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They did this for possible sizes of the House from 275 total seats to 350 total seats. Apportionment can be thought of as dividing a group of people (or other resources) and assigning them to different places. The Alabama paradox was first noticed in 1881 when the seats in the U.S. House of Representatives were reapportioned after the 1880 census. The Quota Method of Apportionment, The American Mathematical Monthly, Volume 82, Number 7, August-September 1975, pages 701-730. You have, aptly, named these new states Stars, White Stripe, and Red Stripe (the stars and stripes, for short). That means that d = 11,000 is much too big. Using Excel to do Apportionment. The terminology we use in apportionment reflects this history. Webster’s method divides all populations by a modified divisor and then rounds the results up or down following the usual rounding rules. One of the most heated and contentious apportionment debates in U.S. history took place in 1832. Start by dividing each population by the standard divisor and rounding each standard quota down. Unfortunately for Hamilton, President Washington vetoed its selection. The total number of seats, 23 is too small. In many situations the five methods give the same results. Jefferson's Method causes violations. Find the standard divisor and the standard quotas for each of the states of Hamiltonia. George Washington exercised his first veto power on a bill that introduced a new plan for dividing seats in the House of Representatives that would have increased the number of seats for northern states. Find the standard divisor,. Dividing by a larger modified divisor will make each quota smaller so the sum of the lower quotas will be smaller. If the sum is too small, make the divisor smaller. There is no formula for this, just guess something. Let’s try the modified divisor, d = 10,000. On moving day, four of his friends come to help and stay until the job is done since Tom promised they will split a case of beer afterwards. Use Adams’s method to apportion the 25 seats in Hamiltonia from Example \(\PageIndex{2}\). Round each modified quota to the nearest integer using the geometric mean as the cut off. The “ D ” here is the same as it was for Hamilton. Our guess for the first modified divisor should be the standard divisor. In this video, we learn how to use Jefferson's Method to solve apportionment problems. Guess #1: d = 1600. From Example \(\PageIndex{2}\) we know the standard divisor is 9480 and the sum of the lower quotas is 20. 5 In Example \(\PageIndex{2}\) the total number of seats allocated would be 26 if we used the usual rounding rule. Sometimes the total number of seats allocated is too high and other times it is too low. Jefferson’s method divides all populations by a modified divisor and then rounds the results down to the lower quota.   A) True B) False   6. Starting with the state that has the largest fractional part and working toward the state with the smallest fractional part, allocate one additional seat to each state until all the seats have been allocated. This situation has not happened in any of the previous examples. Example \(\PageIndex{3}\): Comparison of all Apportionment Methods. As your first act in office, you have decided to help middle school students all over the U.S. by consolidating the states into just three, easy to remember states. Guess #2: d = 1550. Hamiltonia, a small country consisting of six states is governed by a senate with 25 members. Guess #1: d = 1654. The results are summarized below in Table \(\PageIndex{9}\). Because some quotas will be rounded up and other quotas will be rounded down we do not know immediately whether the total number of seats is too big or too small. Guess #3: d = 1625. This took the politics out of apportionment and made it a purely mathematical process. ‹ Apportioning Representatives in the United States Congress - Jefferson's Method of Apportionment up Apportioning Representatives in the United States Congress - Lowndes' Method of Apportionment › Author(s): Michael J. Caulfield (Gannon University) The fact that the affected states in the discrepancy just mentioned are Virginia and Delaware is no coincidence. This resulted in a House of 105 seatswith 19 seats for Virginia even though its quota of 105 seats was only 18.310. For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. Round each modified quota down to its lower quota. Thomas Jefferson, who lived before any of these paradoxes, proposed a different method for apportionment. Try d = 10,500. The question is how to divide the four remaining beers among the five friends assuming they only get whole beers. Since 1792, five different apportionment methods have been proposed and four of these methods have been used to apportion the seats in the House of Representatives. For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. If the sum is the same as the number of seats to be apportioned, you are done. It states that the number of seats that should be allocated to a given party should be between the upper or lower roundings (called upper and lower quotas) of its fractional proportional share (called natural quota). Jefferson’s Method violates the Quota Rule. Legal. After Washington vetoed Hamilton’s method, Jefferson’s method was adopted and used in Congress from 1791 through 1842. An apportionment method exists which satisfies the quota condition and is free from both the population paradox and the Alabama paradox. Note that we must use more decimal places in this example than in the last few examples. No more memorizing 50 states and capitals. Webster’s method was later chosen to be used in 1842 but Adams’s method was never used. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Apportionment Hamilton's Method Jefferson's Method Adams's Method Webster's Method Lowndes's Method Huntingdon-Hill's Method Dean's Method Equal Proportions Method. Have questions or comments? This time the standard divisor will be 24.19. Assign this firefighter to District D since D has the largest fractional part. Tom is moving to a new apartment. Just like Jefferson’s method we keep guessing modified divisors until the method assigns the correct number of seats. None of the apportionment methods is perfect. The difference between the three methods is the rule for rounding off the quotas. As with the other apportionment methods, the method of rounding off the quotas is what distinguishes this method from the others. Because some quotas are rounded up and others down we do not know if the standard divisor will give a sum that is too large or too small. The apportionment methods we will look at in this chapter were all created as a way to divide the seats in the U.S. House of Representatives among the states based on the size of the population for each state. This forces us to use the standard divisor as the first modified divisor. The Jefferson and the D'Hondt methods are equivalent. That is due to rounding and is negligible. However, 2.48 should be rounded down to 2 while 2.53 should be rounded up to 3 according to Webster’s method. Jefferson’s method rounds the quotas down to their lower quotas, Adams’ method rounds the quotas up to their upper quotas, and Webster’s method rounds the quotas either up or down following the usual rounding rule. The geometric mean \(G\) of two positive numbers \(A\) and \(B\) is, Example \(\PageIndex{1}\): Geometric Mean. Guess #2: d = 1600. Unlike the methods of Hamilton, Jefferson, and Webster, Lowndes’s method has never been used to apportion Congress. The U.S. Constitution requires that the seats for the House of Representatives be apportioned among the states every ten years based on the sizes of the populations. In 1941, the number of seats in the House was fixed at 435 and an official method was chosen. This mathematical analysis has its roots in the US Constitution specifically in 1790 when the House of Representatives attempted to apportion themselves. Divide each state’s population by the modified divisor to get its modified quota. Our guess for the first modified divisor should be a number smaller than the standard divisor. This video focuses on Jefferson's method. If fractional seats were possible, Alpha would get 2.532 seats and Beta would get 5.907 seats. Use Hamilton’s method to finish the allocation of seats in Hamiltonia. Jefferson’s method was the first method used to apportion the seats in the U.S. House of Representatives in 1792. Alexander Hamilton proposed the first apportionment method to be approved by Congress. If the sum is the same as the number of seats to be apportioned, you are done. However, if the house size was increased to 300 total seats, Alabama would only receive 7 seats. Watch the recordings here on Youtube! Missed the LibreFest? If the sum is too large, pick a new modified divisor that is larger than d. If the sum is too small, pick a new modified divisor that is smaller than d. Repeat steps three through six until the correct number of seats are apportioned. 4 - Is the Jefferson apportionment method susceptible... Ch. Guess #2: d = 1900. Hamilton’s, Adams’s, Webster’s, and Huntington-Hill’s methods all gave the same apportionment: 15 firefighters to District A, five to District B, seven to District C, six to District D, and nine to District E. Jefferson’s method gave a different apportionment: 16 firefighters to District A, five to District B, seven to District C, five to District D, and nine to District E. The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. Round each modified quota down to its lower quota. This veto was the first presidential veto utilized in the new U.S. government. District B has a standard quota of 68.969 so it should get either its lower quota, 68, or its upper quota, 69, seats. If the sum is too big, pick a new modified divisor that is larger than d. If the sum is too small, pick a new modified divisor that is smaller than d. Repeat steps three through six until the correct number of seats are apportioned. The next step is to find the standard quota for each state. Have questions or comments? However, by the tradition established after 1842, Congress fixes the number of seats up front, with 435 seats being the norm since 1931. An apportionment method that guarantees that this will happen is said to satisfy the Quota Rule.) For this to happen we have to adjust the standard divisor either up or down. The Jefferson method of apportionment can display the Alabama paradox. The sum of 41 is still too small so make the modified divisor smaller. Use Jefferson’s method with a modified divisor of d = 9.83 to apportion the 100 representatives among the departments. The number of senators for each state is proportional to the population of the state. Later, Hamilton’s method was used off and on between 1852 and 1901. They are to be assigned to the five firehouses in town in a manner proportional to the population in each fire district. 5 Distribute the surplus to the states with the largest fractional parts. 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