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Mentor Exercise

 
Directions: Each group of questions is based on a set of conditions. In answering some of the questions, it may be useful to draw a rough diagram. Choose the response that most accurately and completely answers each question. Hints, insights, partial solutions, and answers are provided in the right-hand column.

On Auto Row there are seven dealerships: Audi, Chrysler, Ford, Hyundai, Mazda, Toyota, Volkswagen. All the dealerships are on the same side of the street, which runs from west to east.
 
Ford is not next to Mazda.
 
Audi is the fourth dealership from the west end of the street.
 
Ford is next to Audi.
 
Toyota is west of both Audi and Ford but east of Chrysler.

Begin by symbolizing the conditions. “Ford is not next to Mazda” is naturally symbolized as ~(FM). “Toyota is west of both Audi and Ford but east of Chrysler” can be symbolized as
C—>T—>AF, where the arrow points from west to east. The remaining conditions can be symbolized in like manner, which gives the following schematic:
 
ACFHMTV (V, H “wild”)
 
~(FM)
 
A = 4th
 
FA
 
C—>T—>AF

The diagram will consist of seven dashed lines. To place the elements on the diagram, follow the guidelines on page 33. First, look for a condition that fixes the position of an element. It is A = 4th. This gives
 
 1  2  3  4
A
 5  6  7
 
Next, look for a condition that limits the position of a dealership. There is none. Now, look for a condition connecting two or more dealerships. The third condition, “Ford is next to Audi,” forces the Ford dealership into either space 3 or 5. This generates two alternate diagrams:

Diagram I
 
 1  2  3
F
 4
A
 5  6  7
 
Diagram II
 
 1  2  3  4
A
 5
F
 6  7

Finally, look for a condition that states where a dealership cannot be. There is none. No significant properties can be derived from the initial conditions, so we turn to the questions.
 
Question-1
Which one of the following dealerships CANNOT be next to Chrysler?
  1. Toyota
  2. Ford
  3. Volkswagen
  4. Hyundai
  5. Mazda
Solution
Hint: Use the condition C—>T—>AF.
The answer is B.

 
 
Question-2
If Ford is east of Audi, then Hyundai CANNOT be next to both
  1. Toyota and Ford
  2. Chrysler and Toyota
  3. Ford and Mazda
  4. Ford and Volkswagen
  5. Toyota and Audi
Solution
Since Ford is east of Audi, use Diagram II and again the condition C—>T—>AF.
 
The answer is (A).
 
 
Question-3
If Volkswagen is west of Audi, then which one of the following must be false?
  1. Ford is east of Audi.
  2. Volkswagen is west of Toyota.
  3. Volkswagen is east of Toyota.
  4. Hyundai is west of Mazda.
  5. Hyundai is east of Mazda.
Solution
If Volkswagen is west of Audi, then, from the condition C—>T—>AF, we know that Ford must be east of Audi—otherwise Ford, Chrysler, Toyota, and Volkswagen would all be west of Audi, which would violate the condition A = 4th. Finally, use the condition ~(FM).
 
The answer is (E).
 
 
Question-4
Which one of the following is a possible arrangement of the dealerships from west to east?
  1. C, F, T, A, H, M, V
  2. C, T, F, H, A, M, V
  3. V, C, T, A, F, M, H
  4. C, V, F, A, H, T, M
  5. H, C, T, A, F, V, M
Solution
This is a straightforward elimination problem: Take a condition. Test it against each answer-choice, eliminating any that violate it. Then take another condition; test it against the remaining answer-choices, eliminating any that violate it. Continue until only one answer-choice remains.
 
The answer is (E).
 
 
Question-5
If Hyundai is west of Ford, which one of the following pairs of dealerships must be next to each other?
  1. Chrysler and Hyundai
  2. Volkswagen and Mazda
  3. Ford and Mazda
  4. Toyota and Audi
  5. Hyundai and Mazda
Solution
Hint: If Hyundai is west of Ford, then, from the condition C—>T—>AF, Ford must be east of Audi—otherwise Chrysler, Toyota, Hyundai, and Ford would all be west of Audi, which would violate the condition A = 4th.
 
The answer is B.
 
 
Question-6
If the Volkswagen dealership is on the east end of the street, then which one of the following must be false?
  1. Chrysler is second from the west end of the street.
  2. Ford is east of Audi.
  3. Hyundai is on the west end of the street.
  4. Ford is west of Audi.
  5. Hyundai is fifth from the west end of the street.
Solution
This question is hard because it does not give us much information to work with. Volkswagen was a “wild card”. That is, its position on the street was independent of the other dealerships—except for Audi. So knowing where the Volkswagen dealership is located will probably tell us little, if anything, about where the other dealerships are located. Furthermore, the ques­tion leads us astray by asking “Which one of the following must be false?” This prompts us to look directly for the false answer. In problems of this type, however, it is often better to reword the question as “All of the following could be true except.” Then look for and eliminate the true answer-choices.
 
We’ll use an indirect proof to solve this problem. That is, for each answer-choice, we attempt to construct a possible ordering of the dealerships along the street. The one for which this is not possible will be the answer. Clearly you should save questions like this for last, or skip them all together.
 
Begin with choice (A). In Diagram II, place Chrysler second from the west end of the street:
 
 1  2
C
 3  4
A
 5
F
 6  7
V
 
Next, the condition C—>T—>AF forces T into space 3:
 
 1  2
C
 3
T
 4
A
 5
F
 6  7
V
 
We can place M in space 1 without violating the initial conditions:
 
 1
M
 2
C
 3
T
 4
A
 5
F
 6  7
V
 
Finally, this forces the “wild card”, H, into space 6:
 
 1
M
 2
C
 3
T
 4
A
 5
F
 6
H
 7
V
 
This diagram does not violate any initial condi­tion, so (A) could be true. This eliminates (A).
 
Now apply this method to the remaining answer-choices until you find the one that violates one or more of the conditions or until you have elimi­nated four of the five choices.

The answer is (C).
 




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