## Thinking Mathematically (6th Edition)

(a) 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. (b) We can find the percent increase in the population in each state. State A: $\frac{680-530}{530}\times 100\% = 28.3\%$ State B: $\frac{1250-990}{990}\times 100\% = 26.3\%$ State C: $\frac{2570-2240}{2240}\times 100\% = 14.7\%$ (c) After the population increase, we can find the new standard divisor. $standard~divisor = \frac{total ~population}{number~of~ seats}$ $standard~divisor = \frac{4500}{24}$ $standard~divisor = 187.5$ 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: $standard ~quota = \frac{population}{standard~divisor}$ $standard~quota = \frac{680}{187.5}$ $standard~quota = 3.63$ State B: $standard ~quota = \frac{population}{standard~divisor}$ $standard~quota = \frac{1250}{187.5}$ $standard~quota = 6.67$ State C: $standard ~quota = \frac{population}{standard~divisor}$ $standard~quota = \frac{2570}{187.5}$ $standard~quota = 13.71$ 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 13 seats. The total number of seats which have been apportioned is 3 + 6 + 13 = 22 seats Since there is a total of 24 seats, there are two surplus seats. The first seat is given to State C because it has the largest decimal part (0.71) in its standard quota. The second seat is given to State B because it has the second largest decimal part (0.67) 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 13 + 1 = 14 seats. The population paradox occurs. Originally, State A was given 4 seats and State B was given 6 seats. Then State A's population increased by 28.3% and State B's population increased by 26.3%. After this, State A was given only 3 seats but State B was given 7 seats. That is, State A lost a seat to State B even though the percent increase of State A's population was larger than the percent increase of State B's population. Therefore, the population paradox occurs.