Section 15.3
Apportionment
Methods
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
INB Table of Contents
2.3-2
Date
Topic
October 22, 2014
Section 15.3 Examples
46
October 22, 2014
Section 15.3 Notes
47
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Page #
What You Will Learn

Standard Divisor

Standard Quota

Lower Quota

Upper Quota

Hamilton’s Method

The Quota Rule

Jefferson’s Method

Webster’s Method

Adam’s Method
15.3-3
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Apportionment
The goal of apportionment is to
determine a method to allocate the
total number of items to be
apportioned in a fair manner.
15.3-4
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Apportionment
Four Methods
• Hamilton’s method
• Jefferson’s method
• Webster’s method
• Adams’s method
15.3-5
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Standard Divisor
To obtain the standard divisor when
determining apportionment, use the
following formula.
total population
Standard divisor =
number of items to be allocated
15.3-6
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Standard Quota
To obtain the standard quota when
determining apportionment, use the
following formula.
population for the particular group
Standard quota =
standard divisor
15.3-7
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Example 1: Determining
Standard Quotas
The Shanahan Law Firm needs to apportion 60 new fax machines to be
distributed among the firm’s five offices. Since the offices do not all have
the same number of employees, the firm’s managing partner decides to
apportion the fax machines based on the number of employees at each
office. Find the standard divisor given there are 1080 employees.
15.3-8
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Example 1: Determining
Standard Quotas
Determine the standard quotas for offices B, C, D, and E of the
Shanahan Law Firm and complete the table.
15.3-9
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Lower and Upper Quota

The lower quota is the standard quota
rounded down to the nearest integer.

The upper quota is the standard quota
rounded up to the nearest integer.
15.3-11
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Hamilton’s Method
To use Hamilton’s method for apportionment, do the
following.
1.
Calculate the standard divisor for the set of data.
2.
Calculate each group’s standard quota.
3.
Round each standard quota down to the nearest
integer (the lower quota). Initially, each group receives
its lower quota.
4.
Distribute any leftover items to the groups with the
largest fractional parts until all items are distributed.
15.3-12
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Example 2: Using Hamilton’s Method
for Apportioning Fax Machines
Use Hamilton’s method to distribute the 60 fax machines for
the Shanahan Law Firm discussed in Example 1.
Office
A
B
C
D
E
Total
Employees
246
201
196
211
226
1080
Standard quota
13.67
11.17
10.89
11.72
12.56
60.01
Lower quota
Hamilton’s
Apportionment
15.3-13
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The Quota Rule
An apportionment for every group
under consideration should always be
either the upper quota or the lower
quota.
15.3-16
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Jefferson’s Method
1. Determine a modified divisor, d, such that when each
group’s modified quota is rounded down to the nearest
integer, the total of the integers is the exact number of
items to be apportioned. We will refer to the modified
quotas that are rounded down as modified lower
quotas.
2. Apportion to each group its modified lower quota.
15.3-17
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Example 3: Using Jefferson’s Method for
Apportioning Legislative Seats
The Republic of Geranium needs to apportion 250 seats in the
legislature. Suppose that the population is 8,800,000 and that there are
five states, A, B, C, D, and E. The 250 seats are to be divided among the
five states according to their respective populations, given in the table.
Use Jefferson’s method to apportion the 250 legislature seats among the
five states. The standard divisor is calculated to be 35,200.
15.3-18
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Webster’s Method
1.
Determine a modified divisor, d, such that when each group’s
modified quota is rounded to the nearest integer, the total of the
integers is the exact number of items to be apportioned. We will
refer to the modified quotas that are rounded to the nearest
integer as modified rounded quotas.
2.
Apportion to each group its modified rounded quota.
15.3-26
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Example 4: Using Webster’s Method for
Apportioning Legislative Seats
Consider the Republic of Geranium and apportion the 250 seats among
the five states using Webster’s method.
15.3-27
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Adams’s Method
1. Determine a modified divisor, d, such that when
each group’s modified quota is rounded up to
the nearest integer, the total of the integers is
the exact number of items to be apportioned.
We will refer to the modified quotas that are
rounded up as modified upper quotas.
2. Apportion to each group its modified upper
quota.
15.3-32
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Example 5: Using Adams’s Method for
Apportioning Legislative Seats
Consider the Republic of Geranium. Apportion the 250 seats among the
five states using Adams’s method.
15.3-33
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Apportionment Methods
Of the four methods we have discussed in
this section,

Hamilton’s method uses standard quotas.

Jefferson’ s method, Webster’ s method,
and Adams’ s method all make use of a
modified quota and can all lead to
violations of the quota rule.
15.3-37
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Apportionment Methods
15.3-38
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