A small country consists of four states. There are 100 seats in the legislature. The populations are: A: 44,800 B: 52,200 C: 49,200 D: 53,800

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1 Finite Math A Mrs. Leahy Chapter 4 Test Review Practice Questions Vocabulary: Be prepared to identify/define the following terms and methods. Standard Divisor, Standard Quota, Hamilton s Method, Webster s Method, Huntington-Hill Method, The Quota Rule, Upper Quota Violation, Lower Quota Violation, Alabama Paradox, Population Paradox, New States Paradox, Geometric Means, Huntington Hill Rounding Rules A small country consists of four states. There are 100 seats in the legislature. The populations are: A: 44,800 B: 52,200 C: 49,200 D: 53, What is the standard divisor? 2. What does the standard divisor represent in this problem? 3. What is the standard quota for state A? 4. What is the standard quota of state C? 5. Use Hamilton s Method to apportion the 100 seats. What is the final apportionment? A country has five states and 240 seats in the legislature. The populations of the states are as follows: A: 427,000 B: 754,000 C: 4,389,000 D: 3,873,000 E: 157, What is the standard divisor? 7. What is the standard quota for state D? 8. What is the standard quota for state E? 9. Use Hamilton s Method to apportion the 240 seats. What is the final apportionment? 10. Use a divisor of D= 40,100 to find each state s modified quotas to three decimal places. 11. Use Webster s Method (and your answers from #10) to apportion the 240 seats. What is the final apportionment? A bus company operates four bus routes (A, B, C, D) and 100 buses. The buses are apportioned among the routes on the basis of average daily passengers per route which is given below: A: 12,444 B: 36,503 C: 19,581 D: 31, What is the standard divisor? 13. What is the standard quota for each state? 14. Use Hamilton s Method to find the apportionment. What is the final apportionment? 15. In this problem, the SD works as a suitable divisor for Webster s method. Apportion using Webster s method. (Use your answers from #13 to help you.)

2 The following information refers to a country with six states. There are 250 seats in the legislature and the populations are given below. A: 1456 B: 6733 C: 557 D: 2081 E: 685 F: What is the standard divisor? 17. Use Hamilton s Method to find the final apportionment. 18. Which state received the first extra seat in #17 s apportionment? 19. Use a modified divisor of D=50.1 to find each state s modified quotas. Round to three decimal places. 20. Use Webster s Method (and your answers from #19) to find the final apportionment. Find the geometric mean of the following and and and and 26 Round the given quota using the Huntington Hill Rounding Rules. 25. q = q = q = Q = For the following, write ALL THAT APPLY: Hamilton, Jefferson, Adams, Webster, no method exists 29. Which method(s) does not violate the quota rule? 30. Which method(s) does not suffer from mathematical paradoxes? 31. Which method(s) does not violate the quota rule and does not suffer from paradoxes? Quota Rule and Paradox Practice: 32. In a certain apportionment problem, State X has a standard quota of The final apportionment to State X is 50 seats. This is called A) an upper-quota violation. B) the Alabama paradox. C) the population paradox. D) a lower-quota violation. E) none of these 33. In a certain apportionment problem, State X has a standard quota of The final apportionment to State X is 47 seats. This is called A) an upper-quota violation. B) the Alabama paradox. C) a lower-quota violation.

3 34. In a certain apportionment problem, State X has a standard quota of The final apportionment to State X is 75 seats. This is called A) a lower-quota violation. B) an upper-quota violation. C) the Alabama paradox. 35. In a certain apportionment problem, State X has a standard quota of The final apportionment to State X is 72 seats. This is called A) the Alabama paradox. B) an upper-quota violation. C) a lower-quota violation. 36. Under a certain apportionment method, a state receives an apportionment of 52 seats when the total number of seats in the legislature is 334 but only 51 seats when the total number of seats in the legislature is 335. This is called A) the Alabama paradox. B) the new states paradox. C) the population paradox. D) a violation of the quota rule. E) none of these 37. Under a certain apportionment method, a state receives an apportionment of 42 seats when the total number of seats in the legislature is 234 but only 41 seats when the total number of seats in the legislature is 235. This is called A) the population paradox. B) a violation of the quota rule. C) the new states paradox. D) the Alabama paradox. E) none of these 38. Which apportionment method does not violate the quota rule, and does not suffer from any of the paradoxes? A) Hamilton's method B) Webster's method C) Adam's method D) Jefferson's method E) There is no such method. 39. Under a certain apportionment method State X receives 51 seats and State Y receives 19 seats. Ten years later, the population of State X has increased by 3%, while the population of State Y remains unchanged. Under the same apportionment method, State X now receives 50 seats and State Y receives 20 seats. This is called A) the population paradox. B) a violation of the quota rule. C) the Alabama paradox. 40. Under a certain apportionment method State X receives 41 seats and State Y receives 29 seats. Ten years later, the population of State X has increased by 5%, while the population of State Y remains unchanged. Under the same apportionment method, State X now receives 40 seats and State Y receives 30 seats. This is called A) the Alabama paradox. B) a violation of the quota rule. C) the population paradox.

4 41. A father wishes to distribute 16 pieces of candy among his 3 children (Abe, Betty, and Cindy) based on the number of hours each child spends doing chores around the house. Using a certain apportionment method, he has determined that Abe is to get 9 pieces of candy, Betty is to get 4 pieces, and Cindy is to get 3 pieces. However, just before he hands out the candy, he discovers that he has 17 pieces (not 16) of candy. When he apportions the 17 pieces of candy using the same apportionment method, Abe ends up with 10 pieces, Betty with 5 pieces, and Cindy with 2 pieces. This is an example of A) the Alabama paradox. B) a violation of the quota rule. C) the population paradox. 42. A mother wishes to distribute 11 pieces of candy among her three children (Abe, Betty, and Cindy) based on the number of hours each child spends doing chores around the house. Using a certain apportionment method, she has determined that (based on 703 minutes work for Abe, 243 minutes work for Betty, and 54 minutes work for Cindy) that Abe is to get 8 pieces of the candy, Betty is to get 3 pieces, and Cindy is to get 0 pieces. However, just before she hands out the candy, she discovers that she forgot to include 86 minutes of work that Abe did, 12 minutes of work that Betty did, and 2 minutes of work that Cindy did. When she reapportions the 11 pieces of candy using the corrected times and using the same apportionment method, Abe ends up with 8 pieces, Betty with 2 pieces, and Cindy with 1 piece. This is an example of A) the Alabama paradox. B) the population paradox. C) a violation of the quota rule. 43. A mother wishes to distribute 16 pieces of candy among her three children (Abe, Betty, and Cindy) based on the number of hours each child spends doing chores around the house. Using a certain apportionment method, she has determined that Abe is to get 9 pieces of the candy, Betty is to get 4 pieces, and Cindy is to get 3 pieces. However, just before she hands out the candy, she discovers that a neighbor (Dave) has been helping her children with the household chores. Since Dave has worked the same number of hours as Cindy, the mother decides to add 3 more pieces of candy and distribute 19 pieces. When she apportions the 19 pieces using the same apportionment method, Abe ends up with 10 pieces, Betty with 3 pieces, Cindy with 3 pieces, and Dave with 3 pieces. This is an example of A) the Alabama paradox. B) the population paradox. C) the new states paradox. D) a violation of the quota rule. E) none of these 44. A mother wishes to distribute 16 pieces of candy among her three children (Abe, Betty, and Cindy) based on the number of hours each child spends doing chores around the house. Based on the time each child spent doing chores, Abe deserves 7.8 pieces of candy, Betty deserves 4.8 pieces, and Cindy deserves 3.4 pieces. Using a certain apportionment method, Abe ends up with 9 pieces of the candy, Betty ends up with 4 pieces, and Cindy ends up with 3 pieces. This is an example of A) the new states paradox. B) a violation of the quota rule. C) the population paradox. D) the Alabama paradox. E) none of these

5 Solutions people per seat 1 seat for every 2000 people A = 22, B = 26, C = 25, D = , A = 10, B = 19, C = 110, D = 97, E = A = B = C = D = E = A =11, B = 19, C = 109, D = 97, E = A 33. C 34. B 35. C 36. A 37. D 38. E 39. A 40. C 41. A 42 B 43. C 44. B A = B = C = D = A = 12, B = 37, C = 20, D = A = 12, B = 37, C = 20, D = A = 29, B = 135, C = 11, D = 41, E = 14, F = F 19. A = B = C = D = E = A = 29, B = 134, C = 11, D = 42, E = 14, F = Hamilton 30. Jefferson, Adams, Webster 31. No such method exists