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28 February 2016

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Methods of apportionment are mathematical techniques used to allocate resources such as police officers in a certain city or congressional seats. These techniques are quite complicated and are based on several variables depending on which method one is choosing to use. Two of the most famous methods for solving apportionment problems are known as The Hamilton Method and The Huntington-Hill Principle. In this paper we will start by discussion the Hamilton Method by pretending that 10 different states are to be assigned 100 congressional seats by using apportionment. The Hamilton Method of Apportionment

The Hamilton Method is a “common sense” method that Alexander Hamilton used to apportion the very first United States congress. With that being said, one could pretend that they have to divide or apportion 100 congressional seats among 10 states of the Union. To do this using The Hamilton Method the population for each of the 10 states would have to be known. Then the population for all 10 states would need to be totaled. Once this total is received, then the total population will need to be divided into each individual states population. For example, state 1 has a population of 1500 and state 2 has a population of 2000 for a population total of 3500 (Pirnot, n.d.). 1500/3500 = 0.42857143 (state 1)

2000/3500 = 0.57142857 (state 2)
Next the decimal places in the numbers above will need to be moved two places to the right and round to the nearest hundred if necessary. This should give the answers 42.86 for state 1 and 57.14 for state 2. These numbers are known as your Hamilton numbers. Now in The Hamilton Method the numbers before the decimal are known as the Integers and they represent how many seats each state gets, and the decimal numbers are known as the fractional numbers determine who will get the remaining seats, if there are any. The remaining seats are given to the states that have the largest fractional numbers first and work their way down. Therefore, assuming there are a 100 seats to be apportioned, then 42 seats will go to state one and 57 seats will go to state 2. However, we must remember that there are 100 seats to apportion. 42+57 = 99, therefore there is 1 remaining seat to be apportioned. Since state 1 has a fractional part of .86 and state 2 has a fractional part of 14, state 1 receives the extra seat because it has the larger fractional number (Pirnot, n.d.).

Now let us get back to the original problem of 10 states apportioning 100 seats. Seeing how this is a rather large problem with large numbers one might want to use a calculator or spread sheet to determine how many seats are assigned to each start. By using a spread sheet one can see that the seats are assigned as followed:



Assign Additional
Insert Below
% Representation
Integer Part
Fractional Part
Members Manually

The question now becomes, are these seats all apportioned fairly? To find out we need to know the “Average Constituency” of each state.” The Average Constituency measures the fairness of an apportionment (Pirnot, n.d. pg. 534).” To find the Average Constituency one would take the population of a state and divide it by the assigned seats, and the compare them to determine fairness. Giving an example from the calculations above, one can see that state 1 has a population of 15475 and state 2 has a population of 35644. State 1 has 3 assigned seats and state 2 has 7 (Pirnot, n.d.). 15457/3 = 5158


35644/7 = 5092 Constituents
In comparison, just by looking at the number of constituent verses the number of seats; one would assume that the states are not really represented fairly, because state one has more constituents and fewer representatives than state 2. Below is the average constituency of all 10 states in the given problem above (Pirnot, n.d.).

Having these numbers to compare helps us get a better understanding of how poorly some state can be represented. One would like to think that having the same amount of constituents in each state would be the sure-fire answer to solving that problem, but according to (Pirnot, n.d., pg. 535), “it is usually not possible to achieve this ideal when making and actual apportionment.” Therefore we should at least try to make average constituencies as equal as possible. One can actually measure this by using what is called “Absolute Unfairness” (Pirnot, n.d.).

Absolute Unfairness

Absolute Unfairness is defined as being “the difference in average constituencies” (Pirnot, n.d). To find the absolute unfairness of two of the states given above, we should use this simple formula. (average
constituencies of state A) – (average constituencies of state B) =

Now to use this formula to see if any of the states in our problem has any absolute unfairness, we will pick states 3 and 2 to use as a comparison. (state 3) 5486 – (state 2) 5092 = 394 Absolute Unfairness

One can now see that the absolute unfairness of constituencies between states 3 & 2 is 394. Therefore, according to absolute unfairness these two states are not equally represented. The constituencies would have to have been the same in both states in order for the states to be equally represented, and this is rarely the case. With that being said, absolute unfairness is not what one would want to use to measure the unfairness of two apportionments, because it really show the imbalance of an apportionment of two states. In other words, absolute unfairness might give some people the wrong conclusion about the imbalance. Meaning, just because there is a large absolute unfairness doe not predict a greater imbalance. In all actuality, the sized of the state needs to be taken into consideration as well, when measuring unfairness. For example, in a state with a larger amount of voters like Texas, if a politician loses by 100,000 to 1,500,000 votes, it is considered a close race, in a small town election where the votes tally as 100 to 30 then the difference is considered to be quite large. This is why it is important to measure the “relative unfairness” (Pirnot, n.d).

Relative Unfairness

“Relative unfairness considers the size of constituencies in a calculating absolute unfairness (Pirnot, n.d. pg. 356).” To calculate the relative unfairness of apportioned seats between two states one would use this formula. absolute unfairness of apportionment / smaller average constituency of the two states =

So, using the two states were given to figure out the absolute unfairness we can say that 0.08 is the relative unfairness of the two states. 394 (absolute unfairness) / 5092 (state 2) = 0.07737628 (rounded to the nearest hundred) = 0.08 relative unfairness
To get a comparison we will use two other states. State 1 has 5158 average constituencies, and state 4 has 5196 for a total of 38 absolute unfairness. Remember to subtract the state with the smallest amount of constituencies from the larger state’s constituencies to get the absolute unfairness. To find the relative unfairness, take the absolute unfairness and divide it by the state with the lowest constituency number which was state 1. 38/5158 = 0.007367197 (rounded to the nearest hundred) = 0.007 relative unfairness
The relative unfairness of states 1 and 4 is 0.007. Therefore in comparison with states 2 and 3’s larger relative unfairness of 0.08, it tells us that there is more of an unfair apportionment for states 2 and 3 than the states of 1 and 4. In other words, when comparing relative unfairness the larger number in comparison means it’s apportioned more unfairly. However, due to the fact that all of these calculations were based on The Hamilton Method all of the information could possibly change if there were a sudden population change due to growth. This is called a population paradox (Pirnot, n.d.).

Population Paradox

A population paradox occurs when one state grows in population faster than the other, and the state with the faster growth loses a seat or representative to the other state (Pirnot, n.d.). For example, state 6 has a population of 85663 and state 8 has a population of 84311 for a total population of 169974. Now we want to assign these two states 100 seats of congress using The Hamilton Method. First take the total population and divide by 100 seats to get our standard divisor (Pirnot, n.d.). 169976/100 = 1699.74 (standard divisor)

Now divide each state by 1699.74 to get your Hamilton Number. 85663/1699.74 = 50.4 (state 6)
84311/1699.74 = 49.6 (state 8)
Hamilton Numbers Lower Quota (Integer) Fractional Part Assigned Seats state 6: 50.6 50 0.4 50 state 8: 49.6 49 0.6 50 = 100
seats (Notice that the total for the integer or lower quota is 99, so therefore there was one extra seat to assign and it went to the state with the highest fractional part which was state 8.)

Now if we increase state 6’s population by 1000 and state 8’s population by 100 you will get a population paradox. To find out how this happens you will need to make the same calculations by using The Hamilton Methods, except you will need to increase the population of both states to get the new totals, integers, fractional parts, and assigned seats (Pirnot, n.d.). (state 6) 85663 + 1000 = 86663 (new population)

(state 8) 84311 + 100 = 84411 (new population)
86663 + 84411 = 171074 (total population)
171074/100 = 1710.74 (standard divisor)
86663/ 1710.74 = 50.66 (Hamilton number)
84411 / 1710.74 = 49.34 (Hamilton number)
Notice that the fractional part has changed for the two states Hamilton numbers. Therefore since state 6 now has the larger fractional part due to the population change it will take the extra seat from state 8 for a total of 100 seats. State 6 will have 51 and state 8 will have 49. To find out which state received the greatest amount of growth we simply divide the growth by the original population (Pirnot, n.d.). 1000/85663 = 1.16% (state 6) and 100/84311 (state 8) = 1.19% One can now see that this is a population paradox that occurs when using The Hamilton Method, because the state that had the most growth in population lost a seat to the state with the least of amount of growth due to how the fractional part of the Hamilton numbers changed. However, a population paradox is not the only paradox associated with The Hamilton Method. The Alabama Paradox has also shown its ugly face when using The Hamilton Method of apportionment (Pirnot, n.d.).

Alabama Paradox

In 1870, after the census, the Alabama paradox surfaced. This occurred when a house of 270 members increased to 280 members of the House of Representatives causing Rhode Island to lose one of its 2 seats. Later on after the census a man by the name of C.W. Seaton calculated the apportionments for all House sizes that ranged from 275 to 350 members. According to (, n.d.), “He then wrote a letter to Congress pointing out that if the House of Representatives had 299 seats, Alabama would get 8 seats but if the House of Representatives had 300 seats, Alabama would only get 7 seats.” This became known as the Alabama paradox. It is simply when the total number of seats to be apportioned increases, and in turn causes a state to lose a seat. There is a method called the Huntington-Hill Principle that helps avoid the Alabama paradox. This method only apportions the new seats when the House of Representatives increases in size. This is what avoids the Alabama paradox. To apply the Huntington-Hill Principle we would use this simple algebraic formula below for each of the states for comparison that are in question of gaining the extra seat (Pirnot, n.d.). (population of y)^2 / y * (y + 1)

Let us say that Y has a population of 400 and let Y equal 5, and let’s say that X has a population of 300 and let X equal 2. Now let us see which one of these gets the extra seat. (400)^2 / 5 * (5 + 1) and (300)^2 / 2 * (2 + 1)

160,000 / 5 * 6 = 90,000 / 2 * 3 =
= 160,000 / 30 = 90,000 / 6
= 5333.33 = 15,000
By using the Huntington-Hill Principle method of apportionment we can now compare the two states to see which one will get the extra seat. Notice that state X with the Huntington -Hill number of 15,000 is great than that of state Y, therefore state X should get the extra seat. With this being said, if I were to use apportionment as my way of assigning seats to the House of Representatives, I would definitely choose to use The Huntington-Hill Principle method of apportionment (Pirnot, n.d.).

Apportionment is a great way to achieve fair representation as long as we are not using the Hamilton Method. The Hamilton Method has the possibility of cause three types of paradoxes: the Alabama paradox, the population paradox, and the new states paradox. Even though the Hamilton Method does not violate the quota rule, avoiding these paradoxes are more important when trying to give equal representation to each state of the Union. There are other apportionment methods that are equally as great as The Huntington-Hill Principle, such as Webster’s method (Pirnot, n.d.).

Webster’s Method of apportionment

What really sets Webster’s method apart from Huntington-Hill is that Webster uses modified divisor instead of a standard divisor to calculate what is called a modified quota or Integer. A modified divisor is a divisor that is smaller than the standard divisor. A modified quota is a quota that is larger than the standard quota. One would basically pick a number smaller than the standard divisor and work their way down until they end up with one that will give them and modified quota. Once that quota or Integer is found then it will need to be rounded either up or down depending on the number (the standard way of rounding) to determine who will get the allotted seats. Webster’s method is actually exactly like Huntington-Hill except for the rounding part, and it was the apportionment method used until it was replaced by Huntington-Hill (Pirnot, n.d.)


Apportionment methods are a great way to equally divide certain numbers of substances among varying numbers, as long as one stays away from the Hamilton Method. Sure the Hamilton Method is quite simple to use, but causes many problems such as paradoxes. The Alabama paradox, the population paradox, and the new state paradox are among the ones that the Hamilton Method can cause. This causes states to lose seats due to new Representatives, new population growth and even a new border or state joining the Union. Thankfully there were some people out there that were smart enough to come up with new methods of apportionment that eliminated the issues of the paradoxes, such as the Huntington-Hill method and Webster’s method. Both of these methods are the best apportionment methods out there to help make sure that states are represented equally by congress. , and considering the fact that I live in a very poor, poverty stricken state, I want to make sure that our state gets the best representation possible, so that maybe our representatives will be able to listen to all of their constituents and do something to help boost our economy, increase employment rates, and bring people out of poverty.

Apportionment Paradoxes. Alabama Paradox. Retreived from the Alabama Paradox Pirnot, T. Mathematics All Around, Fourth Addition. Apportionment. Retrieved from

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Apportionment. (28 February 2016). Retrieved from

"Apportionment" StudyScroll, 28 February 2016,

StudyScroll. (2016). Apportionment [Online]. Available at: [Accessed: 30 September, 2023]

"Apportionment" StudyScroll, Feb 28, 2016. Accessed Sep 30, 2023.

"Apportionment" StudyScroll, Feb 28, 2016.

"Apportionment" StudyScroll, 28-Feb-2016. [Online]. Available: [Accessed: 30-Sep-2023]

StudyScroll. (2016). Apportionment. [Online]. Available at: [Accessed: 30-Sep-2023]

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