# Topic 14: Apportionment

The number of representatives from each state in the U.S. House of Representatives is proportional to the state's population, and this number is updated every 10 years. Several different methods, called methods of apportionment, have been used to decide how many representatives each state should have.

Apportionment methods are useful in any situation where a small number of delegates (or representatives) are chosen to represent a larger population that consists of separate groups, in a manner consisten with the size of those groups.

Usually it is not possible to find a perfect method of apportionment, so the resulting choice is a combination of math and politics.

The standard divisor is the total population divided by the total number of seats:
standard divisor = total population / total number of seats

It's the average number of people represented by any one House member.

A state's standard quota (is also called its exact quota), is the number obtained by dividing the state's population by the standard divisor:
standard quota = state's population / standard divisor

If we substitute the standard divisor into the above equation, we see the alternate formula:
standard quota = total number of seats * (state's population / total population)

Quota methods of apportionment work with the standard divisor and round-off standard quotas, according to some rules. Rounding down the standard quota to an integer gives the lower quota and rounding up gives the upper quota.

Hamilton's Method
Hamilton's method consists of assigning the lower quota (except when it is 0) to each state at first, and then assigning any remaining seats to the states whose standard quotas have the largest fractional part.
1. Find the standard divisor
2. Find each state's standard quota
3. Assign the lower quota to each state, unless it is 0. In this case, assign 1.
4. If there are any seats remaining, find the state with the largest fractional part of the standard quota (excluding states that had 0 as lower quotas). If there are still remaining seats, assign the next seat to the state with the next largest fractional part. Continue this way until all are assigned.

Lowndes' Method
Lowndes' method is similar to Hamilton's method, except it uses the relative fractional part to decide on the extra seats. The relative fractional part of a number is its fractional part divided by its integer part. So, steps #1 - #3 of Lowndes' method are the same as in Hamilton's method, but step #4 is modified.
1. Find the standard divisor
2. Find each state's standard quota
3. Assign the lower quota to each state, unless it is 0. In this case, assign 1.
4. If there are any seats remaining, compute the relative fractional parts of the standard quota (excluding states that had 0 as lower quotas), and assign one of the remaining seats to the state with the largest relative fractional part. If there are still remaining seats, assign the next seat to the state with the second largest relative fractional part. Continue this way until all are assigned.