1.	The Hamilton method of apportionment can display the population paradox.	
	A)  True    B)  False	

2.	The Jefferson method of apportionment can display the population paradox.	
	A)  True    B)  False	

3.	The Webster method of apportionment can display the population paradox.	
	A)  True    B)  False	

4.	The Jefferson method of apportionment can display the Alabama paradox.	
	A)  True    B)  False	

5.	An apportionment method exists which satisfies the quota condition and is free from both the population paradox and the Alabama paradox.	
	A)  True    B)  False	

6.	Which method of apportionment always satisfies the quota condition?	
	A)  Hamilton    B)  Hill-Huntington    C)  Jefferson    D)  Webster	

7.	For a given set of populations and house sizes, different methods of apportionment may lead to different apportionments.	
	A)  True    B)  False	

8.	For a given set of populations and house sizes, different methods of apportionment may lead to the same apportionment.	
	A)  True    B)  False	

9.	Which of the following is a true statement?	
A)	Jefferson's method of apportionment is not biased with respect to a state's population.	
B)	Jefferson's method of apportionment is biased toward states with smaller populations.	
C)	Jefferson's method of apportionment is biased toward states with larger populations.	

10.	A county is divided into three districts with the populations shown below.  There are 10 seats on the county council that need to be apportioned.  Find the quota for the Applewood district.

District	
Population	

Applewood	
8280	

Boxwood	
4600	

Central	
5220	

	A)  4.57    B)  4    C)  5    D)  8.28	

11.	We are scheduling seven course sections for a total of 217 students.  Enrollments are: 109 in Calculus I, 79 in Calculus II, and 29 in Advanced Calculus.  Find the quota of sections for Calculus II.	
	A)  2    B)  2.55    C)  3.64    D)  3	

12.	A small county has populations in three districts as shown below.  They are to apportion 10 seats on the county council.  Find the quota for Riverdale. 

District	
Population	

Parkview	
43,000	

Hillside	
32,800	

Riverdale	
24,200	

	A)  4.13    B)  2    C)  4    D)  2.42	

13.	A county is divided into three districts with the populations shown below.  There are 10 seats on the county council that need to be apportioned, using the Hamilton method.  Find the apportionment for the Applewood district.

District	
Population	

Applewood	
8280	

Boxwood	
4600	

Central	
5220	

	A)  4    B)  3    C)  5    D)  2	

14.	We are scheduling seven course sections for a total of 217 students.  Enrollments are: 109 in Calculus I, 79 in Calculus II, 29 in Advanced Calculus.  Find the apportionment for Advanced Calculus using the Hamilton method.	
	A)  0    B)  1    C)  2    D)  3	

15.	A county is divided into three districts with the populations shown below.  There are 10 seats on the county council that need to be apportioned, using the Hamilton method.  Find the apportionment for the Boxwood district.

District	
Population	

Applewood	
8280	

Boxwood	
4600	

Central	
5220	

	A)  2    B)  3    C)  1    D)  4	

16.	A county is divided into three districts with the populations shown below.  There are 10 seats on the county council that need to be apportioned. If the county uses the Jefferson method, what would be the first critical multiplier for the Applewood district?

District	
Population	

Applewood	
8280	

Boxwood	
4600	

Central	
5220	

	A)  1.457    B)  1.1435    C)  1.093    D)  1	

17.	A county has three districts with the populations shown below.  The 11 seats on the county council are to be apportioned using the Jefferson method.  Find the apportionment for each district.

District	
Population	

A	
43,000	

B	
32,800	

C	
24,200	

	A)  5, 3, 3    B)  5, 4, 2    C)  6, 3, 2    D)  4, 4, 3	

18.	A state has four districts with the populations shown below.  The House of Representatives has 20 seats that are to be apportioned using the Jefferson method.  Find the first critical multiplier for District A.

District	
Population	

A	
87,000	

B	
56,000	

C	
72,000	

D	
35,000	

	A)  1.0057    B)  1.16    C)  1.348    D)  1.696	

19.	A county is divided into three districts with the populations shown below.  There are 10 seats on the county council that need to be apportioned. If the county uses the Jefferson method, what would be the initial multiplier used?

District	
Population	

Applewood	
8280	

Boxwood	
4600	

Central	
5220	

	A)  1.3575    B)  1.221    C)  1.093    D)  1.040	

20.	A county has three districts with the populations shown below.  The 11 seats on the county council are to be apportioned using the Jefferson method.  Find the initial multiplier that would be used.

District	
Population	

A	
43,000	

B	
32,800	

C	
24,200	

	A)  1.226    B)  1.127    C)  1.109    D)  1.057	

21.	A county has three districts with the populations shown below.  The 11 seats on the county council are to be apportioned using the Webster method.  Find the first critical multiplier for District A.

District	
Population	

A	
43,000	

B	
32,800	

C	
24,200	

	A)  1.0511    B)  1.0309    C)  0.9701    D)  0.9514	

22.	A county has three districts with the populations shown below.  The 11 seats on the county council are to be apportioned using the Webster method.  Find the apportionment for each district.

District	
Population	

A	
43,000	

B	
32,800	

C	
24,200	

	A)  5, 4, 2    B)  5, 3, 3    C)  4, 4, 3    D)  6, 3, 2	

23.	A state has four districts with the populations shown below.  The House of Representatives has 20 seats that are to be apportioned using the Jefferson method.  Find the apportionment for each district.

District	
Population	

A	
87,000	

B	
56,000	

C	
72,000	

D	
35,000	

	A)  6, 5, 6, 3    B)  7, 4, 5, 4    C)  6, 4, 6, 4    D)  7, 4, 6, 3	

24.	A country has four states with the populations shown below.  The House of Representatives is to have 15 members.  Use the Webster method of apportionment to find the number of seats for each state.

State	
Population	

A	
52,600	

B	
39,900	

C	
34,000	

D	
23,500	

	A)  6, 4, 3, 2    B)  5, 4, 4, 2    C)  6, 3, 3, 3    D)  5, 4, 3, 3	

25.	A small county has populations in three districts as shown below.  They are to apportion 15 seats on the county council using the Webster method.  Find the apportionment for each district.

District	
Population	

Parkview	
43,000	

Hillside	
32,800	

Riverdale	
24,200	

	A)  7, 5, 3    B)  6, 6, 3    C)  6, 5, 4    D)  7, 4, 4	

26.	A small county has populations in three districts as shown below.  They are to apportion 15 seats on the county council using the Hill-Huntington method.  Find the apportionment for each district.

District	
Population	

Parkview	
43,000	

Hillside	
32,800	

Riverdale	
24,200	

	A)  7, 5, 3    B)  6, 6, 3    C)  6, 5, 4    D)  7, 4, 4	

27.	A country has four states with the populations shown below.  The House of Representatives is to have 15 members, apportioned by the Hill-Huntington method.  Find the first critical multiplier for District A.

State	
Population	

A	
52,600	

B	
39,900	

C	
34,000	

D	
23,500	

	A)  1.041    B)  1.052    C)  1.141    D)  1.570	

28.	A country has four states with the populations shown below.  The House of Representatives is to have 15 members.  Use the Hill-Huntington method of apportionment to find the number of seats for each state.

State	
Population	

A	
52,600	

B	
39,900	

C	
34,000	

D	
23,500	

	A)  5, 4, 4, 2    B)  6, 4, 3, 2    C)  5, 4, 3, 3    D)  6, 3, 3, 3	

29.	A county has four districts with the populations shown below.  They are to use the Hill- Huntington method of apportionment to distribute 20 seats on a county council.  Find the first critical multiplier for District B.

District	
Population	

A	
87,000	

B	
56,000	

C	
72,000	

D	
35,000	

	A)  0.9800    B)  0.9956    C)  0.9977    D)  0.9982	

30.	Given three states with the populations shown below, and a national senate with 10 seats, use the Hill-Huntington method of apportionment to distribute the seats to the states.

State	
Population	

Apathy	
69,000	

Bliss	
43,500	

Confusion	
37,500	

	A)  4, 3, 3    B)  4, 4, 2    C)  5, 4, 1    D)  5, 3, 2	

31.	Find the geometric mean of 5 and 6.	
	A)  5.5    B)  5.48    C)  4.69    D)  3.32	

32.	Find the geometric mean of 8 and 9.	
	A)  5.83    B)  8.5    C)  8.74    D)  8.49	

33.	Find the geometric mean of 3 and 7.	
	A)  4.38    B)  5.00    C)  4.58    D)  3.16	

34.	We are scheduling seven course sections for a total of 217 students.  Enrollments are: 109 in Calculus I,  79 in Calculus II, and 29 in Advanced Calculus.  Find the apportionment for each course using the Jefferson method.	
	A)  4, 2, 1    B)  3, 3, 1    C)  4, 3, 0    D)  3, 2, 2	

35.	We are scheduling seven course sections for a total of 217 students.  Enrollments are: 109 in Calculus I,  79 in Calculus II, and 29 in Advanced Calculus.  Find the apportionment for each course using the Hill-Huntington method.	
	A)  4, 2, 1    B)  4, 3, 0    C)  3, 3, 1    D)  3, 2, 2	

36.	Suppose a country has four states with the populations shown below and a parliament with 30 seats.  Use the Jefferson method of apportionment to apportion the seats between the states.

State	
Population	

A	
182,575	

B	
243,170	

C	
322,115	

D	
252,140	

	A)  5, 7, 10, 8    B)  6, 7, 9, 8    C)  5, 7, 11, 7    D)  6, 8, 8, 8	

37.	A state has a population of 24,000 and holds 3 of 28 seats in a parliament.  What is the district population of the state?	
	A)  857    B)  2571    C)  4320    D)  8000	

38.	A county council has 15 seats that are apportioned among four regions. The population of the West region is 3560, and the West region holds four seats on the council.  What is the district population for the West region?	
	A)  949    B)  890    C)  237    D)  59	

39.	A county council has 15 seats that are apportioned among four regions. The population of the West region is 3560, and the West region holds four seats on the council.  What is the representative share for the West region?	
	A)  0.004213    B)  0.001124    C)  0.2667    D)  0.01685	

40.	A small country has four states and a parliament with 23 seats.  One state has a population of 4277 and holds five seats in the parliament.  Find the representative share for the state.	
	A)  0.001169    B)  0.2174    C)  0.005378    D)  0.02689	

41.	A country has three states and a national senate with 10 seats.  The state populations and the apportionment of the senate seats are shown below.  Find the difference in representative shares of states A and B.

State	
Population	
Seats	

A	
69,000	
5	

B	
43,500	
3	

C	
37,500	
2	

	A)  0.2    B)  0.0000035    C)  0.00000125    D)  0.0000725	

42.	A country has three states and a national senate with 10 seats.  The state populations and the apportionment of the senate seats are shown below.  Find the difference in district population of states A and B.

State	
Population	
Seats	

A	
69,000	
5	

B	
43,500	
3	

C	
37,500	
2	

	A)  25,500    B)  20,000    C)  1400    D)  700	

43.	A small county has populations in three districts as shown below.  They have apportioned 15 seats on the county council and obtained the apportionment for each district shown.  Find the difference in representative share between the Parkview and Hillside districts.

District	
Population	
Apportionment	

Parkview	
43,000	
6	

Hillside	
32,800	
5	

Riverdale	
24,200	
4	

	A)  0.06667    B)  0.00004237    C)  0.0000129    D)  0.0000008522	

44.	A small county has populations in three regions as shown below.  They have apportioned 15 seats on the county council and obtained the apportionment for each region shown.  Find the difference in district population between the Parkview and Hillside regions.

District	
Population	
Apportionment	

Parkview	
43,000	
6	

Hillside	
32,800	
5	

Riverdale	
24,200	
4	

	A)  10,200    B)  607    C)  6267    D)  2867	

45.	Find the relative difference of 7 and 9.	
	A)  28.57%    B)  2%    C)  22.22%    D)  43.75%	

46.	Find the relative difference of 12 and 17.	
	A)  41.67%    B)  70.58%    C)  29.41%    D)  17.24%	

47.	A country has three states and a national senate with 10 seats.  The state populations and the apportionment of the senate seats are shown below.  Find the relative difference in representative shares of states A and B.

State	
Population	
Seats	

A	
69,000	
5	

B	
43,500	
3	

C	
37,500	
2	

	A)  66.7%    B)  4.83%    C)  5.07%    D)  6.70%	

48.	A country has three states and a national senate with 10 seats.  The state populations and the apportionment of the senate seats are shown below.  Find the relative difference in district population of states A and B.

State	
Population	
Seats	

A	
69,000	
5	

B	
43,500	
3	

C	
37,500	
2	

	A)  4.83%    B)  1.61%    C)  1.01%    D)  5.07%	

49.	A small county has populations in three regions as shown below.  They have apportioned 15 seats on the county council and obtained the apportionment for each region shown.  Find the relative difference in district population between the Parkview and Hillside regions.

Region	
Population	
Apportionment	

Parkview	
43,000	
6	

Hillside	
32,800	
5	

Riverdale	
24,200	
4	

	A)  9.25%    B)  8.47%    C)  1.85%    D)  1.41%	

50.	A small county has populations in three regions as shown below.  They have apportioned 15 seats on the county council and obtained the apportionment for each region shown.  Find the relative difference in representative share between the Parkview and Hillside regions.

Region	
Population	
Apportionment	

Parkview	
43,000	
6	

Hillside	
32,800	
5	

Riverdale	
24,200	
4	

	A)  9.2%    B)  8.5%    C)  16.7%    D)  20.0%	

51.	Use the Hamiltonian method to round each of the numbers in the sum 
0.85 + 0.8 +2.16 + 5.52 + 0.67 = 10 to a whole number, preserving the total of 10.	
A)	1 + 1 + 2 + 6 + 0 = 10	C)	0 + 1 + 3 + 5 + 1 = 10	
B)	0 + 1 + 2 + 6 + 1 = 10	D)	1 + 1 + 2 + 5 + 1 = 10	

52.	Use the Hamiltonian method to round each of the numbers in the sum 
2.50 + 1.65 + 0.54 + 1.63 + 3.68 = 10 to a whole number, preserving the total of 10.	
A)	3 + 2 + 0 + 2 + 3 = 10	C)	2 + 2 + 1 + 2 + 3 = 10	
B)	2 + 2 + 0 + 2 + 4 = 10	D)	3 + 2 + 0 + 1 + 4 = 10	

53.	Round the following to whole percentages using the Jefferson method.
36.3% + 11.7% + 7.9% + 5.6% + 10.4% + 28.1% = 100%	
A)	37 + 12 + 8 + 5 + 10 + 28	C)	37 + 12 + 8 + 6 + 11 + 29	
B)	36 + 12 + 8 + 6 + 10 + 28	D)	37 + 12 + 8 + 6 + 10 + 28	

54.	Round the following to whole percentages using the Jefferson method.
6.5% + 38.1% + 1.2% + 17.0% + 12.5% + 24.7% = 100%	
A)	7 + 38 + 1 + 17 + 13 + 24	C)	6 + 38 + 1 + 17 + 12 + 24	
B)	7 + 38 + 1 + 17 + 13 + 25	D)	6 + 39 + 1 + 17 + 13 + 24	

55.	Round the following to whole percentages using the Webster method.
36.3% + 11.7% + 7.9% + 5.6% + 10.4% + 28.1% = 100%	
A)	36 + 11 + 8 + 6 + 11 + 28	C)	37 + 12 + 8 + 5 + 9 + 28	
B)	36 + 12 + 8 + 6 + 10 + 28	D)	37 + 11 + 8 + 6 + 9 + 28	

56.	Round the following to whole percentages using the Webster method.
6.6% + 38.1% + 1.1% + 17.0% + 12.6% + 24.6% = 100%	
A)	7 + 38 + 1 + 17 + 13 + 25	C)	7 + 38 + 1 + 17 + 13 + 24	
B)	7 + 38 + 1 + 17 + 12 + 25	D)	6 + 38 + 1 + 17 + 13 + 24	

57.	The Anysville Transit System (ATS) has four bus routes, as identified below: 

Route	
Average passengers per day	

1	
682	

2	
2383	

3	
857	

4	
1425	

The ATS is replacing the entire old bus fleet with 35 new buses. Use the Webster method to determine how many buses each route should get.	
	A)  4, 14, 8, 9    B)  4, 15, 7, 9    C)  4, 16, 6, 9    D)  4, 16, 5, 10	

58.	The Anysville Transit System (ATS) has four bus routes, as identified below: 

Route	
Average passengers per day	

1	
682	

2	
2383	

3	
857	

4	
1425	

The ATS is replacing the entire old bus fleet with 25 new buses. Use the Hill-Huntington method to determine how many buses each route should get.	
	A)  4, 14, 8, 9    B)  4, 15, 7, 9    C)  4, 16, 6, 9    D)  4, 16, 5, 10	

59.	The Collie Creek high school district received a gift of 28 computers. The computers are to be divided among the three schools in the district.

School	
Number of students	

Central High	
384	

East High	
296	

West High	
204	

Use the Jefferson method to determine how many computers each school should get.	
	A)  12, 9, 7    B)  12, 10, 6    C)  11, 10, 7    D)  11, 9, 8	

60.	The Collie Creek high school district received a gift of 28 computers. The computers are to be divided among the three schools in the district.

School	
Number of students	

Central High	
384	

East High	
296	

West High	
204	

Use the Hamilton method to determine how many computers each school should get.	
	A)  12, 9, 7    B)  12, 10, 6    C)  11, 10, 7    D)  11, 9, 8	

Answer Key

1.	A	
2.	B	
3.	B	
4.	B	
5.	B	
6.	A	
7.	A	
8.	A	
9.	C	
10.	A	
11.	B	
12.	D	
13.	C	
14.	B	
15.	A	
16.	C	
17.	B	
18.	A	
19.	D	
20.	D	
21.	D	
22.	B	
23.	D	
24.	B	
25.	C	
26.	C	
27.	A	
28.	A	
29.	D	
30.	D	
31.	B	
32.	D	
33.	C	
34.	A	
35.	C	
36.	A	
37.	D	
38.	B	
39.	B	
40.	A	
41.	B	
42.	D	
43.	C	
44.	B	
45.	A	
46.	A	
47.	C	
48.	D	
49.	A	
50.	A	
51.	D	
52.	B	
53.	A	
54.	D	
55.	B	
56.	C	
57.	C	
58.	C	
59.	B	
60.	A