math 132: foundations of mathematics

May 24, 2010 Math 132: Foundations of Mathematics

Math 132:Math 132:Foundations of MathematicsFoundations of Mathematics

Amy LewisMath Specialist

IU1 Center for STEM Education


14.3 Apportionment Methods

• Find standard divisors and standard quotas.

• Understand the apportionment problem.• Use Hamilton’s, Jefferson’s, Adam’s, and

Webster’s methods.


Apportionment• Why do we have 2 houses in Congress?• How many members of the House of

Representatives are there?• How do we decide how many

representatives each state gets?


Sampling & Populations

The Republic of Margaritaville by StateState A B C D TotalPopulation

(in thousands)275 383 465 767 1890

If Margaritaville has 30 seats in the Congress, how many seats should each state get?


Allocating Seats

• Standard Divisor– The quotient of the total population under

consideration and the number of items to be allocated.

• What is the standard divisor for Margaritaville?

• What does the standard divisor mean?


Allocating Seats

• Standard Quota– The quotient of the group’s population and

the standard divisor.• What is the standard divisor for each state

in Margaritaville?


Sampling & Populations

The Republic of Margaritaville by StateState A B C D TotalPopulation

(in thousands)275 383 465 767 1890

Standard Quota 4.37 6.08 7.38 12.17 30

So, what’s the problem?


Sampling & PopulationsThe Republic of Margaritaville by State

State A B C D TotalPopulation

(in thousands)275 383 465 767 1890

Standard Quota 4.37 6.08 7.38 12.17 30

Lower Quota 4 6 7 12 29

Upper Quota 5 7 8 13 33


Hamilton’s Method

• Calculate each group’s standard quota.• Round each standard quota down to the

nearest whole number (the lower quota). Initially, give each group its lower quota.

• Give the surplus items, one at a time, to the groups with the largest decimal parts until there are no more surplus items.


Other Methods

• Jefferson’s Method• Adam’s Method• Webster’s Method


Jefferson’s Method

• Find a modified divisor, d, such that when each group’s modified quota is rounded down to the nearest whole number, the sum of the whole numbers for all the groups is the number of items to be apportioned. The modified quotients that are rounded down are called modified lower quotas.

• Apportion to each group its modified lower quota.


Adam’s Method

• Find a modified divisor, d, such that when each group’s modified quota is rounded up to the nearest whole number, the sum of the whole numbers for all the groups is the number of items to be apportioned. The modified quotients that are rounded up are called modified lower quotas.

• Apportion to each group its modified upper quota.


Webster’s Method

• Find a modified divisor, d, such that when each group’s modified quota is rounded to the nearest whole number, the sum of the whole numbers for all the groups is the number of items to be apportioned. The modified quotients that are rounded are called modified rounded quotas.

• Apportion to each group its modified rounded quota.


No Homework!

Next Session: Thursday, May 27

Last Class: Friday, May 28th!!!

math 132: foundations of mathematics

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