Problem 48 - Thinking Mathematically

Question

A small country is composed of three states, \(A, B\), and \(C\). The country's constitution specifies that congressional seats will be divided among the three states according to their respective populations. Write an apportionment problem satisfying the given criterion. Hamilton's method and Adams's method result in the same apportionment.

Step-by-Step Solution

Answer

An example of an apportionment problem for which Hamilton's method and Adams's method give the same result can be a country composed of three states with populations of 750,000, 550,000 and 700,000, with 100 seats to be allocated according to population. The final using both methods would be state A receiving 37 seats, state B receiving 28 seats, and state C receiving 35 seats.

Step 1: Understanding Hamilton's and Adams's methods

Hamilton's method involves the use of quotas to divide seats between parties or states. The quota here is calculated by dividing the total population by the total number of seats. Each state gets a number of seats equal to the integer part of its standard quota, and the remaining seats are distributed to the states with the highest fractional parts. Adams's method, on the other hand, will not allocate a state a number of seats lower than its lower quota, which is the greatest whole number less than the quota.

Step 2: Develop an apportionment problem

Invent a small country composed of three states, \(A, B\), and \(C\), with populations of 750,000, 550,000 and 700,000, respectively. The country's total population is 2,000,000. The constitution specifies that, say, 100 congressional seats will be divided among the three states according to their respective populations.

Step 3: Apportion the seats using both methods

With Hamilton's method, each 20,000 people (the total population divided by the total seats) would be represented by one congress seat. Thus, state A would receive 37 seats, state B would receive 27 seats, and state C would receive 35 seats. This distribution uses 99 of the available 100 seats, with state B having the largest remaining fractional part. The last seat would then go to state B, making the final distribution 37 seats for A, 28 for B, and 35 for C. Applying Adams's method would yield the same results, as there's no controversial rounding.

Key Concepts

Hamilton's Method

Hamilton's Method is a classic approach to apportioning seats based on population size. It uses the concept known as the 'Standard Quota'. This quota is calculated by dividing the total population by the total number of seats. For each state, the Standard Quota determines how many seats the state should receive if it were possible to divide seats into fractional parts.

• Calculate each state's initial seat allocation by taking the integer part of the Standard Quota.
• If there are still remaining seats, distribute these seats one by one to the states with the largest fractional parts.

For example, if a state has a population of 750,000 and there are 100 seats available for a total population of 2,000,000, each seat would represent 20,000 people. We divide 750,000 by 20,000 which gives the Standard Quota of 37.5. The state initially gets 37 seats and may potentially get an additional seat if it has one of the largest fractional remainders.

Adams's Method

Adams's Method takes a slightly different approach to apportioning seats compared to Hamilton's. This method is more cautious in applying its quotas to ensure fairness and often avoids giving a state fewer seats than its lower quota.

• The lower quota is the greatest integer less than the quota, while the upper quota is the smallest integer greater than the quota.
• Adams's method rounds up only if necessary, starting a bit stricter than Hamilton's in handling decimal approximations.

This method initially uses a divisor that ensures no state receives fewer seats than its lower quota. It incrementally adjusts its counting until all seats are allocated without violating any state’s lower limit. It's particularly useful in scenarios with a need for more conservative seat allocation, like when seats cannot simply be rounded down.

Congressional Seats Allocation

Congressional Seats Allocation is the process of distributing a fixed number of representatives or seats among various states based on population data. It aims to accurately reflect population sizes in legislative bodies.

• This process begins with determining a divisor, the population divided by the number of seats, known as the Standard Divisor in Hamilton's method.
• Each state’s population divided by this divisor gives its Standard Quota, representing its fair share of seats.

There are various methods, such as Hamilton's and Adams's, to allocate these seats, each handling fractional seat distribution differently to maintain fairness and proportionality. The ultimate goal is to ensure each state's representation aligns as closely as possible with its population size, while accounting for fractional quotas in a systematic manner.