91Ó°ÊÓ

The Bandana Republic is a small country consisting of four states: Apure (population 3,310,000 ), Barinas (population 2,670,000 ), Carabobo (population 1,330,000 ), and Dolores (population 690,000 ). Suppose that there are \(M=160\) seats in the Bandana Congress, to be apportioned among the four states based on their respective populations. (a) Find the standard divisor. (b) Find each state's standard quota.

The Republic of Wadiya is a small country consisting of four provinces: \(A\) (population 4,360.000\(), B\) (population \(2,280,000), C\) (population 729,000\()\), and \(D\) (population 2,631,000 ). Suppose that there are \(M=200\) seats in the Wadiya Congress, to be apportioned among the four provinces based on their respective populations. (a) Find the standard divisor. (b) Find each province's standard quota.

In the 2010 apportionment of the U.S. House of Representatives, Missouri had a standard quota of 8.458641 and a modified quota of \(8.483 .\) How many seats were apportioned to Missouri? Explain your answer.

If the standard quota of state \(Y\) is 78.24 , then which of the following apportionments to state \(Y\) is (or are) possible under Hamilton's method? (a) 78.2 or 78.3 (b) 78 or 79 (c) 78 only (d) 79 only (e) any positive integer less than 79

If the standard quota of state \(Y\) is 78.24 , then which of the following apportionments to state \(Y\) is (or are) possible under Adams's method? (a) 77,78 or 79 (b) 77,78,79 or 80 (c) 78,79,80 or 81 (d) 79 only (e) 78 only

Consider an apportionment problem with \(N\) states. The populations of the states are given by \(p_{1}, p_{2}, \ldots, p_{N},\) and the standard quotas are \(q_{1}, q_{2+\ldots} \ldots q_{N}\), respectively. Describe in words what each of the following quantities represents. (a) \(q_{1}+q_{2}+\cdots+q_{N}\) (b) \(\frac{p_{1}+p_{2}+\cdots+p_{N}}{q_{1}+q_{2}+\cdots+q_{N}}\) (c) \(\left(\frac{P_{N}}{P_{1}+p_{2}+\cdots++p_{N}}\right) \times 100\)

Lowndes's Method Exercises 73 and 74 refer to a variation of Hamilion's method known as Lowndes's method, first proposed in 1822 by South Carolina Representative William Lowndes. The basic difference between Hamilton's and Lowndes's methods is that in Lowndes's method, after each state is assigned the lower quota, the sarphus seats are handed out in order of relative fractional parts. (The relative fractional part of a number is the fractional part divided by the integer purt. For example, the relative fractional part of 41.82 is \(\frac{0.82}{4 !}=0.02,\) and the relative fractional part of 3.08 is \(\frac{a 08}{3}=0.027 .\) Notice that while 41.82 would have priority oser 3.08 umder Hamilton's method, 3.08 has priority over 41.82 under Lonwndes's method because 0.027 is greater than \(0.02 .)\). Consider an apportionment problem with two states, \(A\) and \(B\), Suppose that state \(A\) has standard quota \(q_{t}\) and state \(B\) has standard quota \(q_{2}\), neither of which is a whole number.(Of course, \(q_{1}+q_{2}=M\) must be a whole number.) Let \(f_{1}\) represent the fractional part of \(q_{1}\) and \(f_{2}\) the fractional part of \(q_{2}\) (a) Find values \(q_{1}\) and \(q_{2}\) such that Lowndes's method and Hamilton's method result in the same apportionment. (b) Find values \(q_{1}\) and \(q_{2}\) such that Lowndes's method and Hamilton's method result in different apportionments. (c) Write an inequality involving \(q_{1}, q_{2}, f_{1},\) and \(f_{2}\) that would guarantee that Lowndes' method and Hamilton's method result in different apportionments.

Explain why Webster's method cannot produce (a) the Alabama paradox (b) the new-states paradox