MATH 131--Class Notes--Spring 08

Part I: Apportionment

Wednesday, 01/23/08, and Friday, 01/25/08: We started discussing the procedure used to elect the

President of U.S. That process is laid out in Article II, Section I, of the U.S. Constitution (Underlined text

means that it was amended. By clicking on any part of the underlined text you will get the respective

amendment.), which states:

“Each state shall appoint, in such manner as the Legislature thereof may direct, a number of electors,

equal to the whole number of Senators and Representatives to which the State may be entitled in the

Congress: but no Senator or Representative, or person holding an office of trust or profit under the

United States, shall be appointed an elector.”

We need to now figure out how many members of Congress each state has. The number of Senators

is 2 per state, which leaves us with the problem of figuring out how many Representatives each state

has. Again, the U.S. Constitution tells us how to go about it. Article I, Section 2, states:

“Representatives and direct taxes shall be apportioned among the several states which may be included

within this union, according to their respective numbers, …. The actual Enumeration shall be made within

three years after the first meeting of the Congress of the United States, and within every subsequent term

of ten years, in such manner as they shall by law direct. The number of Representatives shall not exceed

one for every thirty thousand, but each state shall have at least one Representative;”

(I left out here the mention of the infamous 3/5-rule which we discussed in class.)

Mathematically, there are two stipulations that we need to respect: 1) Apportionment has to be based on

the population numbers and 2) No more than 1 Representative for every 30,000.

After the 1790 census, the 15 states had the following apportionment populations:

CT	236,841
DE	55,540
GA	70,835
KY	68,705
MA	475,327
MD	278,514
NC	353,523
NH	141,822
NJ	179,570
NY	331,589
PA	432,879
RI	68,446
SC	206,236
VA	630,560
VT	85,533

Note: Remember that these numbers do NOT necessarily represent the number of people in each state. 
Due to the 3/5-rule slave states had actually a larger population than the number in the table indicates. 
 
The first suggestion as to how to approach this problem came from one of you was to divide each state’s 
apportionment population by 30,000. E.g., NH’s apportionment population in 1790 is 141,822. Dividing 
by 30,000 gives 141,822/30,000 = 4.7274. To make sure NH does NOT get “more than one for every 
30,000” we round DOWN the quotient 4.7274 to 4. With 4 representatives the average for NH is 
141,822/4 = 35,455.5, which is allowable under the Constitution. If we carry out this calculation for 
each of the 15 states we get:
 

		÷30,000	rounded down
CT	236,841	7.8947	7
DE	55,540	1.851333	1
GA	70,835	2.361167	2
KY	68,705	2.290167	2
MA	475,327	15.84423	15
MD	278,514	9.2838	9
NC	353,523	11.7841	11
NH	141,822	4.7274	4
NJ	179,570	5.985667	5
NY	331,589	11.05297	11
PA	432,879	14.4293	14
RI	68,446	2.281533	2
SC	206,236	6.874533	6
VA	630,560	21.01867	21
VT	85,533	2.8511	2
			112

 
 
Thus this process apportions a House with 112 seats. 
Fortunately (if you’re inclined to think of mathematics as being fun) or unfortunately (if ….) the story 
doesn’t end here because this method causes big problems. (we’ll get to the problems in a bit.) 
 
Let’s start over and ask ourselves the following question: How should we apportion a House with 112
seats given the apportionment populations of the 1790 census? A suggestion from on of you was to 
divide the total population of all states, which is 3,615,920, by 112. This quotient is 
3,615,920/112 = 32,285 and it tells us that ideally each representative represents 32285. Now given 
NH’s apportionment population of 141,822 we divide that number by 32,285 and get 4.3928. Thus 
NH has 4.3928 ideal districts. If we carry out this process for all the states we get:
 

CT	236,841	7.33595
DE	55,540	1.7203
GA	70,835	2.19405
KY	68,705	2.12808
MA	475,327	14.7228
MD	278,514	8.62673
NC	353,523	10.9501
NH	141,822	4.39281
NJ	179,570	5.56203
NY	331,589	10.2707
PA	432,879	13.4081
RI	68,446	2.12006
SC	206,236	6.38798
VA	630,560	19.5311
VT	85,533	2.64931
Total	3,615,920	112

 
Of course we can only apportion integer number of representatives. One we to deal with this problem 
is to apportion each state at first its integer part of ideal number of districts. 
 

CT	236,841	7.33595	7
DE	55,540	1.7203	1
GA	70,835	2.19405	2
KY	68,705	2.12808	2
MA	475,327	14.7228