Problem

In 1832, a professor at Dartmouth, James Dean, proposed yet another apportionment method...

In 1832, a professor at Dartmouth, James Dean, proposed yet another apportionment method called Dean’s method. Dean’s method has never been used to apportion seats for the House of Representatives. Nonetheless, it has been a part of the apportionment debate in the United States. After the 1990 reapportionment, Montana and Massachusetts challenged the constitutionality of the Huntington–Hill method that has been used for years. Faced with losing one of its two House seats, Montana favored Adams’ method or Dean’s method, either of which would have allowed Montana to retain its two seats but would have left Massachusetts with only 10 seats. Massachusetts suggested using Webster’s method, which would have allocated 11 seats to Massachusetts and 1 seat to Montana.

Dean’s method is a divisor method and uses the harmonic mean of two numbers. For two whole numbers a and b, the harmonic mean is

In Dean’s method, also called the harmonic mean method, a modified divisor is chosen so that all modified quotas can be rounded upward or downward, and the resulting whole numbers will add up to the number of seats to be apportioned. If the modified quota is less than the harmonic mean of the nearest whole numbers above and below the modified quota, then round down; otherwise, round up. For example, the modified quota 3.15 is between the whole numbers 3 and 4, and the harmonic mean of 3 and 4 is Because 3.15 is less than 3.43, we round the modified quota down to 3.

Use Dean’s method to apportion the 105 House seats in problem 17.

Problem 17:

In 1791, Thomas Jefferson helped to convince President George Washington to veto a bill that established a 120-member House of Representatives to be apportioned using Hamilton’s method. When the House could not override Washington’s veto, a new bill was passed that established a 105-member House to be apportioned using Jefferson’s method. In 1800, there were 141 seats in the House. The following table contains apportionment population totals for the states that were part of the United States in the years 1790 and 1800.