Math 115†††††††††††††††††††††††††††††††††††††††††††††††††††††††† Name:††††††††††††††††††††††† Key†††††††††††††††††††††††††††††

Quiz 4††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† February 27, 2013

As always, you must SHOW ALL WORK to receive credit

 

The Kingdom of Freedonia has 4,500,000 residents living in 3 states - Groucho (G), Chico (C), and Harpo(H).

The population of G is 2,000,000; the population of C is 1,660,000; and the population of H is 840,000.

The legislature of Freedonia has 120 members, and proportional representation.

1. a. Find the standard divisor, and interpret in a complete sentence what it means in the context of this problem.

 

The total population is 4,500,000, and there are 120 seats, so the standard divisor is 4500000/120 = 37500.

For every 37500 residents of a state, that state is entitled to 1 seat in the legislature.

 

 

b.Find the standard quota for each state (round answers to the nearest hundredth)

 

G:2000000/37500 ≈ 53.33

C:1660000/37500 ≈ 44.27

H:840000/37500 = 22.4

 

c.Use Hamiltonís Method to determine how many legislators each state gets.

Start by giving each state its lower quota.This results in 119 seats being given, so there is 1 left.It goes to the state with the highest fractional part, which is H.So the final apportionment is

G = 53, C=44 and H=23 seats.

 

 

2.If Freedonia changed to a 121 member legislature, the resulting apportionment would be 54 seats for G,

45 seats for C and 22 seats for H (you donít need to show this Ė it is true!).What is paradoxical about this result?What is the name given to this type of paradox?

 

The paradoxical result is that H loses a seat simply because the number of seats was increased.That is the Alabama paradox.

 

 

4.Freedonia decides to keep its legislature at 120 seats, but is going to use Jeffersonís method.

a.Three possible modified divisors are shown below.Which one(s) could be used for Jeffersonís Method?Circle the appropriate modified divisor(s) and then fill in the blanks with the number of seats each state gets.

 

 

G

C

H

Population:

2,000,000

1,660,000

†† 840,000

Modified divisor:††††37000

modified quota:

54.05

44.86

22.70

 

 

 

 

 

Modified divisor:††††37350

modified quota:

53.55

44.44

22.49

 

 

 

 

 

Modified divisor:††††37750

modified quota:

52.98

43.97

22.25

 

†††††† Number of seats:††††††††††††† †††††††† †† †††54††††††† †††††††† 44††††††††22†††† ††††††††††

b.Does Jeffersonís method lead to a quota violation in this example?Explain.

 

No, it does not.While the distribution is not identical to that of Hamiltonís method, each state did get either its lower or upper quota, as it should.

 

Note:The middle option would have worked for Websterís method, and would have led to the same apportionment as Jeffersonís method.