Math 115

T1 practice Questions                                    


Note:  These questions are all taken from tests that I have prepared previously.  Don’t expect to see exactly the same questions (or even necessarily exactly the same set of topics) on this test, but these are good representatives of the types of problems you can expect to see, and do make a good self-test on these particular topics.  The answers are included at the end (but in some cases without the work shown– you’ll need to supply that).


If a problem asks you to EXPLAIN, it should be answered in the context of the question, using complete English sentences.


Questions 1 and 2 refer to the

preference schedule shown to the right:




1) a. Find the winner using the Borda count method.


    b. Find the winner using the plurality-with-elimination method.



2) a.  Rank the candidates using the extended plurality method.


   b.  Rank the candidates using the recursive plurality method.


    c.   Do any of the methods used in question 1 or 2 violate the majority criterion for this preference schedule?  Carefully EXPLAIN your answer.



3)  For the weighted voting system [100:70,50,30,10] , determine with justification who (if anyone) is a dictator, who (if anyone) has veto power, and who (if anyone) is a dummy.



4)  Find the Banzhaf power index of each of the players in the previous problem.



5) Matt and Terry buy a dozen donuts - 6 chocolate, 6 jelly.  Matt likes the chocolate donuts twice as much as jelly.  Find 2 different ways that Matt could fairly (to him) divide the donuts.

(Note:  "different ways" means a different number of each type of donut in each pile)

6) Players A, B, C, and D use the method of markers to divide these 15 objects.

Determine which objects each player gets.


7)  A brother and sister inherit a house, a cabin, and a vintage car.  They agree to divide the items by the method of sealed bids.  They bid as shown:


a.  What does each sibling consider to be their fair share?


b.  What is the final outcome of this fair division problem?



8)  A small town has 4 schools, and 25 school buses.   The schools' enrollments are:

A -


B -


C -


D -


The school buses are apportioned to the schools based on the number of students.

 a.  Find the standard divisor.


b.  EXPLAIN what the standard divisor means in the context of this problem.


c.  How many buses would each school get by Hamilton's method?



9) If Jefferson's method were used to apportion the town's buses (from the previous problem):

a.  Find a suitable modified divisor:


b.  Find the modified quota for each school, and determine how many buses each school gets.


c.  Does this violate the quota rule?  Carefully EXPLAIN why it does or does not.


10)  (continuing the previous problem)

a.  Find a suitable modified divisor for Webster’s Method.


b.  Find the modified quota for each school, and determine how many buses each school gets.








1)  and 2) Borda Count Winner = B

Plurality with elimination: Eliminate B first, then A.  At that point C has a majority so C wins.

Extended plurality:  D/C/A/B

Recursive Plurality:  D/A /B/C

None of these violate the majority criterion, as no candidate won a majority of the 1st place votes.


3)  and 4) No one has 100+ votes, so there is no dictator.  Players 2, 3 and 4 combined do not reach 100 votes, so player 1 has veto power. 

Player 4 is never critical, so it is a dummy. 

The 6 winning coalitions are (naming the players A,B,C, and D) :

A, B

A, B, D

A, C

A, C, D

A, B, C

A, B, C, D

The critical players are underlined and shown in italics.  A’s power is 6/10 = 60%, B and C are 2/10 = 20% each (D is 0%)


5) 6C and 6J to each, or 5C and 8J to one, 7C 4J to the other.  There are 2 other possibilities, too (can you find them?).


6)C gets 1 and 2

    D gets 5,6,7

    B gets 9,10,11

    A gets 13,14,15

   The leftovers, 3,4,8 and 12 can be divided in any way whatsoever and it will still be fair.


7) a) Bro:  170000, Sis:  162000.

     b)The brother gets the house and car, pays 96000.  The sister gets the cabin and 96000.



8)   Std divisor = 51.68.  For every 51.68 students attending a school, it should get one bus.

A-6 / B-10 / C-5 / D-4.


9)  48 works (there are other choices). 

7/ 11/ 4/ 3

No, each gets either its upper or lower quota in this example.


10)   52 works(there are other choices). 

6 / 10 / 5 /4