Thinking Mathematically (6th Edition)

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

Chapter 13 - Voting and Apportionment - 13.3 Apportionment Method - Exercise Set 13.3: 10

Answer

Using Jefferson's method, the doctors are apportioned to the clinics as follows: Clinic A is apportioned 7 doctors. Clinic B is apportioned 15 doctors. Clinic C is apportioned 9 doctors. Clinic D is apportioned 17 doctors. Clinic E is apportioned 10 doctors. Clinic F is apportioned 12 doctors.

Work Step by Step

We can find the total patient load. total load = 316 + 598 + 396 + 692 + 426 + 486 total load = 2914 We can find the standard divisor. $standard ~divisor = \frac{total ~patient~load}{doctors}$ $standard ~divisor = \frac{2914}{70}$ $standard ~divisor = 41.63$ If we use the standard divisor and round each standard quota down to the nearest whole number, the sum of the apportioned doctors will be less than 70 doctors. To obtain a sum of 70 doctors, we need to find a modified divisor that is slightly less than the standard divisor. Using the hint in the question, let's choose a modified divisor of 39.7. Note that it may require a bit of trial-and-error to find a modified divisor that works. We can find the modified quota for each clinic. Clinic A: $modified~quota = \frac{patient~load}{modified~divisor}$ $modified~quota = \frac{316}{39.7}$ $modified~quota = 7.96$ Clinic B: $modified~quota = \frac{patient~load}{modified~divisor}$ $modified~quota = \frac{598}{39.7}$ $modified~quota = 15.06$ Clinic C: $modified~quota = \frac{patient~load}{modified~divisor}$ $modified~quota = \frac{396}{39.7}$ $modified~quota = 9.97$ Clinic D: $modified~quota = \frac{patient~load}{modified~divisor}$ $modified~quota = \frac{692}{39.7}$ $modified~quota = 17.43$ Clinic E: $modified~quota = \frac{patient~load}{modified~divisor}$ $modified~quota = \frac{426}{39.7}$ $modified~quota = 10.73$ Clinic F: $modified~quota = \frac{patient~load}{modified~divisor}$ $modified~quota = \frac{486}{39.7}$ $modified~quota = 12.24$ Using Jefferson's method, we apportion the doctors by rounding the modified quota down to the nearest whole number. The doctors are apportioned to the clinics as follows: Clinic A is apportioned 7 doctors. Clinic B is apportioned 15 doctors. Clinic C is apportioned 9 doctors. Clinic D is apportioned 17 doctors. Clinic E is apportioned 10 doctors. Clinic F is apportioned 12 doctors. Note that the sum of the apportioned doctors is 70 doctors, so using 39.7 as a modified divisor is acceptable.
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