14.3 Apportionment Methods 937 Whenever we use Webster’s method, we first round the modified quota to the nearest hundredth and then determine the modified rounded quota. In Example 5, we used a modified divisor of 35,250 to obtain the desired sum of 250. There are an infinite number of modified divisors (in a small range) that could be used to obtain the modified rounded quotas. In fact, any modified divisor from about 35,207 to about 35,271 would result in the modified rounded quotas we obtained in Table 14.35. If you selected a modified divisor lower than about 35,207, you would obtain modified rounded quotas for which the sum of the seats is too high. If you selected a modified divisor greater than about 35,271, you would obtain modified rounded quotas for which the sum of the seats is too low. Occasionally, the results of Webster’s method agree with the results from Hamilton’s method. With Webster’s method, we can use a modified divisor that is less than, greater than, or equal to the standard divisor used with Hamilton’s method. Webster’s method is similar to Jefferson’s method, since they both make use of a modified divisor. Webster’s method, however, is a little more difficult in practice to apply than Jefferson’s method, since the modified divisor could be less than, equal to, or greater than the standard divisor. Webster’s method may seem the most reasonable because the modified quotas are rounded in the conventional way. Webster’s method, however, does have a flaw in that it can violate the quota rule. Even though Example 5 did not uncover this flaw, there are examples in which Webster’s method violates the quota rule. This violation happens more in theory than in practice. As a result, many experts consider Webster’s method the best overall apportionment method. Let’s discuss one more apportionment method that was proposed (but never used by the House of Representatives) by John Quincy Adams around the same time that Webster proposed his method. Adams’ Method When Jefferson’s method was discovered to be problematic because it violated the quota rule, John Quincy Adams proposed a method that is exactly the opposite of Jefferson’s method. Instead of using modified lower quotas as Jefferson proposed, Adams suggested using modified upper quotas. Now try Exercise 15 Table 14.35 Republic of Hydrangea Population Using a Modified Divisor, d 35,250 = State A B C D E Total Population 1,003,200 1,228,600 4,990,700 813,000 764,500 8,800,000 Standard quota 28.50 34.90 141.78 23.10 21.72 Modified quota 28.46 34.85 141.58 23.06 21.69 Modified rounded quota 28 35 142 23 22 250 Now round each modified quota to the nearest integer to get the modified rounded quotas, as shown in Table 14.35. Our sum of the modified rounded quotas is 250, as desired. Therefore, using Webster’s method, each state is awarded the number of legislative seats listed in Table 14.35 under the category of modified rounded quota. 7 Healy, G. P. A. (George Peter Alexander), 1813-1894, artist, J.C. Tichenor/ Library of Congress Prints and Photographs Division [LC-USZ62-117119] Did You Know? Mathematical Contributions from Political Leaders m John Quincy Adams (1767–1848) John Quincy Adams, sixth president of the United States, was the son of John Adams, second president of the United States. Adams’ varied career included the positions of diplomat, senator, secretary of state, president, and member of the House of Representatives. Adams was instrumental in the expansion of U.S. borders southward and westward. He convinced Spain to cede Florida to the United States and also persuaded Spain to agree that the western border of the Louisiana Purchase extended all the way to the Pacific Ocean.
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