Of a certain group of 100 people, 40 graduated from High School X, 65  [#permalink]

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06 Sep 2018, 03:54
Bunuel wrote:
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

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

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Re: Of a certain group of 100 people, 40 graduated from High School X, 65  [#permalink]

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06 Sep 2018, 07:54
Bunuel wrote:
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

This solution is associated with the image attached.

$$?\,\, = \,\,{\left( {{\text{none}}} \right)_{\,\max }}$$

$$\,\,{\left( {{\text{none}}} \right)_{\,\max }}\,\,\,\,\, \Leftrightarrow \,\,\,{\left( {X \cup Y \cup Z} \right)_{\min }}\,\,\,\,\,\,\,\,\,\,\,\,\,\left[ {{\text{none}} + X \cup Y \cup Z\mathop = \limits^{\left( * \right)} {\text{tot}} = {\text{cte}}} \right]\,\,$$

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

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

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

Regards,
fskilnik.
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06Set18_14q.gif [ 17.64 KiB | Viewed 463 times ]

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Re: Of a certain group of 100 people, 40 graduated from High School X, 65  [#permalink]

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06 Sep 2018, 16:34
Thank You for your response. it's been helpful!
GMATH Teacher
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 488
Re: Of a certain group of 100 people, 40 graduated from High School X, 65  [#permalink]

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06 Sep 2018, 17:55
Shridhyey wrote:
Thank You for your response. it's been helpful!

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.
_________________

Fabio Skilnik :: https://GMATH.net (Math for the GMAT) or GMATH.com.br (Portuguese version)
Course release PROMO : finish our test drive till 30/Nov with (at least) 50 correct answers out of 92 (12-questions Mock included) to gain a 50% discount!

Intern
Joined: 14 Jun 2018
Posts: 47
Location: India
Concentration: International Business, Marketing
Re: Of a certain group of 100 people, 40 graduated from High School X, 65  [#permalink]

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07 Sep 2018, 03:33
1
fskilnik wrote:
Shridhyey wrote:
Thank You for your response. it's been helpful!

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.

Yes fkskilnik. I'll check the test drive surely. Thank you.
Re: Of a certain group of 100 people, 40 graduated from High School X, 65 &nbs [#permalink] 07 Sep 2018, 03:33
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# Of a certain group of 100 people, 40 graduated from High School X, 65

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