Chapter 4, Problem 73E

Lowndes’s Method. Exercises 73 and 74 refer to a variation of Hamilton’s method known as Lowndes’s method, first proposed in 1822 by south Carolina Representative William Lowndes. The basic difference between Hamilton's and Lowndes's methods is that in Lowndes's method, after each state is assigned the lower quota, the surplus seats are handed out in order of relative fractional parts. (The relative fractional part of a number is the fractional part divided by the integer part. For example, the relative fractional part of 41.82 is $\frac{0.82}{41}=0.02$, and the relative fractional part of 3.08 is $\frac{0.08}{3}=0.027$. Notice that while 41.82 would have priority over 3.08 under Hamilton's method, 3.08 has priority over 41.82 under Lowndes's method because 0.027 is greater than 0.02.)

a. Find the apportionment of Parador’s Congress (Example $4.3$) under Lowndes’s method.

b. Verify that the resulting apportionment is different from each of the apportionments found under the other methods discussed in the chapter. In particular, list which states do better under Lowndes’s method than under Hamilton’s method.

Example 4.3 The Congress of Parador

Parador is a small republic located in Central America and consists of six states: Azucar, Bahia, Cafe, Diamante, Esmeralda, and Felicidad (A, B, C, D, E, and F for short). There are 250 seats in the Congress, which, according to the laws of Parador, are to be apportioned among the six states in proportion to their respective populations. What is the “correct” apportionment?

Table $4-3$ shows the population figures for the six states according to the most recent census.

Table $4-3$

Republic of Parador (populations by state)

State	A	B	C	D	E	F	Total
Population	$1,646,000$	$6,936,000$	$154,000$	$2,091,000$	$685,000$	$988,000$	$12,500,000$