Apportionment Problems
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1 Apportionment Problems Lecture 16 Section 4.1 Robb T. Koether Hampden-Sydney College Fri, Oct 4, 2013 Robb T. Koether (Hampden-Sydney College) Apportionment Problems Fri, Oct 4, / 15
2 1 Apportionment Problems Hamilton s Method 2 The Round-Off Method 3 Hamilton s Method 4 Assignment Robb T. Koether (Hampden-Sydney College) Apportionment Problems Fri, Oct 4, / 15
3 Outline 1 Apportionment Problems Hamilton s Method 2 The Round-Off Method 3 Hamilton s Method 4 Assignment Robb T. Koether (Hampden-Sydney College) Apportionment Problems Fri, Oct 4, / 15
4 Apportionment Problems Example I have a class of 10 students and I have 50 pieces of candy to hand out. Robb T. Koether (Hampden-Sydney College) Apportionment Problems Fri, Oct 4, / 15
5 Apportionment Problems Example I have a class of 10 students and I have 50 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, Oct 4, / 15
6 Apportionment Problems Example I have a class of 10 students and I have 50 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 80 correct answers. 5, 5, 5, 7, 7, 8, 10, 11, 11, 11, Robb T. Koether (Hampden-Sydney College) Apportionment Problems Fri, Oct 4, / 15
7 Apportionment Problems Example I have a class of 10 students and I have 50 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 80 correct answers. 5, 5, 5, 7, 7, 8, 10, 11, 11, 11, How many candies should each student get? Robb T. Koether (Hampden-Sydney College) Apportionment Problems Fri, Oct 4, / 15
8 Apportionment Problems Example I have a class of 10 students and I have 50 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 80 correct answers. 5, 5, 5, 7, 7, 8, 10, 11, 11, 11, How many candies should each student get? The ratio of 50 : 80 indicates that each correct answer is worth candies. Robb T. Koether (Hampden-Sydney College) Apportionment Problems Fri, Oct 4, / 15
9 Apportionment Problems Definition (Apportionment Problem) The classic apportionment problem involves a representative body where each state is given a certain number of seats, according to the state s population. Let N be the number of states. Let M the number of seats. Let p 1, p 2, p 3,..., p N be the states populations. Let P = p 1 + p 2 + p p N, the total population. Robb T. Koether (Hampden-Sydney College) Apportionment Problems Fri, Oct 4, / 15
10 Definitions Definition (Standard Divisor) The standard devisor (SD) is P M. It represents the number of people that each seat represents. Definition (Standard Quota) The standard quota of a state is the exact fractional number of seats it should get for its fair share. It is computed as ( pi ) q i = M = p i P SD. Definition (Lower and Upper Quotas) The lower quota and the upper quota are the two whole numbers nearest the standard quota. Robb T. Koether (Hampden-Sydney College) Apportionment Problems Fri, Oct 4, / 15
11 Lower and Upper Quotas Example (Lower and Upper Quotas) Find the standard divisor, standard quota, and the lower and upper quotas for the 10 students, with scores and the 50 pieces of candy. 5, 5, 5, 7, 7, 8, 10, 11, 11, 11, Robb T. Koether (Hampden-Sydney College) Apportionment Problems Fri, Oct 4, / 15
12 Outline 1 Apportionment Problems Hamilton s Method 2 The Round-Off Method 3 Hamilton s Method 4 Assignment Robb T. Koether (Hampden-Sydney College) Apportionment Problems Fri, Oct 4, / 15
13 The Round-Off Method Definition (The Round-Off Method) 1 Calculate each state s standard quota. Robb T. Koether (Hampden-Sydney College) Apportionment Problems Fri, Oct 4, / 15
14 The Round-Off Method Definition (The Round-Off Method) 1 Calculate each state s standard quota. 2 Round each one down to the nearest integer to get that state s quota. Robb T. Koether (Hampden-Sydney College) Apportionment Problems Fri, Oct 4, / 15
15 The Round-Off Method Definition (The Round-Off Method) 1 Calculate each state s standard quota. 2 Round each one down to the nearest integer to get that state s quota. 3 Cross your fingers that it all works out ok. Robb T. Koether (Hampden-Sydney College) Apportionment Problems Fri, Oct 4, / 15
16 Example Example Apply round-off method to the 10 students and the 50 candies. The standard quotas are 3.125, 3.125, 3.125, 4.375, 4.375, 5.0, 6.25, 6.875, 6.875, What is the final apportionment? Did it work out ok? Robb T. Koether (Hampden-Sydney College) Apportionment Problems Fri, Oct 4, / 15
17 Outline 1 Apportionment Problems Hamilton s Method 2 The Round-Off Method 3 Hamilton s Method 4 Assignment Robb T. Koether (Hampden-Sydney College) Apportionment Problems Fri, Oct 4, / 15
18 Hamilton s Method Definition (Hamilton s Method) 1 Calculate each state s standard quota. Historical background. Robb T. Koether (Hampden-Sydney College) Apportionment Problems Fri, Oct 4, / 15
19 Hamilton s Method Definition (Hamilton s Method) 1 Calculate each state s standard quota. 2 Round each one down to the lower quota. Historical background. Robb T. Koether (Hampden-Sydney College) Apportionment Problems Fri, Oct 4, / 15
20 Hamilton s Method Definition (Hamilton s Method) 1 Calculate each state s standard quota. 2 Round each one down to the lower quota. 3 Distribute the surplus to the states with the largest fraction parts, i.e., the largest difference between their standard quota and their lower quota. Historical background. Robb T. Koether (Hampden-Sydney College) Apportionment Problems Fri, Oct 4, / 15
21 Example Example Apply Hamilton s method to the 10 students and the 50 candies. The standard quotas are 3.125, 3.125, 3.125, 4.375, 4.375, 5.0, 6.25, 6.875, 6.875, What is the final apportionment? Robb T. Koether (Hampden-Sydney College) Apportionment Problems Fri, Oct 4, / 15
22 Outline 1 Apportionment Problems Hamilton s Method 2 The Round-Off Method 3 Hamilton s Method 4 Assignment Robb T. Koether (Hampden-Sydney College) Apportionment Problems Fri, Oct 4, / 15
23 Assignment Assignment Page 128: Exercises 4, 6, 7. Page 130: Exercises 18, 19. Robb T. Koether (Hampden-Sydney College) Apportionment Problems Fri, Oct 4, / 15
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