The school district uses the Hamilton method to apportion its 22 board members to the 4 towns. How many board members are assigned to each town, using this method? 2. The following year, 900 people move out of Town D. Two hundred of these people move Town C, and 700 of them move to Town B. Now, how many board members does each town have? (Be careful. Make sure you assign a total of 22 board members). 3. Compare the results from the 2 years. Do you think they make sense? How do you think each town would react? Are they fair? Why or Why not?

Step-by-step explanation:

1. The Hamilton method has you compute the number represented by each board member (total population/# members). Using this factor, the number of board members for each district are computed. This raw value is rounded down.

Because this total does not allocate all board members, the remaining members of the board are allocated to the districts based on the size of the fraction that was truncated when rounding down. Allocations start with the largest fraction and work down until all board members have been allocated.

The attached spreadsheet implements this algorithm using a "threshold" that is adjusted to a value between 0 and 1, signifying the cutoff point between a fraction value that gets an additional member and one that doesn't. (Often, that threshold can be set at 0.5, equivalent to rounding the raw board member value to the nearest integer.)

The resulting allocations are ...

  Town A: 2

  Town B: 2

  Town C: 6

  Town D: 12

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2. The second attachment shows the result after the population move. The allocations of board members are identical.

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3. The "factor" (persons per board member) is about 4500, so we don't expect a move of 900 people to make any difference in the allocation. These results make complete sense.

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4. Of course each town will consider its own interest at the expense of everyone else, so they may or may not consider the results fair. The towns have population ratio of about 9 : 9 : 25 : 56, so the ratios 2 : 2 : 6 : 12 are quite in line. Even in the second year, when the ratios are closer to 9 : 10 : 26 : 56, the changes are small enough that the allocation of board members still makes sense. The results are fair.

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Comment on "fair"

The reason there are different methods of allocation is that each seeks to rectify some perceived flaw in one or more of the others. The reason there is not a general agreement on the method to be used is that some benefit more from one method than from another. "Fair" is in the eye of the beholder. I believe in this case it would be very difficult to justify any other allocations than the ones computed here.