# Of the 50 researchers in a workgroup, 40 percent will…

Solution:

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.