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Chapter 13: Problem 7

A small country is composed of five states, \(A, B, C, D\), and \(E\). The population of each state is given in the following table. Congress will have 57 seats, divided among the five states according to their respective populations. Use Jefferson's method with \(d=32,920\) to apportion the 57 congressional seats. $$ \begin{array}{|l|c|c|c|c|c|} \hline \text { State } & \text { A } & \text { B } & \text { C } & \text { D } & \text { E } \\ \hline \text { Population } & 126,316 & 196,492 & 425,264 & 526,664 & 725,264 \\\ \hline \end{array} $$

Short Answer

Expert verified
The seat allocation using Jefferson's method with the given divisor is A = 3 seats, B = 5 seats, C = 12 seats, D = 15 seats and E = 22 seats.

Step by step solution

01

Calculate Standard Quotas

The first step is to determine the standard quota for each state. This is done by dividing the population of each state by the assigned divisor. The divisor given is \(d=32,920\), so the quotas for states A, B, C, D and E would be \(Q_A = \frac{126316}{32920} = 3.84\), \(Q_B = \frac{196492}{32920} = 5.97\), \(Q_C = \frac{425264}{32920} = 12.92\), \(Q_D = \frac{526664}{32920} = 16.00\) and \(Q_E = \frac{725264}{32920} = 22.03\) respectively. These are the initial quotas, before rounding.
02

Round Down Initial Quotas

Now, according to Jefferson's method, these initial quotas must be rounded down to the nearest whole number. These numbers represent the provisional number of seats for each state. This yields \(P_A = 3\), \(P_B = 5\), \(P_C = 12\), \(P_D = 16\) and \(P_E = 22\).
03

Calculate Total Provisional Seats

Add all the provisional seats together. The total is \(T = P_A + P_B + P_C + P_D + P_E = 3 + 5 + 12 + 16 + 22 = 58\). The total number of seats available is 57, so now there is one surplus seat to deal with.
04

Remove Surplus Seats

To get rid of the surplus, identify the state(s) whose decimal part of their quota was closest to a whole number. In case multiple states have quotas with the same decimal part, choose the state with the largest population. Here, state D had a quota of exactly 16.00, so it loses the surplus seat. So the final seat allocation is \(A = 3\), \(B = 5\), \(C = 12\), \(D = 15\), \(E = 22\) with a total of 57 seats.

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Most popular questions from this chapter

Fifty-three people are asked to taste-test and rank three different brands of yogurt, \(A, B\), and \(C\). The preference table shows the rankings of the 53 voters. $$ \begin{array}{|l|c|c|c|} \hline \text { Number of Votes } & 27 & 24 & 2 \\ \hline \text { First Choice } & \text { A } & \text { B } & \text { C } \\ \hline \text { Second Choice } & \text { C } & \text { C } & \text { B } \\ \hline \text { Third Choice } & \text { B } & \text { A } & \text { A } \\ \hline \end{array} $$ a. Which brand has a majority of first-place votes? b. Suppose that the Borda count method is used to determine the winner. Which brand wins the taste test? c. Is the majority criterion satisfied? Explain your answer.

a. A country has two states, state \(\mathrm{A}\), with a population of 9450 , and state \(B\), with a population of 90,550 . The congress has 100 seats, divided between the two states according to their respective populations. Use Hamilton's method to apportion the congressional seats to the states. b. Suppose that a third state, state \(C\), with a population of 10,400 , is added to the country. The country adds 10 new congressional seats for state C. Use Hamilton's method to show that the new-states paradox occurs when the congressional seats are reapportioned.

Describe what is contained in a preference table. What does the table show?

A university is composed of five schools. The enrollment in each school is given in the following table. $$ \begin{array}{|l|c|c|c|c|c|} \hline \text { School } & \begin{array}{c} \text { Human- } \\ \text { ities } \end{array} & \begin{array}{c} \text { Social } \\ \text { Science } \end{array} & \begin{array}{c} \text { Engi- } \\ \text { neering } \end{array} & \text { Business } & \begin{array}{c} \text { Educa- } \\ \text { tion } \end{array} \\ \hline \text { Enrollment } & 1050 & 1410 & 1830 & 2540 & 3580 \\ \hline \end{array} $$ There are 300 new computers to be apportioned among the five schools according to their respective enrollments. Use Hamilton's method to find each school's apportionment of computers.

A candidate won the election using the plurality-withelimination method, yet lost the election when the votes were counted by the pairwise comparison method.
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