Apportionment Problems
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1 Apportionment Problems Lecture 14 Section 4.1 Robb T. Koether Hampden-Sydney College Fri, Feb 20, 2015 Robb T. Koether (Hampden-Sydney College) Apportionment Problems Fri, Feb 20, / 15
2 1 Apportioning Candies 2 The House of Representatives 3 Hamilton s Solution 4 Assignment Robb T. Koether (Hampden-Sydney College) Apportionment Problems Fri, Feb 20, / 15
3 Outline 1 Apportioning Candies 2 The House of Representatives 3 Hamilton s Solution 4 Assignment Robb T. Koether (Hampden-Sydney College) Apportionment Problems Fri, Feb 20, / 15
4 Apportioning Candies Example I have a class of 10 students and I have 150 pieces of candy to hand out. Robb T. Koether (Hampden-Sydney College) Apportionment Problems Fri, Feb 20, / 15
5 Apportioning Candies Example I have a class of 10 students and I have 150 pieces of candy to hand out. I will distribute the candies according to the number of correct answers they give on the next test, out of 12 questions. Robb T. Koether (Hampden-Sydney College) Apportionment Problems Fri, Feb 20, / 15
6 Apportioning Candies Example I have a class of 10 students and I have 150 pieces of candy to hand out. I will distribute the candies according to the number of correct answers they give on the next test, out of 12 questions. The numbers of correct answers are for a total of 75 correct answers. 2, 4, 5, 6, 7, 8, 10, 10, 11, 12, Robb T. Koether (Hampden-Sydney College) Apportionment Problems Fri, Feb 20, / 15
7 Apportioning Candies Example I have a class of 10 students and I have 150 pieces of candy to hand out. I will distribute the candies according to the number of correct answers they give on the next test, out of 12 questions. The numbers of correct answers are for a total of 75 correct answers. 2, 4, 5, 6, 7, 8, 10, 10, 11, 12, How many candies should each student get? Robb T. Koether (Hampden-Sydney College) Apportionment Problems Fri, Feb 20, / 15
8 Apportioning Candies Example I have a class of 10 students and I have 150 pieces of candy to hand out. I will distribute the candies according to the number of correct answers they give on the next test, out of 12 questions. The numbers of correct answers are for a total of 75 correct answers. 2, 4, 5, 6, 7, 8, 10, 10, 11, 12, How many candies should each student get? The ratio of 150 : 75 indicates that each correct answer is worth 2 candies. Robb T. Koether (Hampden-Sydney College) Apportionment Problems Fri, Feb 20, / 15
9 Apportioning Candies Example But, unfortunately, my dog ate 102 of the candies before I could hand them out, so now I have only 48 candies. Robb T. Koether (Hampden-Sydney College) Apportionment Problems Fri, Feb 20, / 15
10 Apportioning Candies Example But, unfortunately, my dog ate 102 of the candies before I could hand them out, so now I have only 48 candies. Now each correct answer is worth 48 75, or 0.64, of a candy. Robb T. Koether (Hampden-Sydney College) Apportionment Problems Fri, Feb 20, / 15
11 Apportioning Candies Example But, unfortunately, my dog ate 102 of the candies before I could hand them out, so now I have only 48 candies. Now each correct answer is worth 48 75, or 0.64, of a candy. How many candies should each student get? Robb T. Koether (Hampden-Sydney College) Apportionment Problems Fri, Feb 20, / 15
12 Apportioning Candies Example If we tried to give each student his exact number of candies, we would give the students 1.28, 2.56, 3.20, 3.84, 4.48, 5.12, 6.40, 6.40, 7.04, 7.68 candies, respectively. Robb T. Koether (Hampden-Sydney College) Apportionment Problems Fri, Feb 20, / 15
13 Apportioning Candies Example If we tried to give each student his exact number of candies, we would give the students 1.28, 2.56, 3.20, 3.84, 4.48, 5.12, 6.40, 6.40, 7.04, 7.68 candies, respectively. If we rounded those numbers off, we would give them 1, 3, 3, 4, 4, 5, 6, 6, 7, 8 candies, which add up to only 47 candies. Robb T. Koether (Hampden-Sydney College) Apportionment Problems Fri, Feb 20, / 15
14 Apportioning Candies Example If we tried to give each student his exact number of candies, we would give the students 1.28, 2.56, 3.20, 3.84, 4.48, 5.12, 6.40, 6.40, 7.04, 7.68 candies, respectively. If we rounded those numbers off, we would give them 1, 3, 3, 4, 4, 5, 6, 6, 7, 8 candies, which add up to only 47 candies. Who gets the extra candy? Robb T. Koether (Hampden-Sydney College) Apportionment Problems Fri, Feb 20, / 15
15 Outline 1 Apportioning Candies 2 The House of Representatives 3 Hamilton s Solution 4 Assignment Robb T. Koether (Hampden-Sydney College) Apportionment Problems Fri, Feb 20, / 15
16 Apportioning Congressional Seats Apportioning Congressional Seats This is analogous to apportioning congressional seats to the states. There are 435 seats to be distributed among 50 states, according to their populations. How many seats should each state get? Robb T. Koether (Hampden-Sydney College) Apportionment Problems Fri, Feb 20, / 15
17 Outline 1 Apportioning Candies 2 The House of Representatives 3 Hamilton s Solution 4 Assignment Robb T. Koether (Hampden-Sydney College) Apportionment Problems Fri, Feb 20, / 15
18 Alexander Hamilton s Solution Alexander Hamilton s Solution Alexander Hamilton proposed a solution in 1791, right after the first national census. His method, briefly, is Calculate the exact number of seats that each state deserves. Separate each of those numbers into a whole number and the fractional part. Give each state its whole number of seats. Distribute the remaining seats to those states with the largest fractional parts. Robb T. Koether (Hampden-Sydney College) Apportionment Problems Fri, Feb 20, / 15
19 Apply Hamilton s Method Example Let s apply Hamilton s method to the 10 students and the 48 candies. Robb T. Koether (Hampden-Sydney College) Apportionment Problems Fri, Feb 20, / 15
20 Apply Hamilton s Method Example Let s apply Hamilton s method to the 10 students and the 48 candies. The exact numbers of candies are 1.28, 2.56, 3.20, 3.84, 4.48, 5.12, 6.40, 6.40, 7.04, Robb T. Koether (Hampden-Sydney College) Apportionment Problems Fri, Feb 20, / 15
21 Apply Hamilton s Method Example Let s apply Hamilton s method to the 10 students and the 48 candies. The exact numbers of candies are 1.28, 2.56, 3.20, 3.84, 4.48, 5.12, 6.40, 6.40, 7.04, Give the students the whole numbers of candies for a total of 44 candies. 1, 2, 3, 3, 4, 5, 6, 6, 7, 7, Robb T. Koether (Hampden-Sydney College) Apportionment Problems Fri, Feb 20, / 15
22 Apply Hamilton s Method Example Let s apply Hamilton s method to the 10 students and the 48 candies. The exact numbers of candies are 1.28, 2.56, 3.20, 3.84, 4.48, 5.12, 6.40, 6.40, 7.04, Give the students the whole numbers of candies for a total of 44 candies. 1, 2, 3, 3, 4, 5, 6, 6, 7, 7, Which students should get the remaining 4 candies? Robb T. Koether (Hampden-Sydney College) Apportionment Problems Fri, Feb 20, / 15
23 Apply Hamilton s Method Example Let s apply Hamilton s method to the 10 students and the 48 candies. The exact numbers of candies are 1.28, 2.56, 3.20, 3.84, 4.48, 5.12, 6.40, 6.40, 7.04, Give the students the whole numbers of candies for a total of 44 candies. 1, 2, 3, 3, 4, 5, 6, 6, 7, 7, Which students should get the remaining 4 candies? Robb T. Koether (Hampden-Sydney College) Apportionment Problems Fri, Feb 20, / 15
24 Historical Note Historical Note Hamilton s proposal passed Congress in Robb T. Koether (Hampden-Sydney College) Apportionment Problems Fri, Feb 20, / 15
25 Historical Note Historical Note Hamilton s proposal passed Congress in However, the Constitution says, The number of Representatives shall not exceed one for every thirty Thousand. Robb T. Koether (Hampden-Sydney College) Apportionment Problems Fri, Feb 20, / 15
26 Historical Note Historical Note Hamilton s proposal passed Congress in However, the Constitution says, The number of Representatives shall not exceed one for every thirty Thousand. Based on the 1790 census and a House of 120 members, several states would have received more than one representative for 30,000 people. Robb T. Koether (Hampden-Sydney College) Apportionment Problems Fri, Feb 20, / 15
27 Historical Note Historical Note Hamilton s proposal passed Congress in However, the Constitution says, The number of Representatives shall not exceed one for every thirty Thousand. Based on the 1790 census and a House of 120 members, several states would have received more than one representative for 30,000 people. So, Washington vetoed the bill. Robb T. Koether (Hampden-Sydney College) Apportionment Problems Fri, Feb 20, / 15
28 Historical Note Historical Note Hamilton s proposal passed Congress in However, the Constitution says, The number of Representatives shall not exceed one for every thirty Thousand. Based on the 1790 census and a House of 120 members, several states would have received more than one representative for 30,000 people. So, Washington vetoed the bill. Congress failed to override Washington s veto. Robb T. Koether (Hampden-Sydney College) Apportionment Problems Fri, Feb 20, / 15
29 Historical Note Historical Note Hamilton s proposal passed Congress in However, the Constitution says, The number of Representatives shall not exceed one for every thirty Thousand. Based on the 1790 census and a House of 120 members, several states would have received more than one representative for 30,000 people. So, Washington vetoed the bill. Congress failed to override Washington s veto. What happened next? Robb T. Koether (Hampden-Sydney College) Apportionment Problems Fri, Feb 20, / 15
30 Historical Note Historical Note Hamilton s proposal passed Congress in However, the Constitution says, The number of Representatives shall not exceed one for every thirty Thousand. Based on the 1790 census and a House of 120 members, several states would have received more than one representative for 30,000 people. So, Washington vetoed the bill. Congress failed to override Washington s veto. What happened next? To be continued... Robb T. Koether (Hampden-Sydney College) Apportionment Problems Fri, Feb 20, / 15
31 Outline 1 Apportioning Candies 2 The House of Representatives 3 Hamilton s Solution 4 Assignment Robb T. Koether (Hampden-Sydney College) Apportionment Problems Fri, Feb 20, / 15
32 Assignment Assignment Chapter 4: Exercises 11, 12, 13, 14. Skip the terminology; apply Hamilton s method. Robb T. Koether (Hampden-Sydney College) Apportionment Problems Fri, Feb 20, / 15
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