## Thinking Mathematically (6th Edition)

(a) We can find the standard divisor. $standard~divisor = \frac{total ~population}{number~of~ seats}$ $standard~divisor = \frac{20,000}{40}$ $standard~divisor = 500$ 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: $standard ~quota = \frac{population}{standard~divisor}$ $standard~quota = \frac{680}{500}$ $standard~quota = 1.36$ State B: $standard ~quota = \frac{population}{standard~divisor}$ $standard~quota = \frac{9150}{500}$ $standard~quota = 18.3$ State C: $standard ~quota = \frac{population}{standard~divisor}$ $standard~quota = \frac{10,170}{500}$ $standard~quota = 20.34$ 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 1 seat. State B is apportioned 18 seats. State C is apportioned 20 seats. The total number of seats which have been apportioned is 1 + 18 + 20 = 39 seats Since there is a total of 40 seats, there is one surplus seat. One more seat is given to State A because it has the largest decimal part (0.36) in its standard quota. Using Hamilton's method, each state is apportioned the following number of seats: State A is apportioned 1 + 1 = 2 seats. State B is apportioned 18 seats. State C is apportioned 20 seats. (b) We can find the standard divisor. $standard~divisor = \frac{total ~population}{number~of~ seats}$ $standard~divisor = \frac{20,000}{41}$ $standard~divisor = 487.8$ 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: $standard ~quota = \frac{population}{standard~divisor}$ $standard~quota = \frac{680}{487.8}$ $standard~quota = 1.39$ State B: $standard ~quota = \frac{population}{standard~divisor}$ $standard~quota = \frac{9150}{487.8}$ $standard~quota = 18.76$ State C: $standard ~quota = \frac{population}{standard~divisor}$ $standard~quota = \frac{10,170}{487.8}$ $standard~quota = 20.85$ 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 1 seat. State B is apportioned 18 seats. State C is apportioned 20 seats. The total number of seats which have been apportioned is 1 + 18 + 20 = 39 seats Since there is a total of 41 seats, there are two surplus seats. The first seat is given to State C because it has the largest decimal part (0.85) in its standard quota. The second seat is given to State B because it has the second largest decimal part (0.76) in its standard quota. Using Hamilton's method, each state is apportioned the following number of seats: State A is apportioned 1 seat. State B is apportioned 18 + 1 = 19 seats. State C is apportioned 20 + 1 = 21 seats. We can see that the Alabama paradox occurs. Initially, with 40 seats, State A was allocated 2 seats. After the number of seats increased to 41, State A was allocated only 1 seat. Therefore, the Alabama paradox occurs.