Thinking Mathematically (6th Edition)

Published by Pearson
ISBN 10: 0321867327
ISBN 13: 978-0-32186-732-2

Chapter 13 - Voting and Apportionment - Chapter 13 Test: 23

Answer

When there were 10 doctors, Clinic A was apportioned 3 doctors. After the total number of doctors increased from 10 to 11, Clinic A was only apportioned 2 doctors. Therefore, the Alabama paradox occurs.

Work Step by Step

We can find the total patient load. total load = 119 + 165 + 216 = 500 We can find the standard divisor when there are 10 doctors. $standard~divisor = \frac{total~load}{number~of~ doctors}$ $standard~divisor = \frac{500}{10}$ $standard~divisor = 50$ The standard divisor is 50. We can find the standard quota for each clinic. Clinic A: $standard~quota = \frac{patient~load}{standard~divisor}$ $standard~quota = \frac{119}{50}$ $standard~quota = 2.38$ Clinic B: $standard~quota = \frac{patient~load}{standard~divisor}$ $standard~quota = \frac{165}{50}$ $standard~quota = 3.30$ Clinic C: $standard~quota = \frac{patient~load}{standard~divisor}$ $standard~quota = \frac{216}{50}$ $standard~quota = 4.32$ Hamilton's method is an apportionment method that involves rounding each standard quota down to the nearest whole number. Surplus doctors are given, one at a time, to the clinics with the largest decimal parts in their standard quotas until there are no more surplus doctors. Initially, each clinic is apportioned its lower quota. Clinic A is apportioned 2 doctors. Clinic B is apportioned 3 doctors. Clinic C is apportioned 4 doctors. The total number of doctors which have been apportioned is 2 + 3 + 4 = 9 doctors Since there is a total of 10 doctors, there is one surplus doctor. One more doctor is given to Clinic A because it has the largest decimal part (0.38) in its standard quota. Using Hamilton's method, each clinic is apportioned the following number of doctors: Clinic A is apportioned 2 + 1 = 3 doctors. Clinic B is apportioned 3 doctors. Clinic C is apportioned 4 doctors. Let's suppose the number of doctors is increased from 10 to 11. We can find the standard divisor when there are 11 doctors. $standard~divisor = \frac{total~load}{number~of~ doctors}$ $standard~divisor = \frac{500}{11}$ $standard~divisor = 45.45$ The standard divisor is 45.45. We can find the standard quota for each clinic. Clinic A: $standard~quota = \frac{patient~load}{standard~divisor}$ $standard~quota = \frac{119}{45.45}$ $standard~quota = 2.62$ Clinic B: $standard~quota = \frac{patient~load}{standard~divisor}$ $standard~quota = \frac{165}{45.45}$ $standard~quota = 3.63$ Clinic C: $standard~quota = \frac{patient~load}{standard~divisor}$ $standard~quota = \frac{216}{45.45}$ $standard~quota = 4.75$ Hamilton's method is an apportionment method that involves rounding each standard quota down to the nearest whole number. Surplus doctors are given, one at a time, to the clinics with the largest decimal parts in their standard quotas until there are no more surplus doctors. Initially, each clinic is apportioned its lower quota. Clinic A is apportioned 2 doctors. Clinic B is apportioned 3 doctors. Clinic C is apportioned 4 doctors. The total number of doctors which have been apportioned is 2 + 3 + 4 = 9 doctors Since there is a total of 11 doctors, there are two surplus doctors. The first doctor is given to Clinic C because it has the largest decimal part (0.75) in its standard quota. The second doctor is given to Clinic B because it has the second largest decimal part (0.63) in its standard quota. Using Hamilton's method, each clinic is apportioned the following number of doctors: Clinic A is apportioned 2 doctors. Clinic B is apportioned 3 + 1 = 4 doctors. Clinic C is apportioned 4 + 1 = 5 doctors. When there were 10 doctors, Clinic A was apportioned 3 doctors. After the total number of doctors increased from 10 to 11, Clinic A was only apportioned 2 doctors. Therefore, the Alabama paradox occurs.
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