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Mathematics LibreTexts

11.4.0: Exercises

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For the following exercises, use the standard quotas given in the table below.

State A State B State C State D State E State F Total Seats
Scenario X 17.63 26.62 10.81 16.01 13.69 15.24 100
Scenario Y 12.37 7.59 71.71 6.75 5.76 20.81 125
Scenario Z 3.53 31.56 2.95 5.12 9.84 NA 53
Exercise 11.4.0.1

Round the standard quota for each state in Scenario X using traditional rounding. Find the sum of the modified quotas. What is the difference between the sum and the house size?

Exercise 11.4.0.2

Round the standard quota for each state in Scenario Y using traditional rounding. Find the sum of the modified quotas. What is the difference between the sum and the house size?

Exercise 11.4.0.3

Round the standard quota for each state in Scenario Z using traditional rounding. Find the sum of the modified quotas. What is the difference between the sum and the house size?

Exercise 11.4.0.4

Find the lower quota for each state in Scenario Y. If each state is allocated its lower quota, how many seats remain to be allocated?

Exercise 11.4.0.5

Find the lower quota for each state in Scenario X. If each state is allocated its lower quota, how many seats remain to be allocated?

Exercise 11.4.0.6

Find the lower quota for each state in Scenario Z. If each state is allocated its lower quota, how many seats remain to be allocated?

Exercise 11.4.0.7

Find the upper quota for each state in Scenario X and determine how much the sum of the upper quotas exceeds the house size.

Exercise 11.4.0.8

Find the upper quota for each state in Scenario Y and determine how much the sum of the upper quotas exceeds the house size.

Exercise 11.4.0.9

Find the upper quota for each state in Scenario Z and determine how much the sum of the upper quotas exceeds the house size.

Exercise 11.4.0.10

Determine the Hamilton apportionment for Scenario Y.

Exercise 11.4.0.11

Determine the Hamilton apportionment for Scenario X.

Exercise 11.4.0.12

Determine the Hamilton apportionment for Scenario Z.

For the following exercises, use the information in the table below, which gives standard and final quotas for Methods X, Y, and Z.

State A State B State C State D State E
Standard Quota 1.67 3.33 5.00 6.67 8.33
Method X 2 2 5 7 9
Method Y 1 3 5 7 9
Method Z 1 3 5 6 10
Exercise 11.4.0.13

Does the apportionment resulting from method X satisfy the quota rule? Why or why not?

Exercise 11.4.0.14

Does the apportionment resulting from method Z satisfy the quota rule? Why or why not?

Exercise 11.4.0.15

Does the apportionment resulting from method Y satisfy the quota rule? Why or why not?

In the movie Black Panther, the hero lives in the fictional country of Wakanda. Imagine that 111 Vibranium artifacts must be distributed among the fortress cities, or birnin, of Wakanda based on the population of each birnin. Use the population and standard quota information in the table below for the following exercises.

Birnin Djata (D) Birnin T'Chaka (T) Birnin Zana (Z) Birnin S'Yan (S) Birnin Bashenga (B) Birnin Azzaria (A) Total
Residents 26,000 57,000 27,000 18,000 64,000 45,000 237,000
Standard Quota 12.18 26.70 12.65 8.43 29.98 21.08 111
Exercise 11.4.0.16

Modify the standard quota for each state using traditional rounding. Find the sum of the modified quotas. What is the difference between the sum and the house size?

Exercise 11.4.0.17

Find the standard lower quota for each state. If each state is allocated its lower quota, how many seats remain to be allocated?

Exercise 11.4.0.18

Find the standard upper quota for each state and determine how much the sum of the upper quotas exceeds the house size.

Exercise 11.4.0.19

Use the Hamilton method to apportion the artifacts.

Exercise 11.4.0.20

Find the modified lower quota for each state using a modified divisor of 2,000. Is the sum of the modified quotas too high, too low, or equal to the house size?

Exercise 11.4.0.21

Find the modified lower quota for each state using a modified divisor of 2,100. Is the sum of the modified quotas too high, too low, or equal to the house size?

Exercise 11.4.0.22

Use the Jefferson method to apportion the artifacts. Determine whether it is necessary to modify the divisor. If so, indicate the value of the modified divisor.

Exercise 11.4.0.23

Does the Jefferson method result in an apportionment that satisfies or violates the quota rule in this scenario?

Exercise 11.4.0.24

Find the modified upper quota for each state using a modified divisor of 2,250. Is the sum of the modified quotas too high, too low, or equal to the house size?

Exercise 11.4.0.25

Find the modified upper quota for each state using a modified divisor of 2,150. Is the sum of the modified quotas too high, too low, or equal to the house size?

Exercise 11.4.0.26

Use the Adams method to apportion the artifacts. Determine whether it is necessary to modify the divisor. If so, indicate the value of the modified divisor.

Exercise 11.4.0.27

Does the Adams method result in an apportionment that satisfies or violates the quota rule in this scenario?

Exercise 11.4.0.28

Which method of apportionment, Jefferson or Adams, is a resident of Birnin T'Chaka likely to prefer? Justify your answer.

Exercise 11.4.0.29

Use the Webster method to apportion the artifacts. Determine whether it is necessary to modify the divisor. If so,indicate the value of the modified divisor.

Exercise 11.4.0.30

Does the Webster method result in an apportionment that satisfies or violates the quota rule in this scenario?

Exercise 11.4.0.31

Which of the four methods of apportionment from this section (Hamilton, Jefferson, Adams, or Webster) are the residents of Birnin S'Yan likely to prefer? Justify your answer.

Children from five families—the Chorro family, the Eswaran family, the Javernick family, the Lahde family, and the Stolly family—joined a town Easter egg hunt. When they returned with their baskets, they had 827 eggs! They decided to share their eggs amongst the families based on the number of children in each family. Use the population and standard quota information in the table below for the following exercises.

(C) Chorro (E) Eswaran (J) Javernick (L) Lahde (S) Stolly Total
Children 3 2 4 1 5 15
Standard Quota 155.04 103.36 206.72 103.36 258.40 827
Exercise 11.4.0.32

Modify the standard quota for each state using traditional rounding. Find the sum of the modified quotas. What is the difference between the sum and the house size?

Exercise 11.4.0.33

Find the standard lower quota for each state. If each state is allocated its lower quota, how many seats remain to be allocated?

Exercise 11.4.0.34

Find the standard upper quota for each state, and determine how much the sum of the upper quotas exceeds the house size.

Exercise 11.4.0.35

Use the Hamilton method to apportion the Easter eggs.

Exercise 11.4.0.36

Find the modified lower quota for each state using a modified divisor of 0.01800. Is the sum of the modified quotas too high, too low, or equal to the house size?

Exercise 11.4.0.37

Find the modified lower quota for each state using a modified divisor of 0.01810. Is the sum of the modified quotas too high, too low, or equal to the house size?

Exercise 11.4.0.38

Use the Jefferson method to apportion the Easter eggs. Determine whether it is necessary to modify the divisor. If so, indicate the value of the modified divisor.

Exercise 11.4.0.39

Does the Jefferson method result in an apportionment that satisfies or violates the quota rule in this scenario?

Exercise 11.4.0.40

Find the modified upper quota for each state using a modified divisor of 0.0182. Is the sum of the modified quotas too high, too low, or equal to the house size?

Exercise 11.4.0.41

Find the modified upper quota for each state using a modified divisor of 0.01816. Is the sum of the modified quotas too high, too low, or equal to the house size?

Exercise 11.4.0.42

Use the Adams method to apportion the Easter eggs. Determine whether it is necessary to modify the divisor. If so, indicate the value of the modified divisor.

Exercise 11.4.0.43

Does the Adams method result in an apportionment that satisfies or violates the quota rule in this scenario?

Exercise 11.4.0.44

Use the Webster method to apportion the Easter eggs. Determine whether it is necessary to modify the divisor. If so, indicate the value of the modified divisor.

Exercise 11.4.0.45

Does the Webster method result in an apportionment that satisfies or violates the quota rule in this scenario?

For the following exercises, use this information: Suppose that the State of Delaware received 2,000 packs of COVID-19 vaccines, with ten doses per pack. These (unopened) packs must be distributed to the three counties based on total population. Use the population information in the table below to determine how many packs of vaccine will be distributed to each county based on the given apportionment method.

(N) New Castle (K) Kent (S) Sussex
Residents 558,753 180,786 234,225
Exercise 11.4.0.46

Hamilton’s Method

Exercise 11.4.0.47

Jefferson’s Method

Exercise 11.4.0.48

Adams's Method

Exercise 11.4.0.49

Webster’s Method

Exercise 11.4.0.50

Notice that the apportionments found in questions 46, 47, 48, and 49 all satisfy the quota rule. Does this contradict the statement, “The Jefferson, Adams, and Webster methods of apportionment all violate the quota rule”? Why or why not?


11.4.0: Exercises is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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