## Thinking Mathematically (6th Edition)

We can find the standard divisor. $standard~divisor = \frac{total ~population}{number~of~ seats}$ $standard~divisor = \frac{3760}{24}$ $standard~divisor = 156.7$ 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: $standard ~quota = \frac{population}{standard~divisor}$ $standard~quota = \frac{530}{156.7}$ $standard~quota = 3.38$ State B: $standard ~quota = \frac{population}{standard~divisor}$ $standard~quota = \frac{990}{156.7}$ $standard~quota = 6.32$ State C: $standard ~quota = \frac{population}{standard~divisor}$ $standard~quota = \frac{2240}{156.7}$ $standard~quota = 14.29$ 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 3 seats. State B is apportioned 6 seats. State C is apportioned 14 seats. The total number of seats which have been apportioned is 3 + 6 + 14 = 23 seats Since there is a total of 24 seats, there is one surplus seat. One more seat is given to State A because it has the largest decimal part (0.38) in its standard quota. Using Hamilton's method, each state is apportioned the following number of seats: State A is apportioned 3 + 1 = 4 seats. State B is apportioned 6 seats. State C is apportioned 14 seats. We can find the standard divisor when the total number of seats increases from 24 to 25. $standard~divisor = \frac{total ~population}{number~of~ seats}$ $standard~divisor = \frac{3760}{25}$ $standard~divisor = 150.4$ 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: $standard ~quota = \frac{population}{standard~divisor}$ $standard~quota = \frac{530}{150.4}$ $standard~quota = 3.52$ State B: $standard ~quota = \frac{population}{standard~divisor}$ $standard~quota = \frac{990}{150.4}$ $standard~quota = 6.58$ State C: $standard ~quota = \frac{population}{standard~divisor}$ $standard~quota = \frac{2240}{150.4}$ $standard~quota = 14.89$ 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 3 seats. State B is apportioned 6 seats. State C is apportioned 14 seats. The total number of seats which have been apportioned is 3 + 6 + 14 = 23 seats Since there is a total of 25 seats, there are two surplus seats. The first seat is given to State C because it has the largest decimal part (0.89) in its standard quota. The second seat is given to State B because it has the second largest decimal part (0.58) in its standard quota. Using Hamilton's method, each state is apportioned the following number of seats: State A is apportioned 3 seats. State B is apportioned 6 + 1 = 7 seats. State C is apportioned 14 + 1 = 15 seats. We can see that the Alabama paradox occurs. Initially, with a total number of 24 seats, State A was allocated 4 seats. After the total number of seats increased to 25, State A was allocated only 3 seats. Therefore, the Alabama paradox occurs.