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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
By using Jefferson's method of apportionment and a lowered divisor, the following seats for each state is calculated: State A: ..., State B: ..., State C: ..., State D: .... These numbers will all be integers and their sum needs to add up to 400. The exact numbers depend on the correctly chosen lowered divisor and cannot be given in this general example.

Step by step solution

01

Find the Standard Quota

Start by finding the standard quota for each state. This is done by dividing the population of each state by the standard divisor \(d = 7.82\). For example, for state A, the standard quota is \(424/7.82 = 54.22\).
02

Apply Jefferson's method

Once the standard quotas are found, apply Jefferson's method, which involves rounding all quotas down to the nearest whole number. For example, for state A, the quota rounded down is 54.
03

Check the total seats

Add up all the rounded down quotas from step 2. If they add up to the total number of congressional seats, the problem is solved. In case the sum is less than the total number of seats, decrease the divisor slightly and repeat step 1 and 2. The aim is to find the lowered divisor which makes the sum of the rounded down quotas equal to the number of congressional seats.
04

Check the new apportion

After obtaining the new lowered divisor, check if the total number of seats equals 400. If not, adjust the divisor slightly again until the total equals 400.

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Key Concepts

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

Mathematical Apportionment
Mathematical apportionment is a process used to assign a limited number of items, like seats in a congress or parliament, proportionally among groups based on certain criteria, such as population or number of states. This concept is crucial for maintaining a representative democracy or any system requiring fair distribution. Jefferson's method, named after Thomas Jefferson, is a classic technique used for this purpose. It is designed to ensure that the apportionment is fair by adjusting a 'divisor' until the total number of seats allocated is equal to the fixed number of seats available (in this case, 400).

Jefferson's method favors larger populations because it rounds down the quotas during calculation, leading to a possibility of larger states gaining more representation at the expense of smaller states. The method remains historically significant and provides a foundational understanding of apportionment even though modern systems may use different methods in pursuit of greater equality.
Standard Quota Calculation
The standard quota calculation is pivotal in allocating resources or positions based on population or other fair-share criteria. In Jefferson's method, this calculation involves dividing the population of each group by a predetermined 'standard divisor' to obtain the 'standard quota'. One critical characteristic of standard quota is that it typically results in a fractional number, which indicates the fair-share number of seats for each state before any adjustments are made.

The divisor represents the population per seat if seats were distributed perfectly proportionally. Finding the right standard divisor is an iterative process, starting with an initial estimate that typically results in fractional quotas. Adjustments are then made to move closer to the target number of seats, which often involves lowering the divisor to assign unallocated seats. Since the standard quota directly impacts the apportionment outcome, it is a key concept in the study of mathematical apportionment and essential for anyone looking to understand or work with proportional distribution models.
Congressional Seat Distribution
Congressional seat distribution is a practical application of mathematical apportionment, where the 'seats' correspond to the number of representatives each state or constituency has in a legislative body. The distribution must be fair, reflecting the population size of each area to ensure that each citizen has relatively equal representation. Jefferson's method plays a pivotal role in this by providing a systematic way to divide seats among states based on population figures.

To visualize this process, imagine you have a variety of sized containers (states) and a certain amount of liquid (seats) to distribute. The aim is to pour the liquid into the containers so that the proportional volume in each container mirrors the size of each. The complexity arises when we can only pour whole units (seats), which is why the standard quota is rounded down in Jefferson's method, requiring subsequent adjustments to the divisor until every 'drop' of liquid is suitably allocated. This mathematical precision ensures that, despite the rounding, the eventual distribution of seats is as fair as the chosen method allows.

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Most popular questions from this chapter

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.

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.

Describe the difference between how modified quotas are rounded using Jefferson's method and Adams's method.

What is the population paradox?

Citizen-initiated ballot measures often present voters with controversial issues over which they do not think alike. Here's one your author would like to initiate: Please rank each of the following options regarding permitting dogs on national park trails. a. Unleashed dogs accompanied by their caregivers should be permitted on designated national park trails. b. Leashed dogs accompanied by their caregivers should be permitted on designated national park trails. c. No dogs should be permitted on any national park trails. Your author was not happy with the fact that he could not take his dog running with him on the park trails at Point Reyes National Seashore. Of course, that is his issue. For this project, group members should write a ballot measure, perhaps controversial, like the sample above, but dealing with an issue of relevance to your campus and community. Rather than holding an election, use a random sample of students on your campus, administer the ballot, and have them rank their choices. a. Use each of the four voting methods to determine the winning option for your ballot measure. b. Check to see if any of the four fairness criteria are violated.
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