Which one of the following could be true?

Ravi January 22, 2019

@alymathieu,

I'm happy to walk you through this question through writing.

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In the meantime, let's take a look at this game.

we're told that a newly formed company has five employees (F, G, H, K,
and L). Each employee holds exactly one of the following positions:
president, manager, or technician.

P M T

Only the president is not supervised. Other employees are each
supervised by exactly one employee, who is either the president or a
manager.

This is important because we know that one employee can't have 2
supervisors now.

We're also told that each supervised employee holds a different
position than his or her supervisor. This means that a manager can't
supervise another manager, nor can a technician supervise another
technician.

Rule 1 says there's exactly one president

1) P=1

Rule 2 says at least one of the employees whom the president
supervises is a manager.

2) P-M

Rule 3 says that each manager supervises at least one employee.

With rule 3, we know that each manager must supervise at least one
employee and we also know that each supervised employee holds a
different position than his or her supervisor, so we know that each
manager must supervise at least one technician.

3) M-T

Rule 4 says that F doesn't supervise any employee. If F doesn't
supervise any employee, then we know that F can't be president since
the president supervises at least one manager. We also know that F
can't be a manager because managers supervise at least one technician.
Therefore, F must be a technician.

4) F is a technician

Rule 5 says that G supervises exactly 2 employees. We know, therefore,
that G must be either a president or manager since those are the only
two positions that can supervise others.

5) G is a president or manager

Now that we've mapped all of the rules, let's see what other
inferences we can make.

F, G, H, K, and L can go into three different type s of positions, so
let's make columns for the positions.

P M T

We know that there's only one president, so there will only be 1 spot under P

P M T
_

We also know that the president supervises at least one manager, so
there has to be at least one M. We also know that every M supervises
at least one T, so we also know there has to be at least one T

P M T
_ _ _

What else can we figure out? We've already filled out 3 of the 5
possible spots. With the remaining 2 spots we can put on the board,
there are 3 possible boards we can make.

1)

P M T
_ _ _
_
_

However, does this board work? No, it doesn't, because every M has to
supervise at least one T and each employee can only have 1 supervisor.
With this board, the T would have at least 3 supervisors, so this
doesn't work.

This means we only have 2 possible boards for this game

P M T
_ _ _
_ _

and

P M T
_ _ _
_
_

We know that F must be in one of the T spots. What about G? We know
that G supervises exactly 2 employees. We can place G in the P spot in
our first board, and for our second board, G can go in either P or M.

P M T
G _ F
_ _

We're placing F on the board here because G supervises both Ms and
both Ms are supervising Ts, so it doesn't matter which T spot we put F
in

P M T
_ _ _
_
_ F can go in any of the Ts on this board. It's best for us
to wait to place F in one of the Ts because we aren't exactly sure
who's supervising whom on this board.

Question 1 asks us to choose which answer is an acceptable assignment
of employees to the positions.

We know there's only one president, so (D) is out. We also know that F
has to be a T, so (C) and (E) can be eliminated. Lastly, we also know
that we can't have 3 Ms, so (A) is out. This leaves us with (B) as our
correct choice.

Question 2 asks us which of the following must be true?

(A) says there can be at most 3 technicians. We know this to be true
by looking at our game boards, so we can choose this and move on.

Question 3 asks us which of the following is a pair of employees who
could serve as managers together?

From our analysis of the game, we know that F can't be a manager, so
(A) and (B) can be eliminated quickly. Additionally, we also know that
on the game board where we have 2 managers, G must be the president
because G has to supervise exactly two employees and if G were serving
as a manager with someone else, then G would only be supervising one
employee. Therefore, we know G can't serve as a manager with another
employee, so (C) and (D) can be eliminated. This leaves us with (E),
which is our answer choice.

Question 4 asks us which one of the following could be true?

(A) says there is exactly one technician. This can't be true because
in looking at our game boards, we know that there have to be at least
2 technicians. (A) is out.

(B) says there are exactly 2 managers. We know this can be true as
it's one of our game boards, so this is the correct answer.

Question 5 asks us if F is supervised by the president, which one of
the following must be true?

We know from this that we are not concerned with the game board that
has 2 Ms because in that board, F is supervised by a manager and not
the president. Therefore, we are concerned with the game board that
has 1 P, 1 M, and 3 Ts.

(A) says G is P. This doesn't have to be true because G could also be M.

(B) says H is P. This doesn't have to be true because H could also be M or T.

(C) says L is T. This doesn't have to be true because L could also be a P or M.

(D) says there's exactly 1 M. This must be true because we know that
this is the game board we're dealing with.

(E) says exactly 2 T; we know this must be false because this scenario
requires 3 Ts.

Question 6 asks us if K supervises exactly two employees, which one of
the following must be true?

We know if K is supervising 2 people, then we're not dealing with the
game board in which there are 2 Ms because both G and K must be
supervising 2 people and the game board with 2 Ms has only one person
(G) supervising 2 people. Thus, we know that we're dealing with the
game board that has 1 P, 1 M, and 3 Ts

Since both G and K must supervise 2 people, then we know that either
one of them could be president, so they can trade off between being
president and manager. That means that H, F, and L must all be
technicians.

(A) says F is supervised by K. We don't know that this has to be true;
F could also be supervised by G.

(B) says G is a manager. This doesn't have to be true; G could be P.

(C) says L is supervised. We know L has to be a T, so L does have to
be supervised. This must be true, and this is our choice, as it's the
correct answer.

Hope this helps. Let us know if you have any more questions!

Julie-V August 16, 2019

Thank you Ravi, you're a lifesaver!