#### Answer

(a) Using Hamilton's method, each state is apportioned the following number of seats:
State A is apportioned 4 seats.
State B is apportioned 6 seats.
State C is apportioned 14 seats.
(b) State A's population increased 28.3 %
State B's population increased 26.3 %
State C's population increased 14.7 %
(c) Using Hamilton's method, each state is apportioned the following number of seats:
State A is apportioned 3 seats.
State B is apportioned 7 seats.
State C is apportioned 14 seats.
The population paradox occurs. 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.

#### Work Step by Step

(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.