1. APPORTIONMENT An APPORTIONMENT PROBLEM:
is a problem that involves dividing up items so that a sum is maintained.
The items being divided in these types of problems are “indivisible” objects.

2. APPORTIONMENT EXAMPLES:
Awarding seats in a representative government, where the total number of delegates is fixed.
Awarding sections of certain courses based on projected enrollment and available time slots.
Dividing up goods/funds for a certain number of agencies based on the number of people they serve.

3. APPORTIONMENT & U.S. History
The problem of fair representation was a major cause of the Revolutionary War.
During the 1787 Constitutional Convention, our current method of representation was proposed:
each state would have 2 Senators
and a number of Representatives based on the population of the state.
In 1790, the first U.S. census was taken.
This census was the basis for the first apportionment of the House of Representatives

4. APPORTIONMENT & U.S. History
Two methods of apportionment were considered by Congress.
One was proposed by Alexander Hamilton;
The other by Thomas Jefferson.
HAMILTON’S METHOD was very straightforward, and was adopted as the method of choice by the Congress.
President Washington had some reservations about this method, and vetoed the bill.
This was the FIRST veto in American history!

5. APPORTIONMENT & U.S. History
So, the first method that was actually used to apportion the U.S. H.R. was JEFFERSON’S METHOD.
It was used until 1842.
Throughout our country’s history, FOUR different apportionment methods have been used.
The method used today is the HUNTINGTON-HILL METHOD.

6. APPORTIONMENT & U.S. History
The current method was adopted in 1941.
Since that time, the size of the H.R. has been fixed at 435 representatives.

7. A brief example...
A fifth grade class is having a canned food drive to help support 3 local agencies.
At the end of each week, they must deliver all of the cans they have collected.
Suppose that one week, they collect 100 cans.
How many cans would they give to each agency?
Certainly, you could divide:
100/3 = 33.3
And decide to give two of the agencies 33 cans, and the other 34 cans.

8. A brief example... Now suppose that you know:
AGENCY #1: serves 1000 people each week
AGENCY #2: serves 200 people each week
AGENCY #3: serves 100 people each week.
Now you might decide to do things a bit differently!
Maybe the number of cans donated should reflect the number served!?

9. A brief example...
This is a simple example of an apportionment problem.
The things being APPORTIONED (divided up) are the cans.
We can not give ‘fractions’ of cans
So we must decide on a fair method of dividing these cans.
Now suppose that you know:
AGENCY #1: serves 1000 people each week
AGENCY #2: serves 200 people each week
AGENCY #3: serves 100 people each week.

10. 100 **handout** #1 #2 #3 1000 200 100 Total served: # cans: 1300
AGENCY: #1 #2 #3
# served
1000
200
100
Total served:
# cans:
1300
Exact quotient
Quotient without decimal
Rank by largest decimal
# of cans donated
100
A fair division would be: 1300/100 = 13 people served per can donated

11. The extra cans will go to...
AGENCY: #1 #2 #3
# served
1000
200
100
Total served:
# cans:
1300
Exact quotient
Quotient without decimal
Rank by largest decimal
# of cans donated
1000/13= 76
1st 77
200/13= 15 15
100/13= 7.692 7
2nd 8
Two more cans must be distributed...
100 98
A fair division would be: 1300/100 = 13 people served per can donated

12. About the example...
This was the method Hamilton proposed for apportionment applications.
In practice, some problems may occur when using this method over a period of time.
But the simplicity of the method makes it useful for small applications such as this.
This example uses a simple method of apportionment called:
“the method of largest fractions”
Here it seems like conventional ‘rounding’
But it may not ALWAYS be the same as conventional rounding, as it does not require the decimal value to be greater than or equal to .5 in order to award the higher integer value.