ACTIVITY 4
APPORTIONMENT
I. Introduction
Apportionment is a method of dividing a whole into various parts. This
mathematical investigation has its roots in the U.S. Constitution when the House of
Representatives in 1790 first attempted to apportion itself. Various methods have been
used to decide how many voters would be represented by each member of the House
(Aufmann et.al, 2013).
II. Objectives
At the end of this activity, you should be able to:
1. solve apportionment problem by the method of Hamilton.
2. solve apportionment problem by the method of Jefferson.
3. solve apportionment problem by the method of Webster.
4. solve apportionment problem by the method of Huntington-Hill.
5. determine the differences of the aforementioned methods.
III. Concepts
A. The Hamilton Plan
This method of apportionment was proposed by Alexander Hamilton.
The procedure for this method is as follows (Baltazar et.al, 2018):
1. Divide the total population by the number of representatives. This number is called
the Standard Divisor.
Total Population
Standard Divisor=
Number of Representatives
2. Divide the population of each state by the standard divisor. The whole number part
of the quotient is called Standard Quota. If the Standard Quota is rounded down to a
whole number then it is called the Lower Quota (L). If the Standard Quota is
rounded up to the next whole number, then it is called Upper Quota (U).
In this method, we use the Lower Quota (L).
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State Population
Standard Quota=
Standard Divisor
3. When the total number of Standard Quota (L) is less than the required number of
representatives, the Hamilton plan remediates by assigning one representative to the
state with the largest decimal number. This method continues until the required
number of representatives is attained.
Example1: To illustrate how the Hamilton Plan works, consider the first 7 most
populous state in the US in 2018. Suppose the US Constitution calls for 60
representatives, how many representatives are needed for each state.
Table 1
Standard Quota
Representatives
Standard quota
Representative
Population
Number of
Additional
Quotient/
State
(L)
39,557,045
California 39,557,045 =16.22 16 0 16
2,438,966.77
28,701,845
Texas 28,701,845 =11.77 11 1 12
2,438,966.77
21,299,325
Florida 21,299,325 =8.73 8 1 9
2,438,966.77
19,542,209
New York 19,542,209 =8.01 8 0 8
2,438,966.77
12,807,060
Pennsylvania 12,807,060 =5.25 5 0 5
2,438,966.77
12,741,080
Illinois 12,741,080 =5.22 5 0 5
2,438,966.77
11,689,442
Ohio 11,689,442 =4.79 4 1 5
2,438,966.77
Total 146,338,006
57 60
Source: US Census 2017 State Estimates
146,338,006
Standard Divisor= ≈ 2,438,966.77
60
From the results of Table 1, the Standard Quota (L) totaled to 57, three
representatives short as required by the US Constitution.
The highlighted rows in Table1 refer to the states having the first three largest
decimal remainders. Hence, one representative is added to the state of Ohio, Texas
and Florida.
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B. The Jefferson Plan
This method of apportionment was proposed by Thomas Jefferson. Under this
method, the standard quota is computed in the same way as the Hamilton Plan.
However, if the total of the Standard Quotas (L) does not yield the correct number of
representatives, a Modified Standard Divisor (MSD) is employed. This number is
chosen by trial and error. The Modified Standard Divisor must be less than the
Standard Divisor. This is in order to get a large quotient and eventually get a larger
sum for the Standard Quotas (L) (Baltazar et. al, 2018).
Example 2: Based on Example1, suppose we let the Modified Standard Divisor
(MSD) be equal to 2,350,000. Table 2 shows the results when the Modified Standard
Divisor is 2,350,000.
Table 2
Standard Quota
Representatives
Population
Number of
2,350,000
Quotient
MSD =
State
(L)
39,557,045
California 39,557,045 =16.83 16
2,350,000
28,701,845
Texas 28,701,845 =12.21 12
2,350,000
21,299,325
Florida 21,299,325 =9.06 9
2,350,000
19,542,209
New York 19,542,209 =8.32 8
2,350,000
12,807,060
Pennsylvania 12,807,060 =5.45 5
2,350,000
12,741,080
Illinois 12,741,080 =5.42 5
2,350,000
11,689,442
Ohio 11,689,442 =4.97 4
2,350,000
Total 146,338,006 59
Observe that the total number of representatives is one short of the required
number of representatives which is 60. Hence, 2,350,000 is not a correct MSD. In this
case the MSD needed should be smaller than 2,350,000. Let MSD be 2,330,000. The
apportionment calculation for this MSD is shown in Table 3.
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Table 3
Representative
Population
Number of
Quota (L)
2,330,000
Standard
Quotient
MSD =
State
s
39,557,045
California 39,557,045 =16.97 16
2,330,000
28,701,845
Texas 28,701,845 =12.31 12
2,330,000
21,299,325
Florida 21,299,325 =9.14 9
2,330,000
19,542,209
New York 19,542,209 =8.39 8
2,330,000
12,807,060
Pennsylvania 12,807,060 =5.49 5
2,330,000
12,741,080
Illinois 12,741,080 =5.46 5
2,330,000
11,689,442
Ohio 11,689,442 =5.01 5
2,330,000
Total 146,338,006 60
C. The Webster Method
Another apportionment method proposed by Daniel Webster in 1832. In this
method, Standard Quotas are not rounded down to the nearest integer. Instead, they
are rounded to the nearest integer using this mathematical rules: round up for fractions
of 1/2 or more and down for fractions that are less than 1/2.
The procedure on how to apply this method is as follows:
1. Determine the Standard divisor.
Total Population
Standard Divisor=
Number of Representatives
2. Determine the Standard Quota, for each group.
State Population
Standard Quota=
Standard Divisor
3. Round Standard Quota in the usual manner for each group (round up for 0.5 or
more, round down for less than 0.5).
4. The apportionment for each group corresponds to the values obtained in step 3, and
the sum of the apportionments for all groups must equal the total number of items to
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be apportioned. In case the sum of the apportionments for all groups is not equal to
the required number of representatives, a Modified Standard Divisor (MSD) is
employed.
Example 3: Consider the data in Example 1. Use Webster Method to apportion 60
representatives to each state.
Solution:
Using the standard divisor of 2,438,966.77, the number of representatives for
each state is shown in Table 4.
Table 4
Standar Number of
State Population Quotient
d Quota representatives
39,557,045
California 39,557,045 =16.22 16 16
2438966.77
28701845
Texas 28,701,845 =11.77 12 12
2438966.77
21,299,325
Florida 21,299,325 =8.73 9 9
2438966.77
19542209
New York 19,542,209 =8.01 8 8
2438966.77
12,807,060
Pennsylvania 12,807,060 =5.25 5 5
2438966.77
12,741,080
Illinois 12,741,080 =5.22 5 5
2438966.77
11,689,442
Ohio 11,689,442 =4.79 5 5
2438966.77
Total 146,338,006 60 60
D. The Huntington-Hill Apportionment
The Huntington-Hill apportionment method is very similar to the Webster
method. Instead of rounding-off the Standard Quota to the nearest integer, it depends
now on the geometric mean between the Lower Quota (L) and Upper Quota (U) of
each state.
Consider the following steps to simplify the procedure.
1. Determine the Lower Quota (L) and Upper Quota (U).
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2. Determine the Geometric Mean (rounded off to two decimal places) of each state
between the Upper Quota and Lower Quota:
Geometric Mean= √ LU
3. To determine the number of representatives for each subgroup, consider the
following cases:
state population
i. If is less than the geometric mean, the number of
standard divisor
representatives is the Lower Quota (L).
state population
ii. If is greater than the geometric mean, the number of
standard divisor
representatives is the Upper Quota (U).
4. If the sum of number of representatives is equal to the required number of
representatives then you are done. Otherwise, use a Modified Standard Divisor
(MSD) and repeat the process until the required number is achieved.
Example 4: Consider the data in Example 1. Use the Huntington-Hill method to
apportion 60 representatives to each state.
Solution:
Using the Standard Divisor of 2438966.77, the number of representatives for
each state is shown in Table 5.
Table 5
representatives
Population
Number of
Geometric
Quota(U)
Quota(L)
Quotient
Quota/
Lower
Upper
Mean
State
California 39,557,045 16.22 16 17 16.49 16
Texas 28,701,845 11.77 11 12 11.49 12
Florida 21,299,325 8.73 8 9 8.49 9
New York 19,542,209 8.01 8 9 8.49 8
Pennsylvania 12,807,060 5.25 5 6 5.48 5
Illinois 12,741,080 5.22 5 6 5.48 5
Ohio 11,689,442 4.79 4 5 4.47 5
Total 146,338,006 60
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IV. Practice Exercises
1. Apportionment of Computers. The following table shows the enrollment of the
five Colleges in Western Mindanao State University last summer, S.Y. 2018-2019.
There are 350 computers that are to be apportioned by the university among the five
colleges based on the enrollments.
College Enrollment
Engineering 383
Liberal Arts 186
Physical Education, Recreation & Sports 145
Public Administration Development Studies 109
Teacher Education 373
Total 1196
a. Use the Hamilton method to determine the number of computers to be apportioned
to each college.
b. Use the Jefferson method to determine the number of computers to be apportioned
to each college.
c. Explain why the modified standard divisor used in the Jefferson method cannot be
larger than the standard divisor.
d. Compare the results produced using the Hamilton method with the Jefferson
method.
2. Apportionment of Committees. The given table shows the populations of the four
regions in Mindanao Philippines last 2015.
Region Population
IX- Zamboanga Peninsula 3,629,783
X- Northern Mindanao 4,689,302
XI- Davao 4,893,318
XII- SOCCSKSARGEN 4,545,276
Total 17757679
A committee of 60 people from these regions is to be formed.
a. Use the Webster apportionment principle to determine the apportionment of the 60
committee members.
b. Use the Huntington-Hill apportionment principle to determine the apportionment of
the 60 committee members.
REFERENCES:
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1. Aufmann, R. N., Lockwood, J.S., Nation, R. D., & Clegg, D. K. (2013).
Mathematical Excursions (Third Edition). Belmont CA: Brooks/Cole Cengage
Learning.
2. Baltazar, E., Ragasa, C., & Evangelista, J. (2018). Mathematics in the Modern
World. C & E Publishing Inc.,
3. US Census 2017 State Estimates. Retrieved from:
http://worldpopulationreview.com/states/#statesTable.
