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Section 5.5 Chapter 5 Review

Exercises 5.5.1 Review Exercises

In exercises 1-4, determine the apportionment using

  1. Hamilton’s Method

  2. Jefferson’s Method

  3. Webster’s Method

  4. Huntington-Hill Method

1.

A small country consists of four states, whose populations are listed below. If the legislature has 78 seats, apportion the seats.

A: 96,400 B: 162,700 C: 119,900 D: 384,900
Solution

  1. A: 10 seats, B: 17 seats, C: 12 seats, D: 39 seats

  2. A: 10 seats, B: 16 seats, C: 12 seats, D: 40 seats, Modified divisor 9,600

  3. A: 10 seats, B: 17 seats, C: 12 seats, D: 39 seats, Divisor 9,793.6

  4. A: 10 seats, B: 17 seats, C: 12 seats, D: 39 seats, Divisor 9,793.6

2.

Reapportion the previous problem with 90 seats.

Solution

  1. A: 12 seats, B: 19 seats, C: 14 seats, D: 45 seats

  2. A: 11 seats, B: 19 seats, C: 14 seats, D: 46 seats, Modified divisor 8,300

  3. A: 11 seats, B: 19 seats, C: 14 seats, D: 46 seats, Modified divisor 8,400

  4. A: 11 seats, B: 19 seats, C: 14 seats, D: 46 seats, Modified divisor 8,400

3.

A small country consists of five states, whose populations are listed below. If the legislature as 100 seats, apportion the seats.

A: 584,000 B: 226,600 C: 88,500 D: 257,300 E: 104,300
Solution

  1. A: 46 seats, B: 18 seats, C: 7 seats, D: 21 seats, E: 8 seats

  2. A: 47 seats, B: 18 seats, C: 7 seats, D: 20 seats, E: 8 seats, Modified divisor 12,400

  3. A: 47 seats, B: 18 seats, C: 7 seats, D: 20 seats, E: 8 seats, Modified divisor 12,555

  4. A: 47 seats, B: 18 seats, C: 7 seats, D: 20 seats, E: 8 seats, Modified divisor 12,555

4.

Reapportion the previous problem with 125 seats.

Solution

  1. A: 58 seats, B: 22 seats, C: 9 seats, D: 26 seats, E: 10 seats

  2. A: 59 seats, B: 22 seats, C: 8 seats, D: 26 seats, E: 10 seats, Modified divisor 9,890

  3. A: 58 seats, B: 22 seats, C: 9 seats, D: 26 seats, E: 10 seats, Divisor 10,085.6

  4. A: 58 seats, B: 22 seats, C: 9 seats, D: 26 seats, E: 10 seats, Divisor 10,085.6

In exercises 5-8, complete the following:

  1. How many voters voted in this election?

  2. How many votes are needed for a majority?

  3. Find the winner under the plurality method.

  4. Find the winner under the Instant Runoff Voting method.

  5. Find the winner under the Borda Count Method.

  6. Find the winner under Copeland’s method.

5.

A Portland Community College Board member race has four candidates: E, F, G, H. The votes are:

Number of voters 12 16 17 15 34 13 19 8
1st choice G H E E F G H G
2nd choice E F F H G H G F
3rd choice F G G F H E F E
4th choice H E H G E F E H

Solution

  1. There are 134 voters.

  2. A majority is 68 votes.

  3. H wins the plurality method with 35 votes.

  4. F wins the Instant Runoff Method with 71 votes.

  5. The points are: E 275, F 364, G 372 and H 329. G wins in the Borda count method.

  6. The points are: F 3, G 2, H 1. F wins with Copeland’s method.

6.

A Forest Grove School Board position has four candidates: I, J, K, L. The votes are:

Number of voters 15 13 25 16 18 10 7 11 2
1st choice K I J L K L I I L
2nd choice J L L I I J K J K
3rd choice L J I K J I J K J
4th choice I K K J L K L L I

Solution

  1. There are 117 voters.

  2. A majority is 59 votes.

  3. K wins the plurality method with 38 votes.

  4. L wins the Instant Runoff Method with 66 votes.

  5. The points are: I 313, J 304, K 261, L 292. I wins in the Borda count method.

  6. The points are: I 2, J 2, L 2. I, J and L tie with Copeland’s method.

7.

A Multnomah County Commissioner's race has five candidates: M, N, O, P, Q. The votes are:

Number of voters 31 18 35 37 33 12
1st choice M Q O N P Q
2nd choice P O Q P M N
3rd choice O M P O N M
4th choice N P N M Q O
5th choice Q N M Q O P

Solution

  1. There are 166 voters.

  2. A majority is 84 votes.

  3. N wins the plurality method with 37 votes.

  4. P wins the Instant Runoff Method with 101 votes.

  5. The points are: M 486, N 482, O 508, P 590, Q 424. P wins the Borda count method.

  6. The points are: M 1, N 2, O 3, P 4. P wins with Copeland’s method.

8.

The Oregon State Governor's race has five candidates: R, S, T, U, V. The votes are:

Number of voters 22 45 20 47 43 18 26
1st choice R S R U T V V
2nd choice T V S T U S T
3rd choice S T V S V U S
4th choice U R U V R R U
5th choice V U T R S T R

Solution

  1. There are 221 voters.

  2. A majority is 111 votes.

  3. U wins the plurality method with 47 votes.

  4. T wins the Instant Runoff Method with 138 votes.

  5. The points are: R 495, S 705, T 768, U 642, V 705. T wins the Borda count method.

  6. The points are: S 3, T 4, U 2, V 1. T wins with Copeland’s method.

In each fictional country in problems 9-10, use the rules of the U.S. government to complete the table and determine the following:

  1. The total number of electors in the state.

  2. The number of electoral votes needed for a majority and win a presidential election.

9.

In this country there is one representative for every 55,000 residents.

State Population Number of
Representatives		 Number of
Senators		 Number of
Electors
Fonville 825,000
Gurley 550,000
Nevarez 275,000
Total
Solution
  1. State Population Number of
    Representatives    		 Number of
    Senators    		 Number of
    Electors
    Fonville 825,000 15 2 17
    Gurley 550,000 10 2 12
    Nevarez 275,000 5 2 7
    Total 1,650,000 30 6 36

    This state has 36 electors.

  2. A majority of electoral votes would be 19 votes.

10.

In this country there is one representative for every 60,000 residents.

State Population Number of
Representatives		 Number of
Senators		 Number of
Electors
Arbery 720,000
Monterrosa 360,000
Bland 240,000
Davis 480,000
Total
Solution
  1. State Population Number of
    Representatives    		 Number of
    Senators    		 Number of
    Electors
    Arbery 720,000 12 2 14
    Monterrosa 360,000 6 2 8
    Bland 240,000 4 2 6
    Davis 480,000 8 2 10
    Total 1,800,000 30 8 38

    This state has 38 electors.

  2. A majority of electoral votes would be 20 votes.

In each fictional country in problems 11-12, use the rules of the U.S. government (assume that all of a state’s electoral votes go to the candidate who received the majority of the votes in that state) to complete the table and determine the following:

  1. The winner of the popular vote in the country and the percentage of votes they won.

  2. The winner of the electoral college who becomes the president and the percentage of electoral votes they won.

11.

In this country from problem 9, there is one representative for every 55,000 residents.

State Votes for
Candidate A		 Votes for
Candidate B		 Number of Electoral
Votes for A		 Number of Electoral
Votes for B
Fonville 684,750 140,250
Gurley 257,400 292,600
Nevarez 132,275 142,725
Total Votes
Solution
  1. State Votes for
    Candidate A    		 Votes for
    Candidate B    		 Number of Electoral
    Votes for A    		 Number of Electoral
    Votes for B
    Fonville 684,750 140,250 17 0
    Gurley 257,400 292,600 0 12
    Nevarez 132,275 142,725 0 7
    Total Votes 1,074,425 575,575 17 19

    A wins the popular vote with 65.1% of the votes.

  2. B wins the electoral college and becomes the president with 52.8% of the electoral votes.

12.

In this country from problem 10, there is one representative for every 60,000 residents.

State Votes for
Candidate A		 Votes for
Candidate B		 Number of Electoral
Votes for A		 Number of Electoral
Votes for B
Arbery 372,240 347,760
Monterrosa 38,880 321,120
Bland 134,640 105,360
Davis 104,160 375,840
Total
Solution
  1. State Votes for
    Candidate A    		 Votes for
    Candidate B    		 Number of Electoral
    Votes for A    		 Number of Electoral
    Votes for B
    Arbery 372,240 347,760 14 0
    Monterrosa 38,880 321,120 0 8
    Bland 134,640 105,360 6 0
    Davis 104,160 375,840 0 10
    Total 649,920 1,150,080 20 18

    B wins the popular vote with 63.9% of the vote.

  2. A wins the electoral college and becomes the president with 52.6% of the electoral votes.

In each fictional country in problems 13-14, use the rules of the U.S. government to complete the table and determine the following:

  1. The state that has the most electoral power

  2. The state that has the least electoral power

13.

In this country from problem 9, there is one representative for every 55,000 residents.

State Population Number of
Representatives		 Number of
Senators		 Number of
Electors		 Electoral Votes per
55,000 people
Fonville 825,000
Gurley 550,000
Nevarez 275,000
Solution
  1. State Population Number of
    Representatives    		 Number of
    Senators    		 Number of
    Electors    		 Electoral Votes per
    55,000 people
    Fonville 825,000 15 2 17 1.13
    Gurley 550,000 10 2 12 1.20
    Nevarez 275,000 5 2 7 1.40

    The state of Nevarez has the most electoral power.

  2. The state of Fonville has the least electoral power.

14.

In this country from problem 10, there is one representative for every 60,000 residents.

State Population Number of
Representatives		 Number of
Senators		 Number of
Electors		 Electoral Votes per
60,000 people
Arbery 720,000
Monterrosa 360,000
Bland 240,000
Davis 480,000
Solution
  1. State Population Number of
    Representatives    		 Number of
    Senators    		 Number of
    Electors    		 Electoral Votes per
    60,000 people
    Arbery 720,000 12 2 14 1.17
    Monterrosa 360,000 6 2 8 1.33
    Bland 240,000 4 2 6 1.50
    Davis 480,000 8 2 10 1.25

    The state of Bland has the most electoral power.

  2. The state of Arbery has the least electoral power.

For each map in problems 15-16, complete the following:

  1. How many votes are needed for a majority?

  2. How many seats are won by each party?

  3. Calculate the efficiency gap.

  4. Calculate the percentage of the state that each district represents.

  5. Calculate how many district seats the efficiency gap is worth.

  6. Explain whether you think the map is fair and why or why not.

15.

This state has 5 districts with 9 people in each district.

Solution

  1. A majority is 5 votes.

  2. The Democrats won 1 seat and the Republicans won 4 seats.

  3. The efficiency gap is \(20/45= 44.4\%\)

    District D Votes R Votes D Surplus Votes R Surplus Vote
    1 4 5 4 \(5-5=0\)
    2 4 5 4 \(5-5=0\)
    3 9 0 \(9-5=4\) 0
    4 4 5 4 \(5-5=0\)
    5 4 5 4 \(5-5=0\)
    Total 25 20 20 0

  4. Each seat is worth 20% of the voters.

  5. The efficiency gap is worth 2.2 seats.

  6. This map is not fair because the efficiency gap is more than two seats. A more fair map would be D3, R2.

16.

This state has 6 districts with 7 people in each.

Solution

  1. A majority is 4 votes.

  2. The Democrats won 5 seats and the Republicans won 1 seat.

  3. The efficiency gap is \(16/42= 38.10\%\)

    District D Votes R Votes D Surplus Votes R Surplus Vote
    1 4 3 \(4-4=0\) 3
    2 4 3 \(4-4=0\) 3
    3 4 3 \(4-4=0\) 3
    4 4 3 \(4-4=0\) 3
    5 4 3 \(4-4=0\) 3
    6 1 6 1 \(6-4=2\)
    Total 21 21 1 17

  4. Each seat is worth 16.67% of the voters.

  5. The efficiency gap is worth 2.3 seats.

  6. This map is not fair because the efficiency gap is more than two seats. A more fair map would be D3, R3 because the population is evenly split.