Math 115 Name: Key
Quiz 4 February 27, 2013
As always, you must SHOW
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 Hamilton’s 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 don’t need to show this – 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 Jefferson’s method.
a. Three possible modified divisors are shown below. Which one(s) could be used for Jefferson’s 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 Jefferson’s method lead to a quota violation in this example? Explain.
No, it does not. While the distribution is not identical to
that of Hamilton’s method, each state did get either its lower or upper quota,
as it should.
Note: The middle option would have worked for
Webster’s method, and would have led to the same apportionment as Jefferson’s
method.