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 20 bottles of wine in problem 11.

Problem 11:

Three friends have pooled their resources to bid on 20 bottles of vintage red wine at a wine auction. They decide to divide the bottles based on the amount each person contributed, using Jefferson’s method of apportionment. Jaron contributed $295, Mikkel contributed $205, and Robert contributed $390.

a. Fill in the following table, and add rows as needed. Show how you arrived at each modified divisor used.

b. How should the bottles be apportioned among the friends?