Problem 1 - 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. Use this information to solve Exercises 1-4. $$ \begin{array}{|l|c|c|c|c|c|} \hline \text { State } & \text { A } & \text { B } & \text { C } & \text { D } & \text { Total } \\ \hline \begin{array}{l} \text { Population } \\ \text { (in thousands) } \end{array} & 138 & 266 & 534 & 662 & 1600 \\ \hline \end{array} $$ According to the country's constitution, the congress wil have 80 seats, divided among the four states according their respective populations. a. Find the standard divisor, in thousands. How many people are there for each seat in congress? b. Find each state's standard quota. c. Find each state's lower quota and upper quota.

Step-by-Step Solution

Answer

The standard divisor is 20 thousand people. The standard quotas for the states A, B, C, and D are \(6.9, 13.3, 26.7,\) and \(33.1\) respectively. The lower quotas for the states A, B, C, and D are \(6, 13, 26,\) and \(33\) respectively, and the upper quotas are \(7, 14, 27,\) and \(34\) respectively

Step 1: Calculate the standard divisor

Calculate the standard divisor by dividing the total population of the country by the total number of seats in the congress. Here, the total population is 1600 thousand people, and the number of congress seats is 80. So, the standard divisor is \(\frac{1600}{80} = 20\) thousand people.

Step 2: Calculate each state's standard quota

To find the standard quota for each state, divide each state's population by the standard divisor. \n State A's standard quota is \(\frac{138}{20}\), B's is \(\frac{266}{20}\), C's is \(\frac{534}{20}\), and D's is \(\frac{662}{20}\).

Step 3: Determine each state's lower and upper quota

The lower quota of a state is the highest whole number less than or equal to the state's standard quota, and the upper quota is the smallest whole number greater than or equal to the state’s standard quota. So, round the standard quotas for each state down to the nearest whole number for the lower quotas, and up for the upper quotas.

Key Concepts

Standard Divisor

In apportionment, the standard divisor plays a crucial role. It helps us distribute resources, such as seats, equitably among different groups based on their population.
To find the standard divisor, divide the total population by the total number of available seats. For our exercise, the total population is 1600 thousand people, and there are 80 seats in congress. Therefore, the standard divisor becomes \[\text{Standard Divisor} = \frac{1600}{80} = 20 \text{ thousand people per seat.}\] This calculation indicates that each of the 80 seats in congress represents 20 thousand people. Understanding the standard divisor allows us to determine how many people are "represented" by each seat in a legislative body.

Standard Quota

The standard quota tells us how many seats each state would ideally receive if it could be assigned in fractions, which of course is not possible in real-life scenarios.
To calculate the standard quota for each state, divide its population by the standard divisor. Here’s how it's done for each state:

• State A: Standard Quota = \(\frac{138}{20} = 6.9\)
• State B: Standard Quota = \(\frac{266}{20} = 13.3\)
• State C: Standard Quota = \(\frac{534}{20} = 26.7\)
• State D: Standard Quota = \(\frac{662}{20} = 33.1\)

These quotas are decimal values, providing an approximation of how many seats each state should receive based strictly on population size. The next steps convert these into whole numbers through the upper and lower quotas.

Lower Quota

The lower quota for each state is the highest whole number that does not exceed the state’s standard quota.
This rounding down process is essential as it helps ensure that we do not allocate extra, non-existent seats.
Calculating the lower quota involves simply taking the floor or the integer part of each state's standard quota:

• State A: Lower Quota = 6 (since 6.9 rounds down to 6)
• State B: Lower Quota = 13 (since 13.3 rounds down to 13)
• State C: Lower Quota = 26 (since 26.7 rounds down to 26)
• State D: Lower Quota = 33 (since 33.1 rounds down to 33)

These values are crucial because they serve as the absolute minimum number of seats that each state can receive given its population.

Upper Quota

On the other hand, the upper quota represents the smallest whole number greater than or equal to the state's standard quota.
This is found by rounding the state’s standard quota up to the nearest whole number, ensuring that each state is fairly represented given its respective population size. For each state, the upper quotas are calculated as follows:

• State A: Upper Quota = 7 (since 6.9 rounds up to 7)
• State B: Upper Quota = 14 (since 13.3 rounds up to 14)
• State C: Upper Quota = 27 (since 26.7 rounds up to 27)
• State D: Upper Quota = 34 (since 33.1 rounds up to 34)

The upper quota provides a boundary to prevent assigning more seats than a state justifiably deserves based on the standard quota; it guarantees fair allocation while avoiding overrepresentation.