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: 21

Answer

Using Adams's method, each shift is apportioned the following number of nurses: Shift A is apportioned 57 nurses. Shift B is apportioned 81 nurses. Shift C is apportioned 68 nurses. Shift D is apportioned 44 nurses.

Work Step by Step

We can find the total number of patients. total patients = 453 + 650 + 547 + 350 total patients = 2000 We can find the standard divisor. $standard~divisor = \frac{total ~patients}{number~of~ nurses}$ $standard~divisor = \frac{2000}{250}$ $standard~divisor = 8$ We can find each shift's standard quota. The standard quota of each shift is the number of patients during the shift divided by the standard divisor. Shift A: $standard ~quota = \frac{patients}{standard~divisor}$ $standard~quota = \frac{453}{8}$ $standard~quota = 56.63$ Shift B: $standard ~quota = \frac{patients}{standard~divisor}$ $standard~quota = \frac{650}{8}$ $standard~quota = 81.25$ Shift C: $standard ~quota = \frac{patients}{standard~divisor}$ $standard~quota = \frac{547}{8}$ $standard~quota = 68.38$ Shift D: $standard ~quota = \frac{patients}{standard~divisor}$ $standard~quota = \frac{350}{8}$ $standard~quota = 43.75$ If each shift is apportioned its upper quota, the number of nurses apportioned is 57 + 82 + 69 + 44 which is 252 nurses. Since there is a total of 250 nurses available, there are too many apportioned nurses. To obtain a sum of 250 nurses, we need to find a modified divisor that is slightly more than the standard divisor. Let's choose a modified divisor of 8.05. 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 shift. Shift A: $ modified ~quota = \frac{patients}{modified~divisor}$ $ modified ~quota = \frac{453}{8.05}$ $ modified ~quota = 56.27$ Shift B: $ modified ~quota = \frac{patients}{modified ~divisor}$ $modified ~quota = \frac{650}{8.05}$ $modified ~quota = 80.75$ Shift C: $modified ~quota = \frac{patients}{modified ~divisor}$ $modified ~quota = \frac{547}{8.05}$ $modified ~quota = 67.95$ Shift D: $modified ~quota = \frac{patients}{modified ~divisor}$ $modified ~quota = \frac{350}{8.05}$ $modified ~quota = 43.48$ Using Adams's method, each shift is apportioned the upper quota of the modified quota. Each shift is apportioned the following number of nurses: Shift A is apportioned 57 nurses. Shift B is apportioned 81 nurses. Shift C is apportioned 68 nurses. Shift D is apportioned 44 nurses. Note that the total number of nurses apportioned is 250, so using a modified divisor of 8.05 is acceptable.
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