Deck 14: Voting and Apportionment

Question 1

The Mathematics Department is holding an election for department chair. Each member ranks the
Candidates from first to third. The preference table below shows the results of the ballots with
Candidates Clark (C), Jones (J), and Smith (S).
A second election is then held resulting in the following preference table:
If the plurality with elimination method is used to determine the winner, is the montonicity
Criterion satisfied?

A) Yes
B) No

Answer 1

A

Question 2

Computer Specialists is planning a group vacation to one of the following locations: Alaska (A),
Florida (F), San Antonio (S), or Hawaii (H). The employees rank the four possible sites according
To the following preference table:
If the plurality with elimination method is used to determine the winner, is the head-to-head
Criterion satisfied?

A) Yes
B) No

Answer 2

A

Question 3

The Mathematics Department is holding an election for department chair. Each member ranks the
Candidates from first to third. The preference table below shows the results of the ballots with
Candidates Clark (C), Jones (J), and Smith (S).
If the Borda count method is used to determine the winner, is the head-to-head criterion satisfied?

A) Yes
B) No

Answer 3

A

Question 4

Make a preference table for the given voting situation.
The Mathematics Department is holding an election for department chair. Each member ranks the
Candidates from first to third. The preference table below shows the results of the ballots with
Candidates Clark (C), Jones (J), and Smith (S).
Determine the winner using the Borda count method and if the winner received a majority of first
Place votes.

A) Jones; No
B) Smith; Yes
C) Jones; Yes
D) Smith; No

Question 5

Make a preference table for the given voting situation.

-Ten voters are asked to rank 4 brands of cell phones: A, B, C, and D. The ten voters turn in the Following ballots showing their preferences in order: $\mathrm{A} \mathrm{B} \mathrm{B} \mathrm{B} \mathrm{A} \mathrm{B} \mathrm{B} \mathrm{B} \mathrm{B} \mathrm{A}$
$\mathrm{B} \mathrm{C} \mathrm{A} \mathrm{C} \mathrm{B} \mathrm{A} \mathrm{C} \mathrm{A} \mathrm{A} \mathrm{B}$
$\mathrm{D}\mathrm{A}\mathrm{C} \mathrm{A}\mathrm{C}\mathrm{C}\mathrm{A} \mathrm{D}\mathrm{C}\mathrm{C}$
$\mathrm{C}\mathrm{D}\mathrm{D}\mathrm{D}\mathrm{D}\mathrm{D}\mathrm{D}\mathrm{C}\mathrm{D}\mathrm{D}$


Make a preference table for these ballots.

A)

B)

C)

D)

Question 6

An order of award presentations has been devised for seven people: Jeff, Karen, Lyle, Maria, Norm, Olivia, and Paul.
A condominium association is holding an election for president of the board of directors. Each
Member ranks the candidates from first to third. The preference table below shows the results of
The ballots with candidates Abbott(A), Blake (B), Cleary (C), and Downs (D).
Determine the winner using the Borda count method and if the winner received a majority of first
Place votes.

A) Cleary; No
B) Abbott; No
C) Blake; No
D) Downs; No

Question 7

Make a preference table for the given voting situation.
Eight voters are asked to rank 4 brands of ice cream: A, B, C, and D. The eight voters turn in the
Following ballots showing their preferences in order:

Question 8

Make a preference table for the given voting situation.

-Eight voters are asked to rank 3 brands of automobiles: A, B, and C. The eight voters turn in the Following ballots showing their preferences in order: $\text { C B A A C B B A }$
$\text { B C B B B C A B}$
$\text { A A C C A A C C }$

Make a preference table for these ballots.

A)
$\begin{array}{lllll}\hline \text { Number of Votes } & 3 & 1 & 2 & 2 \\\hline \text { First } & \text { A } & B & B & C \\\text { Second } & \text { B } & A & C & A \\\text { Third } & \text { C } & C & A & B \\\hline\end{array}$
B)

C)


D)

Question 9

Make a preference table for the given voting situation.
Computer Specialists is planning a group vacation to one of the following locations: Alaska (A),
Florida (F), San Antonio (S), or Hawaii (H). The employees rank the four possible sites according
To the following preference table.
Determine the winner using the plurality method and if the winner received a majority of first
Place votes.

A) Alaska; Yes
B) Hawaii; Yes
C) Hawaii; No
D) Alaska; No

Question 10

An order of award presentations has been devised for seven people: Jeff, Karen, Lyle, Maria, Norm, Olivia, and Paul.
Computer Specialists is planning a group vacation to one of the following locations: Alaska (A),
Florida (F), San Antonio (S), or Hawaii (H). The employees rank the four possible sites according
To the following preference table.
Determine the winner using the plurality with elimination method and if the winner received a
Majority of first place votes.

A) Alaska; Yes
B) Hawaii; Yes
C) Hawaii; No
D) Alaska; No

Question 11

The Mathematics Department is holding an election for department chair. Each member ranks the
Candidates from first to third. The preference table below shows the results of the ballots with
Candidates Clark (C), Jones (J), and Smith (S).
If the Borda count method is used to determine the winner, is the majority criterion satisfied?

A) Yes
B) No

Question 12

An order of award presentations has been devised for seven people: Jeff, Karen, Lyle, Maria, Norm, Olivia, and Paul.
The Mathematics Department is holding an election for department chair. Each member ranks the
Candidates from first to third. The preference table below shows the results of the ballots with
Candidates Clark (C), Jones (J), and Smith (S).
Determine the winner using the pairwise comparison method and if the winner received a
Majority of first place votes.

A) Smith; Yes
B) Jones; Yes
C) Smith; No
D) Jones; No

Question 13

Make a preference table for the given voting situation.

-Ten voters are asked to rank 3 brands of shoes: A, B, and C. The ten voters turn in the following Ballots showing their preferences in order: $\text { C B B C B B C B C B}$
$\text { B C A B C A B C B C }$
$\text { A A C A A C A A A A}$


Make a preference table for these ballots.

A)
$\begin{array}{lc}\hline \text { Number of Votes } &2 \quad 4 \quad 4 \\\hline \text { First } & \text { B C C } \\\text { Second } & \text { A C B } \\\text { Third } & \text { C A A } \\\hline\end{array}$

B)

C)

D)

Question 14

An order of award presentations has been devised for seven people: Jeff, Karen, Lyle, Maria, Norm, Olivia, and Paul.
Computer Specialists is planning a group vacation to one of the following locations: Alaska (A),
Florida (F), San Antonio (S), or Hawaii (H). The employees rank the four possible sites according
To the following preference table.
Determine the winner using the pairwise comparison method.

A) Florida & Hawaii (tie)
B) Alaska & Florida (tie)
C) Alaska
D) Alaska & Hawaii (tie)

Question 15

Make a preference table for the given voting situation.
A condominium association is holding an election for president of the board of directors. Each
Member ranks the candidates from first to third. The preference table below shows the results of
The ballots with candidates Abbott(A), Blake (B), Cleary (C), and Downs (D).
Determine the winner using the plurality method and if the winner received a majority of first
Place votes.

A) Cleary; Yes
B) Abbott; Yes
C) Abbott; No
D) Cleary; No

Question 16

Computer Specialists is planning a group vacation to one of the following locations: Alaska (A),
Florida (F), San Antonio (S), or Hawaii (H). The employees rank the four possible sites according
To the following preference table:
If the Borda count method is used to determine the winner, is the majority criterion satisfied?

A) Yes
B) No

Question 17

The Mathematics Department is holding an election for department chair. Each member ranks the
Candidates from first to third. The preference table below shows the results of the ballots with
Candidates Clark (C), Jones (J), and Smith (S).
If the plurality method is used to determine the winner, is the head-to-head criterion satisfied?

A) Yes
B) No

Question 18

Make a preference table for the given voting situation.
The Mathematics Department is holding an election for department chair. Each member ranks the
Candidates from first to third. The preference table below shows the results of the ballots with
Candidates Clark (C), Jones (J), and Smith (S).
Determine the winner using the plurality method and if the winner received a majority of first
Place votes.

A) Clark; No
B) Jones; No
C) Clark; Yes
D) Smith; Yes

Question 19

An order of award presentations has been devised for seven people: Jeff, Karen, Lyle, Maria, Norm, Olivia, and Paul.
Computer Specialists is planning a group vacation to one of the following locations: Alaska (A),
Florida (F), San Antonio (S), or Hawaii (H). The employees rank the four possible sites according
To the following preference table.
Determine the winner using the Borda count method and if the winner received a majority of first
Place votes.

A) San Antonio; No
B) Florida; No
C) Alaska; No
D) Hawaii; No

Question 20

An order of award presentations has been devised for seven people: Jeff, Karen, Lyle, Maria, Norm, Olivia, and Paul.
The Mathematics Department is holding an election for department chair. Each member ranks the
Candidates from first to third. The preference table below shows the results of the ballots with
Candidates Clark (C), Jones (J), and Smith (S).
Determine the winner using the plurality with elimination method and if the winner received a
Majority of first place votes.

A) Smith; No
B) Jones; Yes
C) Jones; No
D) Smith, Yes

Question 21

Find the standard divisor for the given situation. Round your answer to two decimals.
A school district receives a grant to purchase 50 new computers to be apportioned among the 6
Schools in the district based on the student population of each school. The student populations are
Given in the following table.
Find the standard quota for school A.

A) 11.56
B) 12.62
C) 13.16
D) 10.01

Question 22

Computer Specialists is planning a group vacation to one of the following locations: Alaska (A),
Florida (F), San Antonio (S), or Hawaii (H). The employees rank the four possible sites according
To the following preference table:
If the four voters who voted F, S, H, A, in that order, change their votes to H, S, A, F, and if the
Plurality with elimination method is used to determine the winner, is the monotonicity criterion
Satisfied?

A) Yes
B) No

Question 23

Find the standard divisor for the given situation. Round your answer to two decimals.
A university has 25 scholarships to be apportioned among the engineering students based on the
Enrollment in each department. There are three departments - Mechanical Engineering (M),
Electrical Engineering (E), and Civil Engineering (C). The number of students in each department
Is given in the following table.

A) 27.32
B) 28.44
C) 31.88
D) 29.72

Question 24

Find the standard divisor for the given situation. Round your answer to two decimals.
The faculty senate of a university has 45 senators to be apportioned among its four colleges based
On the number of faculty in each college. The colleges are Liberal Arts (L), Sciences (S), Business
(B), and Engineering (E). The number of faculty in each college is shown in the following table.

A) 27.18
B) 21.76
C) 24.76
D) 30.42

Question 25

Find the standard divisor for the given situation. Round your answer to two decimals.
A small country consists of seven states; there are 156 seats in the legislature that need to be
Apportioned among the seven states; and the population of each state is shown in the table.
Find the modified quota for state F using the divisor 92.

A) 27.29
B) 26.44
C) 27.93
D) 28.16

Question 26

Find the standard divisor for the given situation. Round your answer to two decimals.
The faculty senate of a university has 40 senators to be apportioned among its four colleges based
On the number of faculty in each college. The colleges are Liberal Arts (L), Sciences (S), Business
(B), and Engineering (E). The number of faculty in each college is shown in the following table.
Find the standard quota for the College of Sciences.

A) 11.61
B) 12.75
C) 14.5
D) 10.37

Question 27

A condominium association is holding an election for president of the board of directors. Each
Member ranks the candidates from first to third. The preference table below shows the results of
The ballots with candidates Abbott (A), Blake (B), Cleary (C), and Downs (D).
If the plurality method if used to determine the winner and Downs drops out, is the irrelevant
Alternatives criterion satisfied?

A) Yes
B) No

Question 28

Find the standard divisor for the given situation. Round your answer to two decimals.
A small country consists of seven states; there are 158 seats in the legislature that need to be
Apportioned among the seven states; and the population of each state is shown in the table.

A) 76.87
B) 84.65
C) 75.52
D) 92.77

Question 29

The Mathematics Department is holding an election for department chair. Each member ranks the
Candidates from first to third. The preference table below shows the results of the ballots with
Candidates Clark (C), Jones (J), and Smith (S).
If the plurality method is used to determine the winner and Clark drops out, is the irrelevant
Alternatives criterion satisfied?

A) Yes
B) No

Question 30

Find the standard divisor for the given situation. Round your answer to two decimals.
A university has 22 scholarships to be apportioned among the engineering students based on the
Enrollment in each department. There are three departments - Mechanical Engineering (M),
Electrical Engineering (E), and Civil Engineering (C). The number of students in each department
Is given in the following table.
Find the standard quota for the Mechanical Engineering Department.

A) 6.9
B) 7.51
C) 7.21
D) 6.43

Question 31

Find the standard divisor for the given situation. Round your answer to two decimals.
A small country consists of seven states; there are 157 seats in the legislature that need to be
Apportioned among the seven states; and the population of each state is shown in the table.
Find the standard quota for state C.

A) 35.86
B) 31.99
C) 35.22
D) 29.19

Question 32

Find the standard divisor for the given situation. Round your answer to two decimals.
A small city has 48 police officers to be apportioned among 8 precincts based on the population of
Each precinct. The populations are given in the following table.

A) 534.65
B) 566.5
C) 432.31
D) 636.98

Question 33

Find the standard divisor for the given situation. Round your answer to two decimals.
A small country consists of 7 provinces with the following populations: There are 320 federal judges to be apportioned according to the population of each province.
Find the standard quota for province G.

A) 53.29
B) 58.14
C) 67.74
D) 45.54

Question 34

Find the standard divisor for the given situation. Round your answer to two decimals.
A school district receives a grant to purchase 66 new computers to be apportioned among the 6
Schools in the district based on the student population of each school. The student populations are
Given in the following table.

A) 18.08
B) 20.58
C) 18.85
D) 23.77

Question 35

Find the standard divisor for the given situation. Round your answer to two decimals.
A school district receives a grant to purchase 55 new computers to be apportioned among the 6
Schools in the district based on the student population of each school. The student populations are
Given in the following table.
Find the modified quota for school B using the divisor 28.5.

A) 7.52
B) 7.55
C) 7.40
D) 7.63

Question 36

The Mathematics Department is holding an election for department chair. Each member ranks the
Candidates from first to third. The preference table below shows the results of the ballots with
Candidates Clark (C), Jones (J), and Smith (S).
If the Borda count method is used to determine the winner and Clark drops out, is the irrelevant
Alternatives criterion satisfied?

A) Yes
B) No

Question 37

Find the standard divisor for the given situation. Round your answer to two decimals.
A small city has 46 police officers to be apportioned among 8 precincts based on the population of
Each precinct. The populations are given in the following table.
Find the modified quota for the Sixth Precinct using the divisor 513.

A) 7.98
B) 8.03
C) 7.79
D) 7.85

Question 38

Find the standard divisor for the given situation. Round your answer to two decimals.
A small city has 45 police officers to be apportioned among 8 precincts based on the population of
Each precinct. The populations are given in the following table.
Find the standard quota for the Third Precinct.

A) 6.76
B) 4.59
C) 5.47
D) 5.16

Question 39

Find the standard divisor for the given situation. Round your answer to two decimals.
A university has 27 scholarships to be apportioned among the engineering students based on the
Enrollment in each department. There are three departments - Mechanical Engineering (M),
Electrical Engineering (E), and Civil Engineering (C). The number of students in each department
Is given in the following table.
Find the modified quota for the Civil Engineering Department using the divisor 27.

A) 5.67
B) 5.47
C) 5.52
D) 5.72

Question 40

Find the standard divisor for the given situation. Round your answer to two decimals.
A small country consists of 7 provinces with the following populations: There are 200 federal judges to be apportioned according to the population of each province.

A) 960.59
B) 755.6
C) 880.38
D) 1123.98

Question 41

Find the standard divisor for the given situation. Round your answer to two decimals.
A small country consists of 7 provinces with the following populations: There are 340 federal judges to be apportioned according to the population of each province.
Find the modified quota for province G using the divisor 600.

A) 52.28
B) 53.85
C) 51.97
D) 53.32

Question 42

Determine whether the specified paradox occurs.
A small country consists of seven states; there are 160 seats in the legislature that need to be
Apportioned among the seven states; and the population of each state is shown in the table.
Does the Alabama paradox occur using Hamiltonʹs method if the number of seats is increased
From 160 to 161?

A) Yes
B) No

Question 43

Find the apportionment asked for.
A small country consists of seven states; there are 160 seats in the legislature that need to be
Apportioned among the seven states; and the population of each state is shown in the table.
Find the apportionment for state D using Jeffersonʹs method.

A) 26
B) 23
C) 25
D) 24

Question 44

Find the apportionment asked for.
A small country consists of 7 provinces with the following populations: There are 300 federal judges to be apportioned according to the population of each province. Find
The apportionment for province G using Adamsʹ method.

A) 54
B) 55
C) 53
D) 56

Question 45

Find the apportionment asked for.
A small city has 50 police officers to be apportioned among 8 precincts based on the population of
Each precinct. The populations are given in the following table.
Find the apportionment for the Seventh Precinct using Websterʹs method.

A) 6
B) 7
C) 5
D) 4

Question 46

Determine whether the specified paradox occurs.
In a small country consisting of 5 provinces, 300 federal judges are apportioned according to the
Population of each province. The population of each province is shown for the years 1995 and
2000.
Does the population paradox occur using Hamiltonʹs method of apportionment?

A) Yes
B) No

Question 47

Find the standard divisor for the given situation. Round your answer to two decimals.
The faculty senate of a university has 60 senators to be apportioned among its four colleges based
On the number of faculty in each college. The colleges are Liberal Arts (L), Sciences (S), Business
(B), and Engineering (E). The number of faculty in each college is shown in the following table.
Find the modified quota for the College of Engineering using the divisor 27.

A) 15.27
B) 15.11
C) 15.43
D) 15.67

Question 48

Find the apportionment asked for.
A small city has 50 police officers to be apportioned among 8 precincts based on the population of
Each precinct. The populations are given in the following table.
Find the apportionment for the Seventh Precinct using Adamsʹ method.

A) 5
B) 4
C) 7
D) 6

Question 49

Determine whether the specified paradox occurs.
A country with two states has 16 seats in the legislature. The population of each state is given by: Does the new-states paradox occur using Hamiltonʹs method of apportionment?

A) Yes
B) No

Question 50

Determine whether the specified paradox occurs.
A city has 204 police officers to be apportioned among 4 precincts based on the population of each
Precinct. The populations are given in the following table.
Does the Alabama paradox occur using Hamiltonʹs method if the number of police officers is
Increased from 204 to 205?

A) Yes
B) No

Question 51

Determine whether the specified paradox occurs.
A small country consists of 7 provinces with the following populations: There are 300 federal judges to be apportioned according to the population of each province. Does
The Alabama paradox occur using Hamiltonʹs method if the number of judges is increased from
300 to 301?

A) Yes
B) No

Question 52

Find the apportionment asked for.
A small country consists of seven states; there are 160 seats in the legislature that need to be
Apportioned among the seven states; and the population of each state is shown in the table.
Find the apportionment for state D using Adamsʹ method.

A) 24
B) 26
C) 25
D) 23

Question 53

Find the apportionment asked for.
A small country consists of 7 provinces with the following populations: There are 300 federal judges to be apportioned according to the population of each province. Find
The apportionment for province G using Jeffersonʹs method.

A) 55
B) 56
C) 54
D) 53

Question 54

Find the apportionment asked for.
A small country consists of seven states; there are 160 seats in the legislature that need to be
Apportioned among the seven states; and the population of each state is shown in the table.
Find the apportionment for state D using Hamiltonʹs method.

A) 25
B) 26
C) 23
D) 24

Question 55

Determine whether the specified paradox occurs.
A town has 13 police officers to be apportioned among 3 precincts based on the population of each
Precinct. The populations for the years 1998 and 1999 are given in the following table.
Does the population paradox occur using Hamiltonʹs method of apportionment?

A) Yes
B) No

Question 56

Find the apportionment asked for.
A small country consists of 7 provinces with the following populations: There are 300 federal judges to be apportioned according to the population of each province. Find
The apportionment for province G using Hamiltonʹs method.

A) 56
B) 54
C) 55
D) 53

Question 57

Determine whether the specified paradox occurs.
A country with two states has 16 seats in the legislature. The population of each state (in
Thousands) is given by:
Does the new-states paradox occur using Hamiltonʹs method of apportionment?

A) Yes
B) No

Question 58

Determine whether the specified paradox occurs.
A small country consists of seven states; there are 160 seats in the legislature that need to be
Apportioned among the seven states; and the population of each state is shown below for the years
1990 and 1995.
Does the population paradox occur using Hamiltonʹs method of apportionment?

A) Yes
B) No

Question 59

Find the apportionment asked for.
A small country consists of 7 provinces with the following populations: There are 300 federal judges to be apportioned according to the population of each province. Find
The apportionment for province G using Websterʹs method.

A) 56
B) 53
C) 54
D) 55

Question 60

Find the apportionment asked for.
A small country consists of seven states; there are 160 seats in the legislature that need to be
Apportioned among the seven states; and the population of each state is shown in the table.
Find the apportionment for state D using Websterʹs method.

A) 23
B) 24
C) 25
D) 26

Question 61

Solve the problem.
Which voting method may violate the majority criterion?

A) plurality
B) plurality with elimination
C) Borda count
D) pairwise comparison

Question 62

Solve the problem.
Which fairness criterion is always satisfied by the plurality with elimination method?

A) monotonicity criterion
B) majority criterion
C) irrelevant alternatives criterion
D) head-to-head criterion

Question 63

Solve the problem.
Which apportionment paradox may be produced by the Adamsʹ method but not the Webster
Method?

A) Alabama
B) population
C) new-states
D) none of these

Question 64

Solve the problem.
Which apportionment method may produce the new-states paradox?

A) Webster
B) Jefferson
C) Hamilton
D) Adams

Question 65

Solve the problem.
Which apportionment method never violates the quota rule?

A) Adams
B) Webster
C) Jefferson
D) Hamilton

Question 66

Solve the problem.
Which apportionment method may produce the Alabama paradox?

A) Webster
B) Jefferson
C) Hamilton
D) Adams

Question 67

Solve the problem.
Which apportionment method may produce the population paradox?

A) Jefferson
B) Webster
C) Adams
D) Hamilton

Question 68

Determine whether the specified paradox occurs.
A country with two states has 75 seats in the legislature. The population of each state (in
Thousands) is given by:
Does the new-states paradox occur using Hamiltonʹs method of apportionment?

A) Yes
B) No

Question 69

Solve the problem.
Which fairness criterion may be violated by any of the voting methods?

A) monotonicity criterion
B) head-to-head criterion
C) majority criterion
D) irrelevant alternatives criterion

Question 70

Solve the problem.
Which voting method may violate the monotonicity criterion?

A) plurality with elimination
B) Borda count
C) plurality
D) pairwise comparison

Question 71

Solve the problem.
Which voting method always satisfies the head-to-head criterion?

A) plurality with elimination
B) Borda count
C) pairwise comparison
D) plurality