## Thinking Mathematically (6th Edition)

(a) We can find the standard divisor. $standard~divisor = \frac{total ~population}{number~of~ seats}$ $standard~divisor = \frac{20,000}{200}$ $standard~divisor = 100$ We can find each state's standard quota. The standard quota of each state is the state's population divided by the standard divisor. State A: $standard ~quota = \frac{population}{standard~divisor}$ $standard~quota = \frac{2224}{100}$ $standard~quota = 22.24$ State B: $standard ~quota = \frac{population}{standard~divisor}$ $standard~quota = \frac{2236}{100}$ $standard~quota = 22.36$ State C: $standard ~quota = \frac{population}{standard~divisor}$ $standard~quota = \frac{2640}{100}$ $standard~quota = 26.40$ State D: $standard ~quota = \frac{population}{standard~divisor}$ $standard~quota = \frac{3030}{100}$ $standard~quota = 30.30$ State E: $standard ~quota = \frac{population}{standard~divisor}$ $standard~quota = \frac{9870}{100}$ $standard~quota = 98.70$ Hamilton's method is an apportionment method that involves rounding each standard quota down to the nearest whole number. Surplus seats are given, one at a time, to the states with the largest decimal parts in their standard quotas until there are no more surplus seats. Initially, each state is apportioned its lower quota. State A is apportioned 22 seats. State B is apportioned 22 seats. State C is apportioned 26 seats. State D is apportioned 30 seats. State E is apportioned 98 seats. The total number of seats which have been apportioned is 22 + 22 + 26 + 30 + 98 = 198 seats Since there is a total of 200 seats, there are two surplus seats. The first seat is given to State E because it has the largest decimal part (0.70) in its standard quota. The first seat is given to State C because it has the largest decimal part (0.40) in its standard quota. Using Hamilton's method, each state is apportioned the following number of seats: State A is apportioned 22 seats. State B is apportioned 22 seats. State C is apportioned 26 + 1 = 27 seats. State D is apportioned 30 seats. State E is apportioned 98 + 1 = 99 seats. (b) We can find the percent increase in the population in each state. State A: $\frac{2424-2224}{2224}\times 100\% = 9.0\%$ State B: $\frac{2436-2236}{2236}\times 100\% = 8.9\%$ State C: $\frac{2740-2640}{2640}\times 100\% = 3.8\%$ State D: $\frac{3130-3030}{3030}\times 100\% = 3.3\%$ State E: $\frac{10,070-9870}{9870}\times 100\% = 2.0\%$ (c) After the population increase, we can find the new standard divisor. $standard~divisor = \frac{total ~population}{number~of~ seats}$ $standard~divisor = \frac{20,800}{200}$ $standard~divisor = 104$ We can find each state's standard quota. The standard quota of each state is the state's population divided by the standard divisor. State A: $standard ~quota = \frac{population}{standard~divisor}$ $standard~quota = \frac{2424}{104}$ $standard~quota = 23.31$ State B: $standard ~quota = \frac{population}{standard~divisor}$ $standard~quota = \frac{2436}{104}$ $standard~quota = 23.42$ State C: $standard ~quota = \frac{population}{standard~divisor}$ $standard~quota = \frac{2740}{104}$ $standard~quota = 26.35$ State D: $standard ~quota = \frac{population}{standard~divisor}$ $standard~quota = \frac{3030}{104}$ $standard~quota = 30.10$ State E: $standard ~quota = \frac{population}{standard~divisor}$ $standard~quota = \frac{9870}{104}$ $standard~quota = 96.83$ Hamilton's method is an apportionment method that involves rounding each standard quota down to the nearest whole number. Surplus seats are given, one at a time, to the states with the largest decimal parts in their standard quotas until there are no more surplus seats. Initially, each state is apportioned its lower quota. State A is apportioned 23 seats. State B is apportioned 23 seats. State C is apportioned 26 seats. State D is apportioned 30 seats. State E is apportioned 96 seats. The total number of seats which have been apportioned is 23 + 23 + 26 + 30 + 96 = 198 seats Since there is a total of 200 seats, there are two surplus seats. The first seat is given to State E because it has the largest decimal part (0.83) in its standard quota. The first seat is given to State B because it has the largest decimal part (0.42) in its standard quota. Using Hamilton's method, each state is apportioned the following number of seats: State A is apportioned 23 seats. State B is apportioned 23 + 1 = 24 seats. State C is apportioned 26 seats. State D is apportioned 30 seats. State E is apportioned 96 + 1 = 97 seats. State A's population increased by 9.0% and it was given one more seat (compared with part a). State B's population increased by 8.9% and it was given two more seats (compared with part a). Since State A was given one more seat after its population increased, State A did not lose any seats to State B. Therefore, the population paradox did not occur.