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Math for Liberal Studies
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The US Senate has 100 members: two for each state
In the US House of Representatives, states are represented based on population
PA has 19 representatives
Delaware has 1
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The process by which seats are assigned based on population is called apportionment
The number of seats each state gets is also called that state’s apportionment
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Consider a fictional country with four states and a representative legislature with 50 seats
State Population
Angria 80,000
Bretonnia 60,000
Curaguay 40,000
Dennenberg 20,000
Total 200,000
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Each state should get a proportion of the seats that is equal to its proportion of the total population
State Population
Angria 80,000
Bretonnia 60,000
Curaguay 40,000
Dennenberg 20,000
Total 200,000
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Divide each state’s population by the total population to get the % population
State Population % Pop.
Angria 80,000 40%
Bretonnia 60,000 30%
Curaguay 40,000 20%
Dennenberg 20,000 10%
Total 200,000 100%
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Multiply that percentage by the total number of seats (in this case 50) to get each state’s fair share of seats
State Population % Pop. Fair Share
Angria 80,000 40% 20
Bretonnia 60,000 30% 15
Curaguay 40,000 20% 10
Dennenberg 20,000 10% 5
Total 200,000 100% 50
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Real world examples rarely work out as nicely as the previous example did
Let’s use more realistic population numbers and see what happens
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We will start the problem in the same way
State Population
Angria 83,424
Bretonnia 67,791
Curaguay 45,102
Dennenberg 17,249
Total 213,566
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Divide each state’s population by the total population to get the % population
State Population % Pop.
Angria 83,424 39.06%
Bretonnia 67,791 31.74%
Curaguay 45,102 21.12%
Dennenberg 17,249 8.08%
Total 213,566 100%
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Multiply that percentage by the total number of seats (in this case 50) to get each state’s fair share of seats
State Population % Pop. Fair Share
Angria 83,424 39.06% 19.53
Bretonnia 67,791 31.74% 15.87
Curaguay 45,102 21.12% 10.56
Dennenberg 17,249 8.08% 4.04
Total 213,566 100% 50
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The problem is that we can’t assign a state 19.53 seats… each apportionment must be a whole number!
State Population % Pop. Fair Share
Angria 83,424 39.06% 19.53
Bretonnia 67,791 31.74% 15.87
Curaguay 45,102 21.12% 10.56
Dennenberg 17,249 8.08% 4.04
Total 213,566 100% 50
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We could try rounding each fair share to the nearest whole number, but we end up with 51 seats, which is more than we have!
State Population % Pop. Fair Share Seats
Angria 83,424 39.06% 19.53 20
Bretonnia 67,791 31.74% 15.87 16
Curaguay 45,102 21.12% 10.56 11
Dennenberg 17,249 8.08% 4.04 4
Total 213,566 100% 50 51
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Article 1, Section 2 (1787)…Representatives and direct taxes shall be
apportioned among the several States which may be included within this Union, according to their respective Numbers …
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Article 1, Section 2 (continued)…The actual Enumeration shall be made
within three Years after the first Meeting of the Congress of the United States, and within every subsequent Term of ten Years…
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Article 1, Section 2 (continued)…The Number of Representatives shall not
exceed one for every thirty Thousand, but each State shall have at Least one Representative;
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State Population
Virginia 630,560
Massachusetts 475,327
Pennsylvania 432,879
North Carolina 353,523
New York 331,589
Maryland 278,514
Connecticut 236,841
South Carolina 206,236
New Jersey 179,570
New Hampshire 141,822
Vermont 85,533
Georgia 70,835
Kentucky 68,705
Rhode Island 68,446
Delaware 55,540
Total 3,615,920
How many Representatives?
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State Population
Virginia 630,560
Massachusetts 475,327
Pennsylvania 432,879
North Carolina 353,523
New York 331,589
Maryland 278,514
Connecticut 236,841
South Carolina 206,236
New Jersey 179,570
New Hampshire 141,822
Vermont 85,533
Georgia 70,835
Kentucky 68,705
Rhode Island 68,446
Delaware 55,540
Total 3,615,920
120000,30
920,615,3
How many Representatives?
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State Population
Virginia 630,560
Massachusetts 475,327
Pennsylvania 432,879
North Carolina 353,523
New York 331,589
Maryland 278,514
Connecticut 236,841
South Carolina 206,236
New Jersey 179,570
New Hampshire 141,822
Vermont 85,533
Georgia 70,835
Kentucky 68,705
Rhode Island 68,446
Delaware 55,540
Total 3,615,920
120000,30
920,615,3
How many Representatives?
How many representatives should each state receive?
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State Population Quota
Virginia 630,560 20.93
Massachusetts 475,327 15.78
Pennsylvania 432,879 14.37
North Carolina 353,523 11.73
New York 331,589 11.00
Maryland 278,514 9.24
Connecticut 236,841 7.86
South Carolina 206,236 6.84
New Jersey 179,570 5.96
New Hampshire 141,822 4.71
Vermont 85,533 2.84
Georgia 70,835 2.35
Kentucky 68,705 2.28
Rhode Island 68,446 2.27
Delaware 55,540 1.84
Total 3,615,920 120
How many representatives should each state receive?
A “Fair” Share (or Quota) =
SeatsTotalationTotalPopul
ationStatePopul#
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State Population Quota
Virginia 630,560 20.93
Massachusetts 475,327 15.78
Pennsylvania 432,879 14.37
North Carolina 353,523 11.73
New York 331,589 11.00
Maryland 278,514 9.24
Connecticut 236,841 7.86
South Carolina 206,236 6.84
New Jersey 179,570 5.96
New Hampshire 141,822 4.71
Vermont 85,533 2.84
Georgia 70,835 2.35
Kentucky 68,705 2.28
Rhode Island 68,446 2.27
Delaware 55,540 1.84
Total 3,615,920 120
=
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Named for Alexander Hamilton, the first U.S. Secretary of the Treasury
Hamilton’s method gives eachstate it’s lower quota (roundeddown shares) at a minimum.
If extra seats exist, they are distributed according to the sizeof each state’s fractionalremainder.
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With Hamilton’s method, each state will always receive either their lower quota or their upper quota: this is the quota rule
If a state is apportioned a number of representatives that is either above its upper quota or below its lower quota, then that is a quota rule violation
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State Population Quota Round Priority
Virginia 630,560 20.93 20
Massachusetts 475,327 15.78 15
Pennsylvania 432,879 14.37 14
North Carolina 353,523 11.73 11
New York 331,589 11.00 11
Maryland 278,514 9.24 9
Connecticut 236,841 7.86 7
South Carolina 206,236 6.84 6
New Jersey 179,570 5.96 5 1
New Hampshire 141,822 4.71 4
Vermont 85,533 2.84 2
Georgia 70,835 2.35 2
Kentucky 68,705 2.28 2
Rhode Island 68,446 2.27 2
Delaware 55,540 1.84 1
Total 3,615,920 120 111
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State Population Quota Round Priority
Virginia 630,560 20.93 20 2
Massachusetts 475,327 15.78 15
Pennsylvania 432,879 14.37 14
North Carolina 353,523 11.73 11
New York 331,589 11.00 11
Maryland 278,514 9.24 9
Connecticut 236,841 7.86 7
South Carolina 206,236 6.84 6
New Jersey 179,570 5.96 5 1
New Hampshire 141,822 4.71 4
Vermont 85,533 2.84 2
Georgia 70,835 2.35 2
Kentucky 68,705 2.28 2
Rhode Island 68,446 2.27 2
Delaware 55,540 1.84 1
Total 3,615,920 120 111
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State Population Quota Round Priority
Virginia 630,560 20.93 20 2
Massachusetts 475,327 15.78 15 7
Pennsylvania 432,879 14.37 14
North Carolina 353,523 11.73 11 8
New York 331,589 11.00 11
Maryland 278,514 9.24 9
Connecticut 236,841 7.86 7 3
South Carolina 206,236 6.84 6 4
New Jersey 179,570 5.96 5 1
New Hampshire 141,822 4.71 4 9
Vermont 85,533 2.84 2 6
Georgia 70,835 2.35 2
Kentucky 68,705 2.28 2
Rhode Island 68,446 2.27 2
Delaware 55,540 1.84 1 5
Total 3,615,920 120 111
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State Population Quota Round Priority Seats
Virginia 630,560 20.93 20 2 21
Massachusetts 475,327 15.78 15 7 16
Pennsylvania 432,879 14.37 14 14
North Carolina 353,523 11.73 11 8 12
New York 331,589 11.00 11 11
Maryland 278,514 9.24 9 9
Connecticut 236,841 7.86 7 3 8
South Carolina 206,236 6.84 6 4 7
New Jersey 179,570 5.96 5 1 6
New Hampshire 141,822 4.71 4 9 5
Vermont 85,533 2.84 2 6 3
Georgia 70,835 2.35 2 2
Kentucky 68,705 2.28 2 2
Rhode Island 68,446 2.27 2 2
Delaware 55,540 1.84 1 5 2
Total 3,615,920 120 111 120
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State Population Quota Round Priority Seats People/Rep.
Virginia 630,560 20.93 20 2 21 30,027
Massachusetts 475,327 15.78 15 7 16 29,708
Pennsylvania 432,879 14.37 14 14 30,920
North Carolina 353,523 11.73 11 8 12 29,460
New York 331,589 11.00 11 11 30,144
Maryland 278,514 9.24 9 9 30,946
Connecticut 236,841 7.86 7 3 8 29,605
South Carolina 206,236 6.84 6 4 7 29,462
New Jersey 179,570 5.96 5 1 6 29,928
New Hampshire 141,822 4.71 4 9 5 28,364
Vermont 85,533 2.84 2 6 3 28,511
Georgia 70,835 2.35 2 2 35,418
Kentucky 68,705 2.28 2 2 34,353
Rhode Island 68,446 2.27 2 2 34,223
Delaware 55,540 1.84 1 5 2 27,220
Total 3,615,920 120 111 120
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State Population Quota Round Priority Seats People/Rep.
Virginia 630,560 20.93 20 2 21 30,027
Massachusetts 475,327 15.78 15 7 16 29,708
Pennsylvania 432,879 14.37 14 14 30,920
North Carolina 353,523 11.73 11 8 12 29,460
New York 331,589 11.00 11 11 30,144
Maryland 278,514 9.24 9 9 30,946
Connecticut 236,841 7.86 7 3 8 29,605
South Carolina 206,236 6.84 6 4 7 29,462
New Jersey 179,570 5.96 5 1 6 29,928
New Hampshire 141,822 4.71 4 9 5 28,364
Vermont 85,533 2.84 2 6 3 28,511
Georgia 70,835 2.35 2 2 35,418
Kentucky 68,705 2.28 2 2 34,353
Rhode Island 68,446 2.27 2 2 34,223
Delaware 55,540 1.84 1 5 2 27,220
Total 3,615,920 120 111 120
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State Population Quota Round Priority Seats People/Rep.
Virginia 630,560 20.93 20 2 21 30,027
Massachusetts 475,327 15.78 15 7 16 29,708
Pennsylvania 432,879 14.37 14 14 30,920
North Carolina 353,523 11.73 11 8 12 29,460
New York 331,589 11.00 11 11 30,144
Maryland 278,514 9.24 9 9 30,946
Connecticut 236,841 7.86 7 3 8 29,605
South Carolina 206,236 6.84 6 4 7 29,462
New Jersey 179,570 5.96 5 1 6 29,928
New Hampshire 141,822 4.71 4 9 5 28,364
Vermont 85,533 2.84 2 6 3 28,511
Georgia 70,835 2.35 2 2 35,418
Kentucky 68,705 2.28 2 2 34,353
Rhode Island 68,446 2.27 2 2 34,223
Delaware 55,540 1.84 1 5 2 27,220
Total 3,615,920 120 111 120
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“There is no one proportion or divisor which, applied to the respective numbers of the States will yield the number and allotment of representatives proposed by the Bill.”
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When we divide the total population by the number of seats, we get the standard divisor.
Hamilton’s method for apportionment of the House of Representatives results in:
standard divisor = 132,30120
000,615,3
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State Population Quota Round Priority Seats People/Rep.
Virginia 630,560 20.93 20 2 21 30,027
Massachusetts 475,327 15.78 15 7 16 29,708
Pennsylvania 432,879 14.37 14 14 30,920
North Carolina 353,523 11.73 11 8 12 29,460
New York 331,589 11.00 11 11 30,144
Maryland 278,514 9.24 9 9 30,946
Connecticut 236,841 7.86 7 3 8 29,605
South Carolina 206,236 6.84 6 4 7 29,462
New Jersey 179,570 5.96 5 1 6 29,928
New Hampshire 141,822 4.71 4 9 5 28,364
Vermont 85,533 2.84 2 6 3 28,511
Georgia 70,835 2.35 2 2 35,418
Kentucky 68,705 2.28 2 2 34,353
Rhode Island 68,446 2.27 2 2 34,223
Delaware 55,540 1.84 1 5 2 27,220
Total 3,615,920 120 111 120
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Thomas Jefferson, 1st Secretary of State, 3rd President.
Jefferson’s idea was to modify the standard divisor so that when the shares for each state are rounded down, there are no leftover seats.
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State Population Quota Round
Virginia 630,560 20.93 20
Massachusetts 475,327 15.78 15
Pennsylvania 432,879 14.37 14
North Carolina 353,523 11.73 11
New York 331,589 11.00 11
Maryland 278,514 9.24 9
Connecticut 236,841 7.86 7
South Carolina 206,236 6.84 6
New Jersey 179,570 5.96 5
New Hampshire 141,822 4.71 4
Vermont 85,533 2.84 2
Georgia 70,835 2.35 2
Kentucky 68,705 2.28 2
Rhode Island 68,446 2.27 2
Delaware 55,540 1.84 1
Total 3,615,920 120 111
A 120 seat Congress implies standard divisor
Making the divisor smaller will increase each state’s share.
132,30120
000,615,3
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State Population Share Round
Virginia 630,560 22.12 22
Massachusetts 475,327 16.68 16
Pennsylvania 432,879 15.19 15
North Carolina 353,523 12.4 12
New York 331,589 11.63 11
Maryland 278,514 9.77 9
Connecticut 236,841 8.31 8
South Carolina 206,236 7.24 7
New Jersey 179,570 6.3 6
New Hampshire 141,822 4.98 4
Vermont 85,533 3 3
Georgia 70,835 2.49 2
Kentucky 68,705 2.41 2
Rhode Island 68,446 2.4 2
Delaware 55,540 1.95 1
Total 3,615,920 120
Modified divisor = 28,500 works!
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State Population Share Round
Virginia 630,560 22.12 22
Massachusetts 475,327 16.68 16
Pennsylvania 432,879 15.19 15
North Carolina 353,523 12.4 12
New York 331,589 11.63 11
Maryland 278,514 9.77 9
Connecticut 236,841 8.31 8
South Carolina 206,236 7.24 7
New Jersey 179,570 6.3 6
New Hampshire 141,822 4.98 4
Vermont 85,533 3 3
Georgia 70,835 2.49 2
Kentucky 68,705 2.41 2
Rhode Island 68,446 2.4 2
Delaware 55,540 1.95 1
Total 3,615,920 120
Modified divisor = 28,500 works!
But it’s unconstitutional
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State Population Quota Round
Virginia 630,560 18.31 18
Massachusetts 475,327 13.8 13
Pennsylvania 432,879 12.57 12
North Carolina 353,523 10.27 10
New York 331,589 9.63 9
Maryland 278,514 8.09 8
Connecticut 236,841 6.88 6
South Carolina 206,236 5.99 5
New Jersey 179,570 5.21 5
New Hampshire 141,822 4.12 4
Vermont 85,533 2.48 2
Georgia 70,835 2.06 2
Kentucky 68,705 2 1
Rhode Island 68,446 1.99 1
Delaware 55,540 1.61 1
Total 3,615,920 105 97
A 105 seat Congress implies standard divisor
As before, making the divisor smaller will increase each state’s share.
437,34105
000,615,3
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State Population Share Round
Virginia 630,560 19.11 19
Massachusetts 475,327 14.4 14
Pennsylvania 432,879 13.12 13
North Carolina 353,523 10.71 10
New York 331,589 10.05 10
Maryland 278,514 8.44 8
Connecticut 236,841 7.18 7
South Carolina 206,236 6.25 6
New Jersey 179,570 5.44 5
New Hampshire 141,822 4.3 4
Vermont 85,533 2.59 2
Georgia 70,835 2.15 2
Kentucky 68,705 2.08 2
Rhode Island 68,446 2.07 2
Delaware 55,540 1.68 1
Total 3,615,920 105
Modified divisor = 33,000 works! (and it’s > 30,000)
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State Population Share Round
Virginia 630,560 19.11 19
Massachusetts 475,327 14.4 14
Pennsylvania 432,879 13.12 13
North Carolina 353,523 10.71 10
New York 331,589 10.05 10
Maryland 278,514 8.44 8
Connecticut 236,841 7.18 7
South Carolina 206,236 6.25 6
New Jersey 179,570 5.44 5
New Hampshire 141,822 4.3 4
Vermont 85,533 2.59 2
Georgia 70,835 2.15 2
Kentucky 68,705 2.08 2
Rhode Island 68,446 2.07 2
Delaware 55,540 1.68 1
Total 3,615,920 105
Modified divisor = 33,000 works! (and it’s > 30,000)
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State Population Jefferson Hamilton
Virginia 630,560 19 18
Massachusetts 475,327 14 14
Pennsylvania 432,879 13 12
North Carolina 353,523 10 10
New York 331,589 10 10
Maryland 278,514 8 9
Connecticut 236,841 7 7
South Carolina 206,236 6 6
New Jersey 179,570 5 5
New Hampshire 141,822 4 4
Vermont 85,533 2 2
Georgia 70,835 2 2
Kentucky 68,705 2 2
Rhode Island 68,446 2 2
Delaware 55,540 1 2
Total 3,615,920 105 105
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State Population Jefferson Hamilton
Virginia 630,560 19 18
Massachusetts 475,327 14 14
Pennsylvania 432,879 13 12
North Carolina 353,523 10 10
New York 331,589 10 10
Maryland 278,514 8 9
Connecticut 236,841 7 7
South Carolina 206,236 6 6
New Jersey 179,570 5 5
New Hampshire 141,822 4 4
Vermont 85,533 2 2
Georgia 70,835 2 2
Kentucky 68,705 2 2
Rhode Island 68,446 2 2
Delaware 55,540 1 2
Total 3,615,920 105 105
Jefferson’s hometown = Monticello, VA
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Let’s look at a fictional country with four states.
State Population
Biglandia 96,974
Medios 45,902
Regula 44,921
Smallirus 32,764
Totals 220,561
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This time the standard divisor is 220,561/50 = 4,411.2
State Population Quota Round
Biglandia 96,974 21.98 21
Medios 45,902 10.41 10
Regula 44,921 10.18 10
Smallirus 32,764 7.43 7
Totals 220,561 50 48
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We have leftover seats, so Jefferson’s method tells us to make the divisor smaller
State Population Quota Round
Biglandia 96,974 21.98 21
Medios 45,902 10.41 10
Regula 44,921 10.18 10
Smallirus 32,764 7.43 7
Totals 220,561 50 48
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After some trial and error, we find that a divisor of 4,200 works.
State Population Share Round
Biglandia 96,974 23.09 23
Medios 45,902 10.93 10
Regula 44,921 10.70 10
Smallirus 32,764 7.80 7
Totals 220,561 50
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The top chart uses the standard divisor (4,411)
The bottom chart is our Jefferson’s method solution
We have a quota rule violation!
State Population Quota Round
Biglandia 96,974 21.98 21
Medios 45,902 10.41 10
Regula 44,921 10.18 10
Smallirus 32,764 7.43 7
Totals 220,561 50 48
State PopulationModified
ShareRound
Biglandia 96,974 23.09 23
Medios 45,902 10.93 10
Regula 44,921 10.70 10
Smallirus 32,764 7.80 7
Totals 220,561 50
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As we just witnessed, Jefferson’s method can violate the quota rule.
This is because lowering the divisor causes the quotient to grow at a faster rate if the numerator is higher.
Thus, this method tends to favor larger states over smaller ones.
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John Quincy Adams James Dean Daniel Webster6th President Mathematician 2-time Secretary of State
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This method is the same as Jefferson’s method, except you always round the shares up instead of down
This gives you too many seats, so you need to increase the divisor until you get the right number of seats
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Let each state’s quota = State Population/Divisor Let n = the rounded down quota. Calculate the harmonic mean of n and n+1
If a state’s quota exceeds the harmonic mean – give the state n+1 representatives. Otherwise, give it n.
Adjust the divisor as necessary to match number of Representatives to the House.
5.0
)1(
n
nn
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In this method, we round the shares to the nearest whole number
It’s possible that this gives us the right number of seats, in which case we’re done
Otherwise, we need to increase (if we have too few seats) or decrease (if we have too many seats) the divisor
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State Population Rank Adams Dean Webster Jefferson
NY 1,918,578 1 37 38 39 40
PA 1,348,072 2 26 27 27 28
VA 1,023,503 3 20 21 21 21
OH 937,901 4 18 19 19 19
NC 639,747 5 13 13 13 13
TN 625,263 6 12 13 13 13
KY 621,832 7 12 12 12 13
VT 280,657 16 6 6 6 5
NH 269,326 17 6 5 5 5
AL 262,508 18 6 5 5 5
LA 171,904 19 4 4 3 3
IL 157,147 20 4 3 3 3
MO 130,419 21 3 3 3 2
DE 75,432 24 (Last) 2 2 2 1
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1842 1852 – 1880s 1880s – 1890s 1900 – 1920s 1929 – 1941 1941 – 2000s
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1842
1852 – 1880s (sort of)
1880s – 1890s 1900 – 1920s 1929 – 2000s
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The 1872 apportionment differed from Hamilton’s method in four states
State Population Rank Hamilton Actual
NY 4,382,759 1 34 33
IL 2,539,891 4 20 19
NH 318,300 31 2 3
FL 187,748 33 1 2
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Rutherford B. Hayes won the electoral vote 185 to 184 over Samuel J. Tilden
State Population Rank Hamilton Actual Winner
NY 4,382,759 1 34 33 Tilden
IL 2,539,891 4 20 19 Hayes
NH 318,300 31 2 3 Hayes
FL 187,748 33 1 2 Hayes
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1842 1852 – 1880s
1880s – 1890s 1900 – 1920s 1929 – 2000s
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Hamilton’s method is relatively simple to use, but it can lead to some strange paradoxes:
The Alabama paradox: Increasing the number of total seats causes a state to lose seats
The New States paradox: Introducing a new state causes an existing state to gain seats
The Population paradox: A state that gains population loses a seat to a state that does not
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When the population of each state stays the same but the number of seats is increased, intuitively each state’s apportionment should stay the same or go up
That doesn’t always happen
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Consider these state populations, with 80 seats available. Watch what happens when we increase to 81 seats
State Population % Pop. Fair ShareLower Quota
Priority Seats
Angria 83,424 39.06% 31.25 31 31
Bretonnia 67,791 31.74% 25.39 25 25
Curaguay 45,102 21.12% 16.89 16 1st 17
Dennenberg 17,249 8.08% 6.46 6 2nd 7
Totals 213,566 100% 80 78 80
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State Population % Pop. Fair ShareLower Quota
Priority Seats
Angria 83,424 39.06% 31.25 31 31
Bretonnia 67,791 31.74% 25.39 25 25
Curaguay 45,102 21.12% 16.89 16 1st 17
Dennenberg 17,249 8.08% 6.46 6 2nd 7
Totals 213,566 100% 80 78 80
State Population % Pop. Fair ShareLower Quota
Priority Seats
Angria 83,424 39.06% 31.64 31 2nd 32
Bretonnia 67,791 31.74% 25.71 25 1st 26
Curaguay 45,102 21.12% 17.11 17 17
Dennenberg 17,249 8.08% 6.54 6 6
Totals 213,566 100% 81 79 81
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After the 1880 Census, it was time to reapportion the House of Representatives.
The chief clerk of the Census Bureau computed apportionments for all numbers of seats from 275 to 350.
Alabama would receive 8 seats if there were 299 total seats, but only 7 seats if 300 were available.
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If a new state is added to our country, but the total number of seats remains the same, then we would expect that the apportionment for the existing states should stay the same or go down
This doesn’t always happen
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Consider this country with 4 states and 70 seats
State Population % Pop. Fair ShareLower Quota
Priority Seats
Elkabar 80,424 39.80% 27.86 27 1st 28
Florin 59,902 29.64% 20.75 20 2nd 21
Gondor 48,338 23.92% 16.75 16 3rd 17
Hyrkania 13,405 6.63% 4.64 4 4
Totals 202,069 100% 70 67 70
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Now suppose the small state of Ishtar is added to this country
State Population % Pop. Fair ShareLower Quota
Priority Seats
Elkabar 80,424 39.80% 27.86 27 1st 28
Florin 59,902 29.64% 20.75 20 2nd 21
Gondor 48,338 23.92% 16.75 16 3rd 17
Hyrkania 13,405 6.63% 4.64 4 4
Totals 202,069 100% 70 67 70
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State Population % Pop. Fair ShareLower Quota
Priority Seats
Elkabar 80,424 39.80% 27.86 27 1st 28
Florin 59,902 29.64% 20.75 20 2nd 21
Gondor 48,338 23.92% 16.75 16 3rd 17
Hyrkania 13,405 6.63% 4.64 4 4
Totals 202,069 100% 70 67 70
State Population % Pop. Fair ShareLower Quota
Priority Seats
Elkabar 80,424 38.72% 27.11 27 27
Florin 59,902 28.84% 20.19 20 20
Gondor 48,338 23.28% 16.29 16 16
Hyrkania 13,405 6.45% 4.52 4 2nd 5
Ishtar 5,611 2.70% 1.89 1 1st 2
Totals 207,680 100% 70 68 70
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In 1907, when Oklahoma was admitted to the Union, the total number of seats in Congress did not change
Oklahoma was assigned 5 seats, but this resulted in Maine gaining a seat!
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As time goes on, populations of states change
States that increase population rapidly shouldgain seats over those that do not
However, this doesn’t always happen
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Consider this country with 4 states and 100 seats
State Population % Pop. Fair ShareLower Quota
Priority Seats
Javasu 28,900 18.09% 18.09 18 18
Karjastan 76,200 47.68% 47.68 47 1st 48
Libria 44,200 27.66% 27.66 27 2nd 28
Malbonia 10,500 6.57% 6.57 6 6
Totals 159,800 100% 100 98 100
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Suppose that the populations of the states change
State Population % Pop. Fair ShareLower Quota
Priority Seats
Javasu 28,900 18.09% 18.09 18 18
Karjastan 76,200 47.68% 47.68 47 1st 48
Libria 44,200 27.66% 27.66 27 2nd 28
Malbonia 10,500 6.57% 6.57 6 6
Totals 159,800 100% 100 98 100
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State Population % Pop. Fair ShareLower Quota
Priority Seats
Javasu 28,900 18.09% 18.09 18 18
Karjastan 76,200 47.68% 47.68 47 1st 48
Libria 44,200 27.66% 27.66 27 2nd 28
Malbonia 10,500 6.57% 6.57 6 6
Totals 159,800 100% 100 98 100
State Population % Pop. Fair ShareLower Quota
Priority Seats
Javasu 28,900 17.97% 17.97 17 2nd 18
Karjastan 76,400 47.51% 47.51 47 47
Libria 45,000 27.99% 27.99 27 1st 28
Malbonia 10,500 6.53% 6.53 6 3rd 7
Totals 160,800 100% 100 97 100
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In the 1901 apportionment, Virginia lost a seat to Maine even though Virginia’s population grew at a faster rate!
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1842 1852 – 1880s 1880s – 1890s
1900 – 1920s 1929 – 2000s
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1842 1852 – 1880s 1880s – 1890s 1900 – 1920s
1929 – 2000s
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Determine a priority value (PV) for each state for successive representative numbers until you run out of seats. Let
where n is current number of allotted seats for a given state.
If seats still exist (<436 have been allotted) the state is given the n+1st representative, else it will have n.
)1(
nn
ationStatePopulPVn
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Seat # State Population n Priority Value Total Representatives
51 California 37,341,989 1 26,404,773 53
52 Texas 25,268,418 1 17,867,469 36
53 California 37,341,989 2 15,244,803 53
54 New York 19,421,055 1 13,732,759 27
55 Florida 18,900,773 1 13,364,864 27
56 California 37,341,989 3 10,779,703 53
434 California 37,341,989 52 711,308 53
435 Minnesota 5,314,879 7 710,230 8
436 North Carolina 9,565,781 13 709,062 13
437 Missouri 6,011,478 8 708,459 8
438 New York 19,421,055 27 706,336 27
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Seat # State Population n Priority Value Reps Need for +1
51 California 37,341,989 1 26,404,773 53 N/A
52 Texas 25,268,418 1 17,867,469 36 N/A
53 California 37,341,989 2 15,244,803 53 N/A
54 New York 19,421,055 1 13,732,759 27 N/A
55 Florida 18,900,773 1 13,364,864 27 N/A
56 California 37,341,989 3 10,779,703 53
434 California 37,341,989 52 711,308 53 N/A
435 Minnesota 5,314,879 7 710,230 8 N/A
436 North Carolina 9,565,781 13 709,062 13 15,800
437 Missouri 6,011,478 8 708,459 8 15,100
438 New York 19,421,055 27 706,336 27 107,100
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In 1982 Michel Balinski and H. Peyton Young proved that it is impossible to find an apportionment system that both:
avoids paradoxes
doesn’t violate the quota rule
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0
50
100
150
200
250
300
350
400
450
1793 1803 1813 1823 1833 1843 1853 1863 1873 1883 1893 1903 1913 1923 1933 1943 1953 1963 1973 1983 1993 2003 2013
U.S. House of Representatives 1793 - 2013
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-
100,000
200,000
300,000
400,000
500,000
600,000
700,000
800,000
1793 1803 1813 1823 1833 1843 1853 1863 1873 1883 1893 1903 1913 1923 1933 1943 1953 1963 1973 1983 1993 2003 2013
Average Number of U.S. Citizens per Representative
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Montana
994,416 people
1 representative
Wyoming
568,300 people
1 representative
Rhode Island
1,055,247 people
2 representatives
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Within the last 20 years three states (Utah, Montana and Massachusetts) and multiple individuals have filed suits against current apportionment methods and/or census counting methods.
The Supreme Court has upheld the validity of the Huntington-Hill Method and the current apportionment procedure in all cases.
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