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

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Re: A group of people were given 2 puzzles. 79% people solved puzzle X and [#permalink]
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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.
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Re: A group of people were given 2 puzzles. 79% people solved puzzle X and [#permalink]
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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)
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Re: A group of people were given 2 puzzles. 79% people solved puzzle X and [#permalink]
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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
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Re: A group of people were given 2 puzzles. 79% people solved puzzle X and [#permalink]
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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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Re: A group of people were given 2 puzzles. 79% people solved puzzle X and [#permalink]
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
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Re: A group of people were given 2 puzzles. 79% people solved puzzle X and [#permalink]
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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

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Re: A group of people were given 2 puzzles. 79% people solved puzzle X and [#permalink]
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.

79 + 89 = 168 > 100

So, both x & y = 68

Now, if all people in sent X solves questions of Y, then max is 79

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Re: A group of people were given 2 puzzles. 79% people solved puzzle X and [#permalink]
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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%

Solution:

We can use the formula:

Total = Puzzle X + Puzzle Y - Both + Neither

Assuming Neither = 0 (since 0 is the minimum value of any nonnegative quantities), we have:

100 = 79 + 89 - Both

100 = 168 - Both

Both = 68

Assuming Neither = 11 (since we know 89 percent solved puzzle Y), we have:

100 = 79 + 89 - Both + 11

100 = 179 - Both

Both = 79

We see that the maximum value is 79 percent and the minimum value is 68 percent.

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Re: A group of people were given 2 puzzles. 79% people solved puzzle X and [#permalink]
Didn't see how I could post tables here, but set matrices are super helpful for these types of questions. First thing that one needs to do is in the absence of any count, assume the total number of entities is 100.

There's two scenarios:
1. Maximum: 79 (This is the highest possible b/c we know that 79 people solved puzzle X)
2. Minimum: 68 (This is the lowest possible b/c we know that 21 people did not solve X)

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Re: A group of people were given 2 puzzles. 79% people solved puzzle X and [#permalink]
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Re: A group of people were given 2 puzzles. 79% people solved puzzle X and [#permalink]
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