1.	Explain the Alabama paradox. 	

2.	Explain the population paradox. 	

3.	The Hamilton method of apportionment always satisfies what is called the "quota condition."  What does this mean? 	

4.	Find the geometric mean of 4 and 9. 	

5.	Round the following numbers by the method used in the Hill-Huntington apportionment procedure:  3.67, 9.42, 2.46, and 6.49	

6.	Given the cities and the populations below, use the Hamilton method of apportionment to distribute 25 seats on a regional board.

City	
Population	

Greenville	
34,569	

Riverdale	
27,943	

Oceanside	
21,350	

Parkview	
16,138	

7.	Use the Hamilton method of apportionment to distribute 15 representatives to the three states with the populations shown below.

State	
Population	

A	
253,000	

B	
182,000	

C	
85,000	

8.	Use the Hamilton method of apportionment to distribute 12 seats on a city council to three districts with the populations shown below.

District	
Population	

North	
30,000	

Central	
10,100	

South	
34,900	

9.	Given the cities and the populations below, use the Jefferson method of apportionment to distribute 25 seats on a regional board.

City	
Population	

Greenville	
34,569	

Riverdale	
27,943	

Oceanside	
21,350	

Parkview	
16,138	

10.	Use the Jefferson method of apportionment to distribute 15 representatives to the three states with the populations shown below.

State	
Population	

A	
253,000	

B	
182,000	

C	
85,000	

11.	Use the Jefferson method of apportionment to distribute 12 seats on a city council to three districts with the populations shown below.

District	
Population	

North	
30,000	

Central	
10,100	

South	
34,900	

12.	Given the cities and the populations below, use the Webster method of apportionment to distribute 25 seats on a regional board.

City	
Population	

Greenville	
34,569	

Riverdale	
27,943	

Oceanside	
21,350	

Parkview	
16,138	

13.	Use the Webster method of apportionment to distribute 15 representatives to the three states with the populations shown below.

State	
Population	

A	
253,000	

B	
182,000	

C	
85,000	

14.	Use the Webster method of apportionment to distribute 12 seats on a city council to three districts with the populations shown below.

District	
Population	

North	
30,000	

Central	
10,100	

South	
34,900	

15.	Given the cities and the populations below, use the Hill-Huntington method of apportionment to distribute 25 seats on a regional board.

City	
Population	

Greenville	
34,569	

Riverdale	
27,943	

Oceanside	
21,350	

Parkview	
16,138	

16.	Use the Hill-Huntington method of apportionment to distribute 15 representatives to the three states with the populations shown below.

State	
Population	

A	
253,000	

B	
182,000	

C	
85,000	

17.	Use the Hill-Huntington method of apportionment to distribute 12 seats on a city council to three districts with the populations shown below.

District	
Population	

North	
30,000	

Central	
10,100	

South	
34,900	

18.	A city is divided into four regions for distributing the 20 representatives on a public utility board.  The regions and their populations are shown below.  The board uses the Hamilton method of apportionment to distribute seats.  Find the district population for each region based on this apportionment.

Region	
Population	

North	
8700	

South	
5600	

East	
7200	

West	
3500	

19.	A country has three states with the populations shown below.  The House of Representatives for the country is to have 15 members.  Find the district population for each state if the country uses the Jefferson method of apportionment.

State	
Population	

A	
245,000	

B	
320,000	

C	
155,000	

20.	A city is divided into four regions for distributing the 20 representatives on a public utility board.  The regions and their populations are shown below.  The board uses the Hamilton method of apportionment to distribute seats.  Find the representative share for each region based on this apportionment.

Region	
Population	

North	
8700	

South	
5600	

East	
7200	

West	
3500	

21.	A country has three states with the populations shown below.  The House of Representatives for the country is to have 15 members.  Find the representative share for each state if the country uses the Jefferson method of apportionment.

State	
Population	

A	
245,000	

B	
320,000	

C	
155,000	

22.	A town board has 10 members, divided among three regions of the town using the Hamilton method of apportionment.  The regions and their populations are shown below. Find the difference in district population for the North and South region.

Region	
Population	

North	
13,400	

Central	
21,900	

South	
8700	

23.	A small country has four states and a House of Representatives with 20 seats.  Given the states, populations, and apportionment below, find the relative difference in district population between states A and C.

State	
Population	
Apportionment	

A	
234,000	
5	

B	
124,000	
3	

C	
346,500	
7	

D	
268,500	
5	

24.	A town board has 10 members, divided among three regions of the town using the Hamilton method of apportionment.  The regions and their populations are given below.  Find the difference in representative share for the North and South region.

Region	
Population	

North	
13,400	

Central	
21,900	

South	
8700	

25.	A small country has four states and a House of Representatives with 20 seats.  Given the states, populations and apportionment below, find the relative difference in representative share between states A and C.

State	
Population	
Apportionment	

A	
234,000	
5	

B	
124,000	
3	

C	
346,500	
7	

D	
268,500	
5	

26.	The manager of a busy restaurant is working on a schedule for the restaurant's 25 food servers. There are three shifts, as identified below:

Shift	
Average number of customers	

1	
235	

2	
610	

3	
453	

Use the Jefferson method to determine how many food servers should be scheduled for each shift. 	

27.	The manager of a busy restaurant is working on a schedule for the restaurant's 25 food servers. There are three shifts, as identified below:

Shift	
Average number of customers	

1	
164	

2	
560	

3	
328	

Use the Webster method to determine how many food servers should be scheduled for each shift. 	

28.	The manager of a busy restaurant is working on a schedule for the restaurant's 25 food servers. There are three shifts, as identified below:

Shift	
Average number of customers	

1	
124	

2	
480	

3	
391	

Use the Hill-Huntington method to determine how many food servers should be scheduled for each shift. 	

29.	The manager of a busy restaurant is working on a schedule for the restaurant's 25 food servers. There are three shifts, as identified below:

Shift	
Average number of customers	

1	
122	

2	
451	

3	
674	

Use the Hamilton method to determine how many food servers should be scheduled for each shift. 	

30.	Find the geometric mean of 10 and 6. 	

Answer Key

1.	The Alabama paradox occurs when a state loses a seat as a result of an increase in the House size.  It can happen when an increase in House size causes the quota for some states to increase more than others and a state that originally assigned its upper quota then receives only its lower quota.	
2.	The population paradox occurs when one state's apportionment decreases and another state's apportionment increases even though the first state has gained population and the second state has lost population.	
3.	The quota in an apportionment problem is the exact share that would be allocated if a whole number were not required.  A method is said to satisfy the quota condition if each state's apportionment is equal to its lower quota (the largest whole number no more than the quota) or its upper quota (the smallest whole number no less than the quota).	
4.	The geometric mean of 4 and 9 is Ö36 = 6.	
5.	3.67 rounds to 4
9.42 rounds to 9
2.46 rounds to 3
6.49 rounds to 7	
6.	Greenville  9
Riverdale   7
Oceanside  5
Parkview   4	
7.	A  7
B  5
C  3	
8.	North    5
Central  2
South    5	
9.	Greenville  9
Riverdale   7
Oceanside  5
Parkview   4	
10.	A  8
B  5
C  2	
11.	North    5
Central  1
South    6	
12.	Greenville  9
Riverdale   7
Oceanside  5
Parkview   4	
13.	A  7
B  5
C  3	
14.	North    5
Central  2
South    5	
15.	Greenville  9
Riverdale   7
Oceanside  5
Parkview   4	
16.	A  7
B  5
C  3	
17.	North    5
Central  2
South    5	
18.	North  1242.86
South  1400
East    1200
West   1166.67	
19.	A  49000
B  45714.29
C  51666.67	
20.	North  .0008046
South  .0007143
East    .0008333
West   .0008571	
21.	A  .0000204
B  .0000219
C  .0000194	
22.	116.67	
23.	5.77%	
24.	6 ´ 10–6	
25.	5.79%	
26.	4, 12, 9	
27.	4, 13, 8	
28.	3, 12, 10	
29.	2, 9, 14	
30.	7.75