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# A group of people were given 2 puzzles. 79% people solved puzzle X and

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Math Expert
Joined: 02 Sep 2009
Posts: 60627
A group of people were given 2 puzzles. 79% people solved puzzle X and  [#permalink]

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26 Aug 2015, 01:28
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Difficulty:

35% (medium)

Question Stats:

75% (01:39) correct 25% (01:42) wrong based on 357 sessions

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A group of people were given 2 puzzles. 79% people solved puzzle X and 89% people solved puzzle Y. What is the maximum and minimum percentage of people who could have solved both the puzzles?

(A) 11%, 0%
(B) 49%, 33%
(C) 68%, 57%
(D) 79%, 68%
(E) 89%, 79%

Kudos for a correct solution.

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Senior Manager
Joined: 15 Sep 2011
Posts: 305
Location: United States
WE: Corporate Finance (Manufacturing)
Re: A group of people were given 2 puzzles. 79% people solved puzzle X and  [#permalink]

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26 Aug 2015, 03:35
6
4
The maximum and minimum are establish through what's input for the neither variable. Below,
Total = Group1 + Group2 - Both + Neither

Maximum:
100 = 79 + 89 - Both + 11, where 11 is the lesser of the two not within a group.
Both = 79

Minimum:
100 = 79 + 89 - Both + 0, in that both of them had atleast one, puzzle x or puzzle y.
Both = 68

Therefore, D 79, 68
##### General Discussion
Director
Joined: 21 May 2013
Posts: 632
Re: A group of people were given 2 puzzles. 79% people solved puzzle X and  [#permalink]

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26 Aug 2015, 05:12
5
Bunuel wrote:
A group of people were given 2 puzzles. 79% people solved puzzle X and 89% people solved puzzle Y. What is the maximum and minimum percentage of people who could have solved both the puzzles?

(A) 11%, 0%
(B) 49%, 33%
(C) 68%, 57%
(D) 79%, 68%
(E) 89%, 79%

Kudos for a correct solution.

To solve this question, we don't need any formulas. Just read the question carefully. If 79% of the people solved puzzle X and 89% solved puzzle Y, then maximum % that can solve both correctly can only be 79%. From the given options, only D looks good

Manager
Joined: 14 Mar 2014
Posts: 141
GMAT 1: 710 Q50 V34
Re: A group of people were given 2 puzzles. 79% people solved puzzle X and  [#permalink]

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27 Aug 2015, 06:08
4
Bunuel wrote:
A group of people were given 2 puzzles. 79% people solved puzzle X and 89% people solved puzzle Y. What is the maximum and minimum percentage of people who could have solved both the puzzles?

(A) 11%, 0%
(B) 49%, 33%
(C) 68%, 57%
(D) 79%, 68%
(E) 89%, 79%

Kudos for a correct solution.

IMO : D

Max Overlap we can obtain from two values i.e. 79 and 89 will be 79.
Only Option D has Max = 79.
Thus We need not calculate the min. value
Intern
Joined: 28 May 2013
Posts: 18
Schools: Mannheim"17
A group of people were given 2 puzzles. 79% people solved puzzle X and  [#permalink]

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26 Aug 2015, 03:36
3
1
consider total 100 people.
we minimize the of people who solved only X puzzle and the of people who solved only Y. This would maximize # people who solved both.
So, X min will be (100-89) =11...(89 is #people solving Y)
Y min will be (100-79) = 21...(79 is #people solving X)

Thus, Xmin + Ymin = 32. Thus XY min(#people solving both)= 100-32, i.e 68 or 68%.

Also, to maximize the number of people solving both, we need to find # people with maximum overlap.
i.e 79 people can be the # people solving X and both. In other words, none of the people solved only X. Then, we have Xonly=0, XY=79, X=79, Y=89, Yonly= 100-79, i.e 21.

Thus, we have XY max =79%

Tx
Kindly correct me if it's wrong.
Math Expert
Joined: 02 Sep 2009
Posts: 60627
Re: A group of people were given 2 puzzles. 79% people solved puzzle X and  [#permalink]

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30 Aug 2015, 08:49
2
Bunuel wrote:
A group of people were given 2 puzzles. 79% people solved puzzle X and 89% people solved puzzle Y. What is the maximum and minimum percentage of people who could have solved both the puzzles?

(A) 11%, 0%
(B) 49%, 33%
(C) 68%, 57%
(D) 79%, 68%
(E) 89%, 79%

Kudos for a correct solution.

VERITAS PREP OFFICIAL SOLUTION:

The first thing to note here is that we do not know the % of people who could not solve either puzzle. All we know is that puzzle X was solved by 79% of the people and puzzle Y was solved by 89% of the people.

Let’s first try to maximize the % of people who solved both the puzzles. We want to make these two sets overlap as much as possible i.e. we need to get them as close to each other as possible. Region of overlap can be 79% at most since we know that only 79% people solved puzzle X. In this case, the venn diagram will look something like this.

Hence, the maximum % of people who could have solved both the puzzles is 79%.

Now, let’s try to minimize the % of people who solved both the puzzles. We want the sets to be as far apart as possible. In this case, the % of people who solved neither puzzle must be 0. Only then will the overlap of the sets be as little as possible.

In this case, 68% people must have solved both the puzzles.

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Joined: 10 May 2014
Posts: 135
A group of people were given 2 puzzles. 79% people solved puzzle X and  [#permalink]

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26 Aug 2015, 20:09
1
1
Overlapping Sets formulas
Formula 1: Total = X total + Y total - Both + Neither
Formula 2: Total = X only + Y only + Both + Neither

You are given that
100 = 79 + 89 - Both + Neither

Maximum "Both"
Total = X only + Y only + Both + Neither
100 = 0 + 79 + 10 + 11

Minimum "Both"
Total = X only + Y only + Both + Neither
100 = 11 + 68 + 21 + 0

Hence, option D (79, 68)
Manager
Joined: 23 Oct 2017
Posts: 61
Re: A group of people were given 2 puzzles. 79% people solved puzzle X and  [#permalink]

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13 Jan 2018, 06:33
2 puzzles: X & Y.
Let x be the the no. of people who solved both.
100= (79+89) - x + neither (=> taking the component of neither is key here)

lets try to express x as f(neither)

x= 68 + neither
Now neither can takes values of 0 till (100-89)
min: x= 68+0 =68
max: x= 68+11 =79
Director
Joined: 24 Oct 2016
Posts: 585
GMAT 1: 670 Q46 V36
GMAT 2: 690 Q47 V38
A group of people were given 2 puzzles. 79% people solved puzzle X and  [#permalink]

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23 Sep 2019, 07:14
Bunuel wrote:
A group of people were given 2 puzzles. 79% people solved puzzle X and 89% people solved puzzle Y. What is the maximum and minimum percentage of people who could have solved both the puzzles?

(A) 11%, 0%
(B) 49%, 33%
(C) 68%, 57%
(D) 79%, 68%
(E) 89%, 79%

Kudos for a correct solution.

Method: Direct Formula (Better)

Max (A&B) = Min(A, B) = 79
Min (A&B) = A + B - Total + None (To minimize Both, minimize None by taking None=0 except when there's a min constraint on None.)
Min (X&Y) = A + Y - 100 + None = 79 + 89 - 100 + 0 = 68 (None has no constraint)

Since there’s only one choice with max=79, no need to calculate min. => D

Alternate Method for Min (A&B):

Min (A & B) = (A + B) % Total + None
Since None has no constraint, minimize None by taking it equal to 0 => Min (A & B) = (79 + 89)%100 + 0 = 168%100 = 68

A group of people were given 2 puzzles. 79% people solved puzzle X and   [#permalink] 23 Sep 2019, 07:14
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