Problem 8 - Thinking Mathematically

Question

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. $$ \begin{array}{|l|c|c|c|c|} \hline \text { State } & \text { A } & \text { B } & \text { C } & \text { D } \\\ \hline \begin{array}{l} \text { Population } \\ \text { (in thousands) } \end{array} & 424 & 664 & 892 & 1162 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

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 1: 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: \(q_A = \frac{424}{7.82} = 54.22\), \(q_B = \frac{664}{7.82} = 84.91\), \(q_C = \frac{892}{7.82} = 114.07\), \(q_D = \frac{1162}{7.82} = 148.59\)

Step 2: 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. \(Seats_A = \lfloor q_A \rfloor = 54\), \(Seats_B = \lfloor q_B \rfloor = 84\), \(Seats_C = \lfloor q_C \rfloor = 114\), \(Seats_D = \lfloor q_D \rfloor = 148\)

Step 3: 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.

Key Concepts

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.