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Mathematics in Contemporary Society: A Work in Progress

Elizabeth O’Neil

Section 6.2 Apportionment

The United States Capitol Building
Figure 6.2.1. US Capitol Building: home of the House of Representatives
Apportionment is the problem of dividing up a fixed number of things among groups of different sizes. In politics, this takes the form of allocating a limited number of representatives to constituent groups. This problem, presumably, is older than the United States, but the best-known ways to solve it have their origins in the problem of assigning each state an appropriate number of representatives in the new Congress when the country was formed. States also face this apportionment problem in defining how to draw districts for state representatives.
In this section, we apply four apportionment methods in social, economic, and political contexts, and compare the results.

Subsection 6.2.1 The Problem of Apportionment

The legislative, or law-making, branch of the United States government consists of the Senate and the House of Representatives. The members of the Senate are called senators, and the members of the House of Representatives are called congresspeople or representatives. Each state is represented by two senators and a number of congresspeople that is proportional to the population of the state.
The total number of congresspeople has changed multiple times over the history of the United States before laws were enacted to fix the number of representatives. The Apportionment Act of 1911 brought the total number of representatives to 435, and the Permanent Apportionment Act of 1929 established that the number would not change again.
It sounds simple enough: a state with \(x\) percent of the population should receive \(x\) percent of the the representatives. The problem is that the calculations usually do not come out neat and simple. For example, Washington State residents make up about 2.325% of the US population, so they should have 2.325% of the 435 representatives. That is 10.11 people. Using standard rounding rules, Washington State should receives 10 representatives. If we continue this approach and assign whole numbers of representatives to each state, we could end up with too few or too many representatives.
This same problem also arises in other contexts, such as figuring out how many school counselors each high school shoud have on staff, or disbursing stocks.
The requirements are as follows:

Apportionment Rules.

  • The things being apportioned can only exist in whole numbers. (A state can’t have, for example, 2.7 representatives.)
  • We must apportion exactly the number of things we have. (The total number of representatives must be exactly 435.)
  • Each group must get at least one of the things being apportioned. (Each state must get at least one representative.)
  • The number of things assigned to each group should be approximately proportional to the population of the group. (Each congressperson should represent close to the same number of people.)
In addition to the mathematical challenges of the calculations themselves, it is important to aknowledge that the population data being used may not be accurate. According the the Census Bureau, the 2020 census overcounted homeowners, non-hispanic white residents, and asian residents. It undercounted renters, black or African American residents, American Indian or Alaska Native residents, and Hispanic or Latio residents.
 1 
US Census Bureau. (2022, March 10). Census Bureau releases estimates of Undercount and overcount in the 2020 census. Census.gov. www.census.gov/newsroom/press-releases/2022/2020-census-estimates-of-undercount-and-overcount.html
Historically, undercounts were intentional: prior to 1868, each enslaved person was counted as 3⁄5 of a resident, and prior to 1940, census counts explicitly excluded people who were classified as "Indians not taxed", meaning Indigenous Americans living on reservations or unsettled lands.

Subsection 6.2.2 Hamilton’s Method

Alexander Hamilton proposed the method that now bears his name. His method was approved by Congress in 1791, but was vetoed by President Washington. It was later adopted in 1852 and used through 1911. He began by determining, to several decimal places, how many representatives each state should get. Then he used the size of the remainders and the total number of representatives to determine whether each state’s apportionment should be "rounded down" or "rounded up". (We are using the terms "round down" and "round up" loosely here, as he did not follow traditional rounding rules.)

Hamilton’s Method for Apportionment.

For historical accuracy, we use the language of states and representatives in these steps. However, this method can be used in any application of apportionment.
  1. Determine the divisor, which is the number of people each representative should represent. It is found by dividing the total population of the country by the number of representatives to be apportioned.
  2. Divide each state’s population by the divisor to determine how many representatives it should have. This number is called the quota.
  3. Truncate ("round down") each state’s quota to a whole number, but be sure to keep a record of the original value. The truncated values are called the lower quotas.
  4. Add the lower quotas and subtract from the total number of representatives to determine how many remain. Assign the remaining representatives, one each, to the states in order of the size of the decimal portion of their quota. (Exception: if a state’s lower quota is 0, they must receive an one of the remaining representatives even if the decimal portion does not warrant it.)

Example 6.2.2.

The state of Delaware has three counties: Kent, New Castle, and Sussex. The Delaware state House of Representatives has 41 members. If Delaware wants to divide this representation along county lines (which is not required, but let’s pretend they do), let’s use Hamilton’s method to apportion them. The populations of the counties are as follows (from the 2020 Census):
County Population
New Castle 570,719
Sussex 237,378
Kent 181,851
Total 989,948
Step 1: Find the divisor. Divide the total population 989,948 by the number of representatives, 41, to obtain 24,145.073. This number means that each representative should represent about 24,145 people. For accuracy, store the divisor in your calculator’s memory or use a spreadsheet.
Step 2: Find each county’s quota. Divide each county’s population by the divisor, using either your calculator’s memory or a cell reference in a spreadsheet.
Step 3: Find each county’s lower quota. Truncate (round down) each county’s quota. If using a spreadsheet, this can be accomplished quickly using the built-in formula, TRUNC. The results of steps 2 and 3 are as follows:
County Population Quota Lower Quota
New Castle 570,719 23.63708 23
Sussex 237,378 9.831322 9
Kent 181,851 7.531599 7
Total 989,948 41
At this point, we can say that New Castle should have at least 23 representatives, Sussex should have at least 9, and Kent should have at least 7.
Step 4: Assign remaining representatives. Adding the lower quotas, we find that only 39 of the 41 have been assigned. The two remaining will be given to the counties whose quotas had the largest decimal portions.
The decimal portion for New Castle’s quota is .63708, Sussex’s is .831322, and Kent’s is .531599. The two largest are New Castle and Sussex, so the two remaining representatives will be apportioned to these counties.
County Population Quota Lower Quota Final Apportionment
New Castle* 570,719 23.63708 23 24
Sussex* 237,378 9.831322 9 10
Kent 181,851 7.531599 7 7
Total 989,948 41 39 2 41

Example 6.2.3.

Use Hamilton’s method to apportion the 75 seats of Rhode Island’s state House of Representatives among its five counties. Their populations are below.
County Population
Providence 660,741
Kent 170,363
Washington 129,839
Newport 85,643
Bristol 50,793
Solution.
We add to find that the total population of the state is 1,097,379 people. Then we divide the population to find that the devisor is about 14,631.72. Storing this divisor in the memory of our calculator or using a cell reference in a spreadsheet, we build the following table to find the apportionment.
County Population Quota Lower Quota Final Apportionment
Providence 660,741 45.15812 45 45
Kent* 170,363 11.6434 11 12
Washington* 129,839 8.8738 8 9
Newport* 85,643 5.8532 5 6
Bristol 50,793 3.4714 3 3
Total 1,097,379 75 72 75
Note that there were three additional representatives which were assigned to Kent, Washington, and Newport counties.
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