14.3 Apportionment Methods 941 Of the four methods we have discussed in this section, Hamilton’s method uses standard quotas. Jefferson’s method, Webster’s method, and Adams’ method all make use of a modified quota and can all lead to violations of the quota rule. As we will see in the next section, Hamilton’s method can also be problematic by producing paradoxes. Table 14.43 summarizes the four apportionment methods. Using d 40,300, = we have a sum of 260 seats, as desired. Therefore, each state receives the number of seats listed under the modified upper quotas in Table 14.41. d) With Webster’s method, we need to determine a modified divisor such that when each state’s modified quota is rounded to the nearest integer, the total of the integers is 260. The modified divisor may be less than, equal to, or greater than the standard divisor. If we round the standard quotas from part (a) to the nearest integer, the sum of the rounded quotas is 260. Therefore, we will use the standard divisor as our modified divisor. Table 14.42 shows the results when we round the modified quotas to the nearest integer. Now try Exercise 43 Table 14.42 Webster’s Method, Modified Divisor, d 40,000 = State A B C D Total Population 3,350,000 1,850,000 2,365,000 2,835,000 10,400,000 Standard quota 83.75 46.25 59.13 70.88 Modified quota 83.75 46.25 59.13 70.88 Modified rounded quota 84 46 59 71 260 Each state receives the number of seats listed under the modified rounded quotas in Table 14.42. Note that in this example, all four methods led to the same apportionment. However, that is not always the case. 7 Table 14.43 Summary of Apportionment Methods Method Divisor Apportionment Hamilton’s Standard divisor total population number of items to be allocated = Round each standard quota down. Distribute any leftover items to the groups with the largest fractional parts until all items are distributed. Favors large states. Jefferson’s The modified divisor is less than the standard divisor. Each group’s modified quota is rounded down to the nearest integer. Apportion to each group its modified lower quota. Favors large states. Webster’s The modified divisor is less than, greater than, or equal to, the standard divisor. Each group’s modified quota is rounded to the nearest integer. Apportion to each group its modified rounded quota. Favors small states. Adams’ The modified divisor is greater than the standard divisor. Each group’s modified quota is rounded up to the nearest integer. Apportion to each group its modified upper quota. Favors small states. Instructor Resources for Section 14.3 in MyLab Math • Objective-Level Videos 14.3 • PowerPoint Lecture Slides 14.3 • MyLab Exercises and Assignments 14.3
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