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# Week 9 - Apportionment Part 1 MATH 1014

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This 3 page Class Notes was uploaded by Amy Brogan on Saturday March 12, 2016. The Class Notes belongs to MATH 1014 at University of Cincinnati taught by Mary Koshar in Spring 2016. Since its upload, it has received 17 views. For similar materials see Mathematics of Social Choice in Mathematics (M) at University of Cincinnati.

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Date Created: 03/12/16

Koshar Amy Brogan March 11, 2016 Week 9: Apportionment How does the House of Representatives determine how many representatives each state gets? It’s by population, but what is the procedure? Consider the following: If there is going to be 1 rep. for each group of 30,000 people, and there are 309 million people living in the states as of 2010, how many reps is that? 309 million people / 30,000 = 309/0.03 = 10,300 reps. If this were true, then Ohio with a population of 10,300,00 (a multiple of the product above) would have 10 representatives, and Delaware with a population of 82,400 would have 8 representatives. In reality, the House of Representatives is capped at 435 members (as of 1929). So with the same population: 309 million people / 435 representatives = 710,000 people per representative. With Ohio’s population in 2010 of 11.54 million this comes out to 16.253 representatives. It’s not possible to send only a portion of a person, so how do we know how many people to send to the House? Example 1: 3 counties want to divide up 15 forensic devices between them evenly by population (in the thousands): A 18 B 30 C 42 Total pop: 90 We then find out how many devices there should be per person: 90 thousand people / 15 devices = 6 thousand people per device Then using this we divide the populations above by the product. In this case it comes out even, but this is not a usual occurrence. County population # of machines A 18 7 devices B 30 5 devices C 42 3 devices Double check your work by making sure the outcome adds up to the number of items being apportioned: 7 + 5 + 3 = 15 Example 2: Hamilton Method 5 colonies are coming together to form a new country and they want to make a Building of Representatives. They want to apportion the reps by population, and want to cap the number of reps at 10 so that there is 1 representative per 2,000 people. Island Population (in thousands) Carian 5.1 Wata 7.8 Hulatos 2.3 Diti 3.5 Guaut 1.5 Total 20.2 Using the steps from before: 90 thousand people / 10 representatives = 2.02 thousand people per rep. This outcome is the standard devisor and next we divide each population by it to get to the number of representatives per island. Island Population (in thousands) Divided by 2.02 Carian 5.1 2.25 Wata 7.8 3.861 Hulatos 2.3 1.139 Diti 3.5 1.733 Guaut 1.5 .743 Total 20.2 In the next step we are going to round down to see if the number of representatives will come out even. Island Pop. (thous.) Divided by 2.02 Round Down Carian 5.1 2.25 2 Wata 7.8 3.861 3 Hulatos 2.3 1.139 1 Diti 3.5 1.733 1 Guaut 1.5 .743 0 Total 20.2 7 (not 10) It does not, so next we look back at the portion values and see which ones were closest to being at the next whole value by rounding up. Island Pop. (thous.) Divided by 2.02 Round Down Round Up End Result Carian 5.1 2.25 2 2 Wata 7.8 3.861 3 + 1 4 Hulatos 2.3 1.139 1 1 Diti 3.5 1.733 1 + 1 2 Guaut 1.5 .743 0 + 1 1 Total 20.2 7 (not 10) 10 Identification Practice: Which of the following are an example of an Apportionment problem? 1. Determine how much water each person is rationed in a life boat. 2. Determine how many computers will go to each school 3. How much money will each child get for doing chores? 4. How many sections of calculous should a school offer in the fall? Answers: 1. Determine how much water each person is rationed in a life boat. a. No, this is not because water can be rationed in smaller and smaller portions that don’t need to be in whole amounts (cups, bottles, etc.) 2. Determine how many computers will go to each school a. Yes, because it is not possible (or desirable) to send only a portion of a computer 3. How much money will each child get for doing chores? a. No, paying for chores done can be in whole dollars, in quarters, nickels, etc. 4. How many sections of calculous should a school offer in the fall? a. Yes, because a school cannot assign only half a class (even though the students might like that).

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