ch04 - Excursions in Modern Mathematics Sixth Edition Peter...

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1 Excursions in Modern Mathematics Sixth Edition Peter Tannenbaum
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2 Chapter 4 The Mathematics of Apportionment Making the Rounds
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3 The Mathematics of Apportionment Outline/learning Objectives To state the basic apportionment problem. To implement the methods of Hamilton, Jefferson, Adams and Webster to solve apportionment problems. To state the quota rule and determine when it is satisfied. To identify paradoxes when they occur. To understand the significance of Balanski and Young’s impossibility theorem.
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4 The Mathematics of Apportionment 4.1 Apportionment Problems
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5 The Mathematics of Apportionment We are dividing and assigning things We are dividing and assigning things. We are doing this on a proportional basis and We are doing this on a proportional basis and in a planned, organized fashion in a planned, organized fashion. Apportion- two critical elements in the definition of the word
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6 The Mathematics of Apportionment State A B C D E F Total Population 1,646,000 6,936,000 154,000 2,091,000 685,000 988,000 12,500,000 Table 4-3 Republic of Parador (Population by State) The first step is to find a good unit of measurement. The most natural unit of measurement is the ratio of population to seats. We call this ratio the standard divisor SD = P/M SD = 12,500,000 / 250 = 50,000
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7 The Mathematics of Apportionment State A B C D E F Total Population 1,646,000 6,936,000 154,000 2,091,000 685,000 988,000 12,500,000 Standard quota 32.92 138.72 3.08 41.82 13.70 19.76 250 Table 4-4 Republic of Parador: Standard Quotas for Each State (SD = 50,000) For example, take state A. To find a state’s standard quota, we divide the state’s population by the standard divisor: Quota = population/SD = 1,646,000/50,000 = 32.92
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The Mathematics of Apportionment The “states.” This is the term we will use to describe the players involved in the apportionment. The “seats.” This term describes the set of M identical, indivisible objects that are being divided among the N states. The “populations.”
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This note was uploaded on 11/11/2011 for the course MATH 110 taught by Professor Staff during the Winter '08 term at BYU.

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ch04 - Excursions in Modern Mathematics Sixth Edition Peter...

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