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Chapter 13: Problem 8

A small country is comprised of four states, A, B, C, and D. The population of each state, in thousands, is given in the following table. Congress will have 400 seats, divided among the four states according to their respective populations. Use Jefferson's method with \(d=7.82\) to apportion the 400 congressional seats. $$ \begin{array}{|l|c|c|c|c|} \hline \text { State } & \text { A } & \text { B } & \text { C } & \text { D } \\\ \hline \begin{array}{l} \text { Population } \\ \text { (in thousands) } \end{array} & 424 & 664 & 892 & 1162 \\ \hline \end{array} $$

Short Answer

Expert verified
The final apportionment of the 400 seats is: State A gets 54 seats, State B gets 84 seats, State C gets 114 seats, and State D gets 148 seats.

Step by step solution

01

Calculate the initial quotas

First, calculate the initial quotas for each state by dividing the state's population by the original standard divisor \(d=7.82\). This results in the following quotas: \(q_A = \frac{424}{7.82} = 54.22\), \(q_B = \frac{664}{7.82} = 84.91\), \(q_C = \frac{892}{7.82} = 114.07\), \(q_D = \frac{1162}{7.82} = 148.59\)
02

Determine the initial apportionment

Next, determine the initial apportionment by rounding the quotas down. This is the distinctive feature of Jefferson's method - the quotas are always rounded down, not to the nearest whole number. \(Seats_A = \lfloor q_A \rfloor = 54\), \(Seats_B = \lfloor q_B \rfloor = 84\), \(Seats_C = \lfloor q_C \rfloor = 114\), \(Seats_D = \lfloor q_D \rfloor = 148\)
03

Check and adjust the apportionment

The total of the initial apportionment is \(54 + 84 + 114 + 148 = 400\). In this case, the initial apportionment happens to be equal to the total number of seats (400). So there is no need to revise the divisor and the initial apportionment will be the final apportionment. However, if the total was not equal to 400, we would need to adjust the divisor, either up or down, and recalculate the quotas until the total of the quotas rounded down is equal to 400.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Apportionment
Apportionment is a method used to distribute a given number of seats (or resources) among different groups or states based on certain criteria, such as population. This process ensures that representation is fair and proportional. For instance, in the context of congressional seats, each state receives a number of representatives that corresponds to its population size.
In Jefferson's method, apportionment involves mathematical calculations to decide how many seats each state in a country or region will receive in a legislative body like Congress. This method helps ensure that the distribution of seats is as fair as possible, based on the relative sizes of the states' populations and often requires careful adjustments to arrive at an exact distribution.
Population
Population is a central concept in the apportionment process, as it's the determining factor when deciding how many seats each state receives. For example, a state with a larger population will typically receive more congressional seats compared to a state with a smaller population. This ensures that all citizens have equal representation in the legislative process.
In the given exercise, each state's population is measured in thousands, and these numbers are crucial when using Jefferson's method to attain the apportionment of congressional seats. The population figures are divided by a divisor to arrive at initial quota values, which form the basis for the seats allotted to each state.
Congressional Seats
Congressional seats represent the positions allocated to states in a legislature, reflecting their share of representation in government. In the context of the United States, these seats are crucial for ensuring each state's population is fairly represented.
Using Jefferson's method, states are given seats based on their population, creating a concrete means by which representation is achieved. In our exercise, the task is to allocate 400 congressional seats among four states. Each seat signifies a vote and a voice in the legislative process, making it critical that they are distributed in alignment with population sizes.

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Most popular questions from this chapter

In this Exercise Set, we have used apportionment methods to divide congressional seats, assign computers to schools, assign doctors to clinics, divide police officers among precincts, divide shares of stock, assign sections of bilingual math, assign buses to city routes, and assign nurses to hospital shifts. Describe another situation that requires the use of apportionment methods.

MTV's Real World is considering three cities for its new season: Amsterdam (A), Rio de Janeiro (R), or Vancouver (V). Programming executives and the show's production team vote to decide where the new season will be taped. The winning city is to be determined by the plurality method. The preference table for the election is shown at the top of the next column. $$ \begin{array}{|l|c|c|c|c|} \hline \text { Number of Votes } & \mathbf{1 2} & \mathbf{9} & \mathbf{4} & \mathbf{4} \\ \hline \text { First Choice } & \text { A } & \text { V } & \text { V } & \text { R } \\ \hline \text { Second Choice } & \text { R } & \text { R } & \text { A } & \text { A } \\ \hline \text { Third Choice } & \text { V } & \text { A } & \text { R } & \text { V } \\ \hline \end{array} $$ a. Which city is favored over all others using a head-tohead comparison? b. Which city wins the vote using the plurality method? c. Is the head-to-head criterion satisfied? Explain your answer.

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.

Twenty sections of bilingual math courses, taught in both English and Spanish, are to be offered in introductory algebra, intermediate algebra, and liberal arts math. The preregistration figures for the number of students planning to enroll in these bilingual sections are given in the following table. Use Webster's method with \(d=29.6\) to determine how many bilingual sections of each course should be offered. $$ \begin{array}{|l|c|c|c|} \hline \text { Course } & \begin{array}{c} \text { Introductory } \\ \text { Algebra } \end{array} & \begin{array}{c} \text { Intermediate } \\ \text { Algebra } \end{array} & \begin{array}{c} \text { Liberal Arts } \\ \text { Math } \end{array} \\ \hline \text { Enrollment } & 130 & 282 & 188 \\ \hline \end{array} $$

a. A country has three states, state A, with a population of 99,000 , state B, with a population of 214,000 , and state C, with a population of 487,000 . The congress has 50 seats, divided among the three states according to their respective populations. Use Hamilton's method to apportion the congressional seats to the states. b. Suppose that a fourth state, state D, with a population of 116,000 , is added to the country. The country adds seven new congressional seats for state D. Use Hamilton's method to show that the new-states paradox occurs when the congressional seats are reapportioned.
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