14.4 Flaws of the Apportionment Methods 951 Exercises Warm Up Exercises In Exercises 1– 6, fill in the blank with an appropriate word, phrase, or symbol(s). 1. When group A loses an item or items to group B even though group A’s population grew at a faster rate than group B’s, the ________ paradox occurs. Population 2. When the addition of a new group and additional items to be apportioned reduces the prior apportionment of another group, the ________ paradox occurs. New-states 3. When an increase in the total number of items to be apportioned results in a loss of an item for a group, the ________ paradox occurs. Alabama 4. Hamilton’s and Jefferson’s apportionment methods, favor ________ states. Large 5. Adams’ and Webster’s apportionment methods favor ________ states. Small 6. The apportionment method that satisfies the quota rule but may produce a paradox is called ________ method. Hamilton’s Practice the Skills/Problem Solving In Exercises 7–18, when appropriate, round quotas and divisors to the nearest hundredth. 7. Doctors Consider the apportionment of 60 doctors for First Physicians Organization given in Example 2 on page 931 of Section 14.3. The apportionment using Hamilton’s method is shown in the table below. Office A B C D Total Employees 739 277 618 958 2592 a) Apportion the printers using Hamilton’s method. 41, 16, 34, 53 b) Does the Alabama paradox occur using Hamilton’s method if the number of new printers is increased from 144 to 145? Explain your answer. * 9. Legislative Seats A country with three states has 30 seats in the legislature. The population of each state is shown in the table below. State A B C Total Population 161 250 489 900 a) Apportion the seats using Hamilton’s method. 6, 8, 16 b) Does the Alabama paradox occur using Hamilton’s method if the number of seats is increased from 30 to 31? Explain your answer. Yes. When the number of seats increases, states B and C gain a seat and state A loses a seat. 10. Legislative Seats A country with three states has 200 seats in the legislature. The population of each state is shown in the table below. State A B C Total Population 247,100 481,900 271,000 1,000,000 a) Apportion the seats using Hamilton’s method. 50, 96, 54 b) Does the Alabama paradox occur using Hamilton’s method if the number of seats is increased from 200 to 201? Explain your answer. No. The new apportionment is 50, 97, 54. No state loses a seat so the Alabama paradox does not occur. In Exercises 11–14, assume that the number of items to be apportioned does not change. 11. Promotions Spectrum has 30,000 employees in three cities as shown in the table below. It wishes to give promotions to 200 employees. Cities A B C Total Employees 9130 6030 14,840 30,000 a) Apportion the promotions using Hamilton’s method. 61, 40, 99 b) Suppose that in 10 years the cities have the following number of employees and the company wishes to again give promotions to 200 employees. Does the population paradox occur using Hamilton’s method? No Cities A B C Total Employees 9150 6030 14,945 30,125 SECTION 14.4 Clinic A B C D ETotal Patients 246 201 196 211 226 1080 Standard quota 13.67 11.17 10.89 11.72 12.56 60.01 Lower quota 13 11 10 11 12 57 Hamilton’s apportionment 14 11 11 12 12 60 Does the Alabama paradox occur using Hamilton’s method if the number of doctors is increased from 60 to 61? Explain your answer. No. The new apportionment is 14, 11, 11, 12, 13. No clinic suffers a loss, so the Alabama paradox doesn’t occur. 8. Color Printers A large company with offices in four cities must distribute 144 new color printers to the four offices. The printers will be apportioned based on the number of employees in each office as shown in the following table. Andrey Popov/Shutterstock *See Instructor Answer Appendix
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