I.Basics

Example 1: Suppose a country consists of five states (North, South, East, West and Central) and representation in the National Assembly should be in proportion to each state’s population. The latest census figures are: |North | 9279 |

|South | 7217 |

|East | 5155 |

|West | 3093 |

|Central | 2062 |

|Total |26,806 |

If there are 26 seats in the Assembly, how should they be apportioned among the five states? Solution: Let’s start by dividing the total population by the number of seats: ____1031_______ . This tells us how many citizens each member of the National Assembly represents and is called the standard divisor.

The number of seats each state is entitled to, called its quota, is

North’s quota is _____9____ South’s quota is __7_______ East’s quota is ____5_____ West’s quota is ______3___ Central’s quota is _____2____ The two formulas above can be combined to yield:

Example 2: Suppose a later census is taken and the following population figures are revealed. How should the 26 seats be apportioned among the 5 states?

Solution: Notice the quotas aren’t whole numbers any more. Round them off, you say? Std. Divisor = 1000

|State |Population |Std. Quota |Quota |Rounded quota | |North | 9061 |9.061 |9 | | |South | 7179 |7.179 |7 | | |East | 5259 |5.259 |5 | | |West | 3319 |3.319 |3 | | |Central | 1182 |1.182 |1 | | |Total |26,000 | |25 | |

That doesn’t work either since the rounded quotas sum to 25, not 26. One state has to gain a seat here. The question is which one? First, some vocabulary. If we round down, we get what is called the lower quota. If we round up, we get what is called the upper quota. The following notation is used: [pic] and [pic]

This problem of the rounded quotas not summing to the desired number is quite common. Sometimes the rounded quotas sum to a number that is too big. This problem became apparent in the late 18th century. The U.S. Constitution, while requiring that the House of Representatives be apportioned proportional to each state’s population, didn’t spell out a precise method for doing this. Nor did it specify the size of the House. The first census was taken in 1790, 3 years after the Constitutional Convention. We are going to study five methods for solving this problem. One method considered by Congress, and the easiest to use, was advocated by Alexander Hamilton, America’s first Secretary of the Treasury.

II.Hamilton’s method consists of three steps:

1. Calculate each state’s standard quota.

2. Give to each state a number of seats equal to its lower quota. 3. Give the remaining seats, one at a time, to the states with the largest decimal parts until there are no more remaining seats.

Applying Hamilton’s method to our example:

|State |Population |Quota |Lower quota |Decimal part |Additional seats |Final apportionment | |North |9061 |9.061 |9 |.061 | |9 | |South |7179 |7.179 |7 |.179 |...