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3 Questions of Mathematics in the World - Assignment 1 | MA 125, Assignments of Mathematics

Material Type: Assignment; Professor: Stickles; Class: Math in the World; Subject: Mathematics; University: Millikin University; Term: Spring 2008;

Typology: Assignments

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MA 125 - In-Class Assignment 1 - Spring 2008
Hello, my name is
1. Five hundred students at a university are voting for a new school mascot. After 300 votes
have been counted, the tallies are as follows.
Purple People Eaters 106
Flying Squirrels 98
Rambling Platypi 96
(a) What is the minimum number of the remaining votes needed to guarantee that the new
mascot will be the Purple People Eaters? Explain.
The closest competitor is Flying Squirrels. To tie, Flying Squirrels would need 8 votes,
leaving 200 8 = 192 votes remaining. Purple People Eaters would need a majority of
these votes. Since 192 2 = 96;Purple People Eaters would need 97 of the remaining
votes to guarantee victory.
(b) What is the minimum number of the remaining votes needed to guarantee that the new
mascot will be the Flying Squirrels? Explain.
To tie Purple People Eaters, Flying Squirrels would need 8 votes, leaving 200 8 = 192
votes remaining. Flying Squirrels would need a majority of these votes. Since 192 2 =
96;Flying Squirrels would need 97 of these remaining votes, plus the additional 8 to tie,
to guarantee victory. So, Flying Squirrels would need 97 + 8 = 105 votes.
(c) What is the minimum number of the remaining votes needed to guarantee that the new
mascot will be the Rambling Platypi? Explain.
To tie Purple People Eaters, Rambling Platypi would need 10 votes, leaving 200 10 =
190 votes remaining. Flying Squirrels would need a majority of these votes. Since
1902 = 95;Flying Squirrels would need 96 of these remaining votes, plus the additional
10 to tie, to guarantee victory. So, Flying Squirrels would need 96 + 10 = 106 votes.
2. Suppose there were 15,364 votes an election involving six candidates.
(a) If a candidate is required to have majority of the votes cast to be considered the winner,
what is the minimum number of votes needed to win? Explain.
A majority win requires a vote tally over 50%. Fifty percent of 15;364 is 15364 :5 =
7682:So, a candidate would need 7683 votes to have over 50% and thus have a majority.
(b) If a candidate is needs a plurality of the votes cast to be considered the winner, what is
the minimum number of votes a winning candidate can have and still win the election?
Explain.
Dividing the votes equally among the candidates, we have 15364 6 = 2560:67:So,
each candidate could get 2560 votes, leaving four remaining votes. One extra vote for
a candidate will not gain that candidate victory as the remaining three votes must go
elsewhere, giving another candidate a total of at least 2561. So, a candidate must have
an addition two of the four votes, or 2562, in order to win. (This does not guarantee
victory, however.)
3. A university has 75 faculty members, and they need to vote for a faculty representative to the
Board of Trustees. There are four candidates running for this position, and the preference
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MA 125 - In-Class Assignment 1 - Spring 2008

Hello, my name is

  1. Five hundred students at a university are voting for a new school mascot. After 300 votes have been counted, the tallies are as follows.

Purple People Eaters 106 Flying Squirrels 98 Rambling Platypi 96

(a) What is the minimum number of the remaining votes needed to guarantee that the new mascot will be the Purple People Eaters? Explain. The closest competitor is Flying Squirrels. To tie, Flying Squirrels would need 8 votes, leaving 200 8 = 192 votes remaining. Purple People Eaters would need a majority of these votes. Since 192  2 = 96; Purple People Eaters would need 97 of the remaining votes to guarantee victory. (b) What is the minimum number of the remaining votes needed to guarantee that the new mascot will be the Flying Squirrels? Explain. To tie Purple People Eaters, Flying Squirrels would need 8 votes, leaving 200 8 = 192 votes remaining. Flying Squirrels would need a majority of these votes. Since 192  2 = 96 ; Flying Squirrels would need 97 of these remaining votes, plus the additional 8 to tie, to guarantee victory. So, Flying Squirrels would need 97 + 8 = 105 votes. (c) What is the minimum number of the remaining votes needed to guarantee that the new mascot will be the Rambling Platypi? Explain. To tie Purple People Eaters, Rambling Platypi would need 10 votes, leaving 200 10 = 190 votes remaining. Flying Squirrels would need a majority of these votes. Since 190 2 = 95; Flying Squirrels would need 96 of these remaining votes, plus the additional 10 to tie, to guarantee victory. So, Flying Squirrels would need 96 + 10 = 106 votes.

  1. Suppose there were 15,364 votes an election involving six candidates.

(a) If a candidate is required to have majority of the votes cast to be considered the winner, what is the minimum number of votes needed to win? Explain. A majority win requires a vote tally over 50%. Fifty percent of 15 ; 364 is 15364  :5 = 7682 : So, a candidate would need 7683 votes to have over 50% and thus have a majority. (b) If a candidate is needs a plurality of the votes cast to be considered the winner, what is the minimum number of votes a winning candidate can have and still win the election? Explain. Dividing the votes equally among the candidates, we have 15364  6 = 2560: 67 : So, each candidate could get 2560 votes, leaving four remaining votes. One extra vote for a candidate will not gain that candidate victory as the remaining three votes must go elsewhere, giving another candidate a total of at least 2561. So, a candidate must have an addition two of the four votes, or 2562, in order to win. (This does not guarantee victory, however.)

  1. A university has 75 faculty members, and they need to vote for a faculty representative to the Board of Trustees. There are four candidates running for this position, and the preference

rankings are as follows.

Number of Voters 10 8 9 4 14 6 11 13 Adkins 1 1 3 2 2 3 2 2 Blythe 2 4 1 1 3 2 3 4 Cassidy 3 2 4 3 1 1 4 3 Dearborn 4 3 2 4 4 4 1 1

(a) Which candidate wins a plurality election?

Adkins = 18 Blythe = 13 Cassidy = 20 Dearborn = 24

So, Dearborn wins. (b) Which candidate wins a plurality election with a runo§ between the top two Önishers? From part (a), we see the runo§ would be between Dearborn and Cassidy. All of those who voted for Adkins and four of those who voted for Blythe prefer Cassidy to Dearborn, giving Cassidy 20 + 18 + 4 = 42 votes, while nine of those who voted for Blythe prefer Dearborn to Cassidy, giving Dearborn 24 + 9 = 33 votes. So, Cassidy wins. (c) Which candidate wins a plurality election with a runo§ between the top three Önishers, and then a runo§ between the top two Önishers of the Örst runo§? From part (a), the top three Önishers are Adkins, Cassidy, and Dearborn. So, in the runo§ among these three, four of the Blythe voters would vote for Adkins, and nine would vote for Dearborn. This makes the new vote count

Adkins = 18 + 4 = 22 Cassidy = 20 Dearborn = 24 + 9 = 33

So, now in a runo§ between Adkins and Cassidy, the Blythe votes would be split the same way as above, and all of the Cassidy votes would go to Adkins. So, we have

Adkins = 22 + 20 = 42 Dearborn = 33

So, Adkins wins. (d) In a plurality election, could those who ranked Adkins as their top candidate have achieved a preferable outcome by voting strategically if the others voted as shown in the table? Explain. The ten Adkins voters who ranked Blythe as their second choice cannot get Blythe elected because Blythe is eleven votes behind Dearborn. However, the eight voters who prefer Cassidy second could throw their support to Cassidy, giving Cassidy enough votes to win. So, yes, those who ranked Adkins as their top candidate can achieve a preferable outcome, namely a victory for Cassidy. (Notice those who ranked Blythe second would also prefer this outcome to Dearborn winning.) (e) In a plurality election with a runo§ between the top two Önishers, could those who ranked Blythe Örst and Dearborn second have achieved a preferable outcome by voting strategically if the others voted as shown in the table? Explain. In the runo§, the nine who voted for Blythe would vote for Dearborn. So, the outcome of the runo§ would not be a§ected. However, if in the initial plurality election these

(i) If Bordaís method is used, could the voters who ranked Blythe Örst and Dearborn second have achieved a preferable outcome by voting strategically if the others voted as shown in the table? Explain. Since Adkins wins and is the third preference to Blythe, the only other option to have a preferable result is for these Blythe supporters to vote for Dearborn instead to try to get Dearborn to win (since Blythe did not). To do this most e¢ ciently, they would need to rank Dearborn Örst and Adkins last. This would change the Borda counts as follows.

Adkins = 18  4 + 42  3 + 6  2 + 9  1 = 219 Dearborn = 33  4 + 0  3 + 8  2 + 34  1 = 182

Adkins would still win in this election. So, no, the voters who ranked Blythe Örst and Dearborn second cannot achieve a preferable outcome to Adkins winning.