#### Answer

(a) 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 27 seats.
State D is apportioned 30 seats.
State E is apportioned 99 seats.
(b) State A's population increased 9.0 %
State B's population increased 8.9 %
State C's population increased 3.8 %
State D's population increased 3.3 %
State E's population increased 2.0 %
(c) Using Hamilton's method, each state is apportioned the following number of seats:
State A is apportioned 23 seats.
State B is apportioned 24 seats.
State C is apportioned 26 seats.
State D is apportioned 30 seats.
State E is apportioned 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.

#### Work Step by Step

(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.