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# Nature of Mathematics

Mathematics

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## Quiz 17 : The Nature of Voting and Apportionment

Use Jefferson's plan. Which state does violate the quota rule? Number of seats: 200 ​ ​
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Multiple Choice

C

Apportion the indicated number of representatives to two states, A, and B, using Hamilton's plan. Next, recalculate the apportionment using Hamilton's plan for the three states, C and the original states. Decide whether the new states paradox occurs.
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Multiple Choice

B

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None of the plans

Use Jefferson's plan. Which state does violate the quota rule? ​ Number of seats: 200 ​ __________ (A, B, C, D, none)
Use Hamilton's plan to apportion the new seats to the existing states. Then increase the number of seats by one and decide whether the Alabama paradox occurs. Assume that the populations are in thousands. Number of seats: 86 ​ ​
Multiple Choice
An elderly rancher died and left her estate to her three children. She bequeathed her 17 prize horses in the following manner: 1/2 to the eldest, 1/3 to the second child, and 1/9 to the youngest. ​ The children decided to call in a very wise judge to help in the distribution of the rancher's estate. They informed the judge that the 17 horses were not of equal value. The children agreed on a ranking of the 17 horses (#1 being the best and #17 being a real dog of a horse). They asked the judge to divide the estate fairly so that each child would receive not only the correct number of horses but horses whose average rank would also be the same. For example, if a child received horses 1 and 17, the number of horses is two and the average value is . How did the judge apportion the horses? ​
Multiple Choice
Use Jefferson's plan. Which state does violate the quota rule? Number of seats: 200 ​ ​ __________ (A, B, C, D, E, none)
A fair apportionment of dividing a leftover piece of cake between two children is to let child #1 cut the cake into two pieces and then to let child #2 pick which piece he or she wants. Consider the following apportionment of dividing the leftover piece of cake among three children. Let the first child cut the cake into two pieces. Then the second child is permitted to cut one of those pieces into two parts. Child #3 can select any of the pieces, followed by child #1 selecting one of the remaining pieces, followed by child #2 who gets the remaining piece. Is this allocation process fair if each child's goal is to maximize the size of his or her own piece of cake? ​
Multiple Choice
An elderly rancher died and left her estate to her three children. She bequeathed her 17 prize horses in the following manner: 1/2 to the eldest, 1/3 to the second child, and 1/9 to the youngest. ​ The children decided to call in a very wise judge to help in the distribution of the rancher's estate. They informed the judge that the 17 horses were not of equal value. The children agreed on a ranking of the 17 horses (#1 being the best and #17 being a real dog of a horse). They asked the judge to divide the estate fairly so that each child would receive not only the correct number of horses but horses whose average rank would also be the same. For example, if a child received horses 1 and 17, the number of horses is two and the average value is . How did the judge apportion the horses?
Essay
An elderly rancher died and left her estate to her three children. She bequeathed her 71 prize horses in the following manner: 1/2 to the eldest, 1/3 to the second child, and 1/9 to the youngest. ​ The children decided to call in a very wise judge to help in the distribution of the rancher's estate. The judge arrived with a horse of his own. He put his horse in with the 71 belonging to the estate, and then told each child to pick from among the 72 in the proportions stipulated by the will (but be careful, he warned, not to pick his horse). The first child took thirty six horses, the second child took twenty four, and the third child, eight. The 71 horses were thus divided among the children. The wise judge took his horse from the corral, took a fair sum for his services, and rode off into the sunset. ​ The youngest son complained that the oldest son received 36 horses (but was entitled to only 71/2 = 35.5 horses). The judge was asked about this, and he faxed the children the following message: "You all received more than you deserved. The eldest received 1/2 of an 'extra' horse, the middle child received 1/3 more, and the youngest, 1/9 of a horse 'extra.'" Apportion the horses according to Adams', Jefferson's, and Webster's plans. Which plan gives the appropriate distribution of horses? ​
Multiple Choice
Apportion the indicated number of representatives to three states, A, B, and C, using Hamilton's plan. Next, use the revised populations to reapportion the representatives. Decide whether the population paradox occurs. ​ ​
Multiple Choice
Use Jefferson's plan. Which state does violate the quota rule? Number of seats: 200 ​ ​
Multiple Choice
Apportion the indicated number of representatives to two states, A, and B, using Hamilton's plan. Next, recalculate the apportionment using Hamilton's plan for the three states, C and the original states. Decide whether the new states paradox occurs.
Multiple Choice
Round the given modified quota ​ 3)57 ​ By comparing it first with the arithmetic mean, and then with the geometric mean of the lower and upper quotas. ​
Multiple Choice
Apportion the indicated number of representatives to two states, A, and B, using Hamilton's plan. Next, recalculate the apportionment using Hamilton's plan for the three states, C and the original states. Decide whether the new states paradox occurs. ​ __________ (A illustrates the new states paradox.; B illustrates the new states paradox.; C illustrates the new states paradox.; A and B illustrate the new states paradox.; The paradox does not occur.)
Apportion the indicated number of representatives to two states, A, and B, using Hamilton's plan. Next, recalculate the apportionment using Hamilton's plan for the three states, C and the original states. Decide whether the new states paradox occurs. __________ (A illustrates the new states paradox.; B illustrates the new states paradox.; C illustrates the new states paradox.; A and B illustrate the new states paradox.; The paradox does not occur.)
Suppose the annual salaries of three people are: What are their salaries if they are given a 4% raise, and then the result is rounded to the nearest $1,000 using Hamilton's plan with a cap on the total salaries of$113,000? ​