Of the 50 researchers in a workgroup, 40 percent will be

SOLUTION

Of the 50 researchers in a workgroup, 40 percent will be assigned to Team A and the remaining 60 percent to Team B. However, 70 percent of the researchers prefer Team A and 30 percent prefer Team B. What is the lowest possible number of researchers who will NOT be assigned to the team they prefer?

(A) 15
(B) 17
(C) 20
(D) 25
(E) 30

The number of researchers assigned to Team A will be 0.40*50 = 20;
The number of researchers assigned to Team B will be 0.60*50 = 30.

The number of researchers who prefer Team A will be 0.70*50 = 35;
The number of researchers who prefer Team B will be 0.30*50 = 15.

Now, if ALL 20 spots in Team A, will be filled by the researchers who prefer Team A, then the remaining 15 researchers who prefer Tam A will be assigned to Team B (along with 15 researchers who DO prefer Team B).

Hence, the minimum number of researchers who will NOT be assigned to the team they prefer is 15.

Answer: A.
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This question would be tough if the wordings of the question are changed from minimum to maximum no of people.

Old Phrasing - What is the lowest possible number of researchers who will NOT be assigned to the team they prefer? = 15 people
New Phrasing - What is the highest possible number of researchers who will NOT be assigned to the team they prefer? = 45 people

I hope these phrasings will help many people to understand the concept in totality

IMO A

15 person prefer B and 30 will be assigned to Team B. All preference for Team B are fulfilled.
35 Person prefer A and only 20 will be assigned to Team A and rest 15 person will be forced to join team B.
So (35-20)=15 will not get their preference.

Hi Bunuel,

How to solve this using Double set Matrix?

Regards,
Sachin
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Even if you use a double set matrix, it will be quite irrelevant as its slightly modified. But if you draw the matrix it will be clear what the question is asking you!

A great way to solve this is to use the Double-Matrix Chart

------------Team A|Team B
prefer A|______|__________=.7 * 50 = 35
prefer B|______|__________=.3 * 50 = 15
________________________________
----------.4*50=20 + .6*50=30-----------= 50

Columns:
20 people should be assigned to A.
30 people should be assigned to B.

Rows:
35 prefer Team A.
15 prefer Team B.

All totaling to 50.

In order to get the lowest number who will not get what they want,
we give the assignment to those who wants it as much as we can.

For example:
If 35 wants Team A... assign as much as we can to Team A.
If 15 wants Team B... assign as much as we can to Team B.

(See table below)

------------Team A|Team B
prefer A|___20_|___?_______=.7 * 50 = 35
prefer B|___?__|__15_______=.3 * 50 = 15
________________________________
---------------20---+----30--------= 50

Now we calculate the balance of those who did not get the slot...

------------Team A|Team B
prefer A|___20_|___15_______=.7 * 50 = 35
prefer B|___0__|__15_______=.3 * 50 = 15
________________________________
---------------20---+----30--------= 50

Note that the row of "prefer A" should always tally to 35. That is why we know the balance is 15.
Note that the row of "prefer B" should always tally to 15. That is why we know the balance is 0.



Now we sum up those who did not get what they want:

15 on those who prefer A did not get it
0 on those who prefer B did not get it.


Answer: 15

This chart works on all overlapping set problems...

Matrix based on the question:


The question is asking us to make x+y minimum.

If you make y=0, x will become 15
So, x+y = 15

[quote="Bunuel"]Of the 50 researchers in a workgroup, 40 percent will be assigned to Team A and the remaining 60 percent to Team B. However, 70 percent of the researchers prefer Team A and 30 percent prefer Team B. What is the lowest possible number of researchers who will NOT be assigned to the team they prefer?

(A) 15
(B) 17
(C) 20
(D) 25
(E) 30

Here Total team =50
Allowed in A = 20
Allowed in B = 30
Want A=35
Want B=15

To find the least that don't get there preference => let us arrange them in a way that all get what they want
here B members are fully fit in B=> no left
but For A => 20 seats and 35 need them .
SO 15 will have no place to go ; hence they will be assigned to team B.
Thus => 15.
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We are given that there are a total of 50 researchers in the workgroup and that 40% will be assigned to team A and 60% assigned to team B. Thus, we know:

Assigned to team A = (0.4)(50) = 20

Assigned to team B = (0.6)(50) = 30

We are also given that 70% of the people PREFER team A and that 30% of the people PREFER team B. Thus we know:

Prefer team A = (0.7)(50) = 35

Prefer team B = (0.3)(50) = 15

We are asked to find the LOWEST POSSIBLE NUMBER of people who will NOT be assigned to the team they prefer. Let’s start with team B.

Currently we have 15 people who PREFER team B and 30 people who will be assigned to team B. Because we are looking for the lowest possible number of people who will not be assigned to the team they prefer, we must assume that all 15 people who prefer team B will indeed be assigned to team B, but this means that, of the 30 people who will be on team B, 15 of them DO NOT WANT TO BE ON THE TEAM.

Turning to team A, we know that we have 35 people who PREFER team A and 20 people who will be assigned to team A. Again we are looking for the lowest possible number of people who will not be assigned to the team they prefer, we must assume that all 20 people who will be on team A actually prefer to be on that team.

Thus the lowest number of people who do not prefer the team in which they have been assigned is 15 people.

Note that the answer is NOT 30. Don't be guilty of double-counting the disappointed people. Think instead that all 20 people assigned to team A wanted to be on it, and of the 30 people assigned to team B, exactly 15 wanted to be on it. Thus, we have 35 people who got the team they wanted, with 15 who were assigned to the team they didn't want.

Answer A.
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fameatop please show the calculation process for:
What is the highest possible number of researchers who will NOT be assigned to the team they prefer?

Dear msu6800
Change the positions of the team; put A into B and conversely.
Hence 35 would be in B in lieu of A, and 15 would be in B in lieu of A.

There are total of 35+15 in the wrong teams.

The number of researchers assigned to Team A will be 0.40*50 = 20;
The number of researchers assigned to Team B will be 0.60*50 = 30.

The number of researchers who prefer Team A will be 0.70*50 = 35;
The number of researchers who prefer Team B will be 0.30*50 = 15.

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