Problem 7 - Thinking Mathematically

Question

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. $$ \begin{array}{|l|c|c|c|c|c|} \hline \text { State } & \text { A } & \text { B } & \text { C } & \text { D } & \text { E } \\ \hline \text { Population } & 126,316 & 196,492 & 425,264 & 526,664 & 725,264 \\\ \hline \end{array} $$

Step-by-Step Solution

Answer

After applying Jefferson's Method, the apportionment of the 57 congressional seats among the states A, B, C, D, and E would be 2, 5, 12, 16, and 22 respectively.

Step 1: Calculating the Standard Quota

Begin with calculating the standard quota for each state. The standard quota is the population of the state divided by the standard divisor. For state A: \( \frac{126,316}{32,920} = 3.83 \),For state B: \( \frac{196,492}{32,920} = 5.97 \), For state C: \( \frac{425,264}{32,920} = 12.92 \), For state D: \( \frac{526,664}{32,920} = 16.00 \), and For state E: \( \frac{725,264}{32,920} = 22.03 \)

Step 2: Calculating the Lower Quotas

Calculate the lower quota for each state by rounding down the quota from the first step to the nearest whole number. For state A, it is 3,For state B, it is 5, For state C, it is 12, For state D, it is 16, and For state E, it is 22.

Step 3: Distribute the Remaining Seats

Up until now, \(3 + 5 + 12 + 16 + 22 = 58\) seats have been apportioned, which is one over the total 57 seats. As per Jefferson's Method, reduce the quota of the state which has the smallest decimal until the total number of seats matches the given total. Here, the state with the smallest decimal is state A, so reduce the quota of state A by one, making it 2 instead of 3. Now, the total number of seats is 3 + 5 + 12 + 16 + 22 - 1 = 57, matching the total number of seats given.

Step 4: Final Apportionment

Each state's final quota is: State A: 2State B: 5 State C: 12 State D: 16 State E: 22

Key Concepts

apportionment

Apportionment is the process by which a government or authority distributes something, such as seats in a legislative body, among different entities or regions. In this context, it involves distributing congressional seats among five states based on their population sizes. Each state is allocated a number of seats that corresponds to its proportion of the total population. To achieve apportionment fairly, different methods can be used, with Jefferson's Method being one approach. This method involves using a specific formula and adjustments to ensure that the distribution of seats reflects the population distribution as accurately as possible. Understanding apportionment is crucial because it ensures that representation in governance is balanced and equitable. The goal is to match each state's influence in Congress to its population size, thereby providing equal representation for all citizens.

congressional seats

Congressional seats refer to the allocated positions in a legislative body that representatives occupy on behalf of their constituents. In the exercise, there are a total of 57 congressional seats that need to be distributed among the states. The method used to allocate these seats is important because it affects representation in government. The more seats a state has, the more influence it can exert. Therefore, ensuring fair and accurate representation by distributing seats according to population is essential. The calculation of congressional seats begins with determining standard quotas and then making necessary adjustments to ensure the total matches the predetermined number of seats, utilizing a method like Jefferson's.

standard quota

The standard quota is a crucial calculation in the apportionment process. It represents how many congressional seats each state should ideally receive based on its population relative to a standard divisor.In simple terms, the standard quota is found by dividing each state's population by the divisor. For example, in the exercise, state A's standard quota is calculated as \( \frac{126,316}{32,920} = 3.83 \). This calculation is done for each state to determine how many seats they would get before rounding or adjustments.Once calculated, the standard quota serves as the baseline for further steps in the apportionment process, such as determining lower quotas and making final adjustments to meet the total number of seats available.

population distribution

Population distribution refers to how people are spread across different states or regions, which impacts political representation. In the context of the exercise, it's the distribution of the population across five states that determines each state's share of the total number of congressional seats. Understanding the population distribution is vital because it shapes the apportionment process. With varying populations in each state, the goal is to ensure seats are apportioned fairly relative to each state's share of the total population. This ensures equitable representation of citizens in the legislative body. The method used must take into account both large and small populations, balancing the representation so that even less populated states receive fair representation compared to those with larger populations.