Hamilton s Method By: Lauren, Lane, Devonte, Sarah, Hannah A, & Katie H
Warm-up: Matrices Review Name the Dimensions 1. [8 3 2] Add the matrices. 2. [ 1 2 3 4] + [-1-2 -3-4] = 3.Find the determinant of the 3x3 matrix
Warm-up Answer Key 1. 1x3 2. [ 0 0 0 0] 3. 2
History of Hamilton s Method -The apportionment method was suggested by Alexander Hamilton was approved by Congress in 1791. -This method was vetoed by President Washington, then was adopted by Congress in 1852 and was replaced by Webster s method in 1911 - A founding father of the U.S.
https://m.youtube.com/watch?v=7bbkd2yxwlq Hamilton's Method Video
Steps to Follow
Formulas Standard Divisor = total population number of seats SD = Total M Meaning: The population represented by 1 seat Standard Quota = State Population SD Meaning: The state s exact fair share of seats if fractional parts were allowed
Example 1 Consider a country with 4 states and 30 seats in a congress and populations distributed as in the table below. Population Quota Quota Lower Quota Final apportionment State A 275 275 63 4.37 4 4 State B 383 383 63 6.08 6 6 State C 465 465 63 7.38 7 +1 8 State D 767 767 63 12.17 12 12 Total 1890 1890 63 30 29 30
Example 1 Solution 1. Find the Standard Divisor: Total number number of items- 1890 30= 63 2. Next divide each population by 63 to determine the quota (See Chart) 3. You cannot have a number like 4.37 as seen in the chart so you must round DOWN. Must do for each quota 4. Examine the quotas and determine which one has the largest fractional part, in our case it is state C. 5. Then add +1 to the quota with the largest fractional part (State C) 6. Final apportionments are determined by the quota
Example 2 Suppose a high-school teacher can teach a maximum of five classes in one day. Suppose that she teaches geometry, Pre-calc, and calculus. 52 students registered for geometry, 33 for Pre-calc, and 15 for calculus. How many of each class should the teacher be assigned?
Example 2 Solution
Example 3 Banana Republic has states Apure, Barinas, Carabobo, and Dolores and 160 seats in the legislature. State Apure Barinas Carabobo Dolores Population (in millions) 3.31 2.67 1.33 0.69
Example 3 Solution 1. Standard Divisor: (3.31 + 2.67 + 1.33 +.69)/160 =.05 2. Next divide each population by.05 to determine the quota. Round Each answer DOWN to the nearest whole number State Standard Lower Quota Quota Apure 3.31/.05=66.2 66 Barinas 2.67/.05=53.4 53 3. Allocate Surplus Total Allocated = 158 Carabobo 1.33/.05=26.6 26 Surplus = 160 158 = 2 Dolores.69/.05=13.8 13 D gets one more seat, C gets one more 4. Total Allocation: Apure = 66 Barinas = 53 Carabobo = 27 Dolores = 14
Example 4 The Scotia Metro Area Rapid Transit operates 6 bus routes (A-F) and 130 buses, apportioned based on the average number of daily passengers: A B C D E F 45,300 31,070 20,490 14,160 10,260 8,720 1. Find the Standard Divisor 2. Calculate Quotas
Example 4 Solution 1. Standard Divisor : Sum of all passengers: 130,000 130,000/130=1000 2. Std Quota: Route Std. Quota Lower Quota A 45,300/1000=45.3 45 B 31,070/1000=31.07 31 C 20,490/1000=20.49 20 D 14,160/1000=14.16 14 E 10,260/1000=10.26 10 F 8,720/1000=8.72 8
3. Allocate Surplus Total Allocated = 128 Surplus = 130 128 = 2 F gets one more, C gets one more Total Allocation: A = 45 B=31 C=21 D= 14 E=10 F= 9
Example 5 There are 25 seats available in congress. Assign a number of seats in congress to each of the following five states. State North Carolina South Carolina Florida Georgia Virginia Total Population (Million) 19.89 4.832 19.89 4.477 8.326 57.415
Example 5 Solution 1.Find total Population 2.Find the Standard Divisor 2.2966 3.Find the Standard Quotas North Carolina 8.66 South Carolina 2.10 Florida 8.66 Georgia 1.94 Virginia 3.625 4.Surplus = 1 5. Virginia Gets 1 more North Carolina 9 South Carolina 2 Florida 9 Georgia 2 Virginia 4
Let s Do An Activity!
Problem 1: 1. A small country consists of four states. The population of state A is 67,200. The population of state B is 78,300. The population of state C is 73,800. The population of state D is 80,700. The total number of seats in the legislature is 100. a) What is the standard divisor? b) Find each state s standard quota.
Problem 1: Solution
Problem 2: Consider three states whose populations are given below: Find apportionments for 24 seats State A 39 State B 70 State C 270 States Population
Problem 2 Solution
Problem 3: The Banana Republic is a small country consisting of four states: Apure, Barinas, Carabobo, and Delores. Suppose there are 160 seats in the Banana Congress to be apportioned among the four states based on their respective populations. The following table show the population per state. State Apure Barinas Carabobo Delores Population 3,310,000 2,670,000 1,330,000 690,000 a) Determine the Standard Divisor (SD) and what it represents in the given situation. b) Use Hamilton s Method to find the apportionment for the given number of seats, M.
Problem 3 Solution
Problem 4: The Interplanetary Federation of Fraternia consists of six planets: Alpha, Kappa, Beta, Theta, Chi Omega, Delta Gamma, Epsilon Tau, and Phi Sigma (A, B, C, D, E, F for short). The federation is governed by the Inter-Franternia Congress, consisting of 200 seats apportioned among the planets according to their populations. The following table gives the planet populations. Planet A B C D E F Population 11,370,000 8,070,000 38,620,000 14,980,000 10,420,000 16,540,000 a) Determine the Standard Divisor (SD) and what it represents in the given situation. b) Use Hamilton s Method to find the apportionment for the given number of seats, M.
Problem 4 Solution
Problem 5 Consider three states whose populations are given below: Find apportionments for 25 seats States Population A 39 B 70 C 270
Problem 5 Solution