936 CHAPTER 14 Voting and Apportionment Webster’s Method Hamilton’s method rounded standard quotas down to the nearest integer. Jefferson’s method rounded modified quotas down to the nearest integer. With Webster’s method, we round modified quotas to the nearest integer such that the rounding gives the exact sum to be apportioned. WEBSTER’S METHOD To use Webster’s method for apportionment, do the following. 1. Determine a modified divisor, d, such that when each group’s modified quota is rounded to the nearest integer , the total of the integers is the exact number of items to be apportioned. We will refer to modified quotas that are rounded to the nearest integer as modified rounded quotas . 2. Apportion to each group its modified rounded quota. PROCEDURE With Webster’s method, we use a modified divisor that may be less than, equal to , or greater than the standard divisor and we use modified rounded quotas . With Webster’s method, as with Jefferson’s method, the modified divisor used must result in all the items being distributed, with no items left over. We will use Webster’s method in Example 5. Table 14.34 Republic of Hydrangea Population State A B C D E Total Population 1,003,200 1,228,600 4,990,700 813,000 764,500 8,800,000 Example 5 Using Webster’s Method for Apportioning Legislative Seats Once again, let’s consider the Republic of Hydrangea and apportion the 250 seats among the five states using Webster’s method. Use the population from Table 14.34. Solution Our first task is to determine the modified divisor, d. We could choose a number either less than, greater than, or equal to the standard divisor. As with Jefferson’s method, there is no formula to determine this new divisor and usually there is more than one divisor that works. Again, we must use trial and error until we determine a divisor that apportions the 250 legislative seats. Let’s use the results from Example 3 as a guide. If we round the standard quotas from Table 14.30 to the nearest integer, the sum is + + + + 29 35 142 23 22, or 251. This number is too high, since we want a sum of 250. If the sum is too high, as it is in this case, we need to use a larger divisor. If the sum is too low, we need to use a smaller divisor. In Example 3, we used the standard divisor of 35,200. Let’s try using 35,250 as our modified divisor, d. Using 35,250 as the modified divisor, the modified quota for state A is 1,003,200 35,250 28.46 ≈ Table 14.35 shows the results using d 35,250. = Did You Know? Mathematical Contributions from Political Leaders Brady-Handy Photograph Collection/ Library of Congress Prints and Photographs Division [LC-DIG-ggbain-33131] m Daniel Webster (1782–1852) Daniel Webster, a native of New Hampshire and educated at Dartmouth College, was a member of the Senate for many years. As a result of his successful defense of several famous constitutional cases argued before the Supreme Court, he was regarded as one of the leading lawyers of the country. He was also regarded as one of the best-known orators of his time.
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