1. Apportionment Methods
14.3
Apportionment Methods

2. Objectives
Find standard divisors and standard quotas.
Understand the apportionment problem.
Use Hamilton’s method.
Understand the quota rule.
Use Jefferson’s method.
Use Adam’s method.
Use Webster’s method.

3. Standard Divisors and Standard Quotas Standard Divisors
The standard divisor is found by dividing the total population under consideration by the number of items to be allocated.
The standard quota for a particular group is found by dividing that group’s population by the standard divisor.

4. Example 1: Finding Standard Quotas
The Republic of Margaritaville is composed of four states, A, B, C, and D. The table of each state’s population (in thousands) is given below.
State A B C D
Total
Population
275
383
465
767
1890
Standard Quota

5. Example 1: Finding Standard Quotas
The standard quotas are obtained by dividing each state’s population by the standard divisor. We previously computed the standard divisor and found it to be 63.

6. Example 1: Finding Standard Quotas
Standard quota for state A
Standard quota for state B
Standard quota for state C
Standard quota for state D

7. Example 1: Finding Standard Quotas
State A B C D
Total
Population
275
383
465
767
1890
Standard Quota
4.37
6.08
7.38
12.17 30
Notice that the sum of all the standard quotas is 30, the total number of seats in the congress.

8. Apportionment Problem
The standard quotas represent each state’s exact fair share of the 30 seats for the congress of Margaritaville.
Can state A have 4.37 seats in congress?
The apportionment problem is to determine a method for rounding standard quota into whole numbers so that the sum of the numbers is the total number of allocated items.
Can we round standard quotas up or down to the nearest whole number and solve this problem?
The lower quota is the standard quota rounded down to the nearest whole number.
The upper quota is the standard quota rounded up to the nearest whole number.

9. Apportionment Problem
Here’s the apportionment problem for Margaritaville.
There are four different apportionment methods called Hamilton’s method, Jefferson’s method, Adam’s method, and Webster’s method.

10. Hamilton’s Method Calculate each group’s standard quota.
Round each standard quota down to the nearest whole number, thereby finding the lower quota. Initially, give to each group its lower quota.
Give the surplus items, one at a time, to the groups with the largest decimal parts until there are no more surplus items.

11. Hamilton’s Method
Example: Consider the lower quotas for Margaritaville. We have a surplus of 1 seat.
The surplus seat goes to the state with the greatest decimal part. The greatest decimal part is 0.38 for state C. Thus, state C receives the additional seat, and is assigned 8 seats in congress.

12. The Quota Rule
A group’s apportionment should be either its upper quota or its lower quota. An apportionment method that guarantees that this will always occur is said to satisfy the quota rule.
A group’s final apportionment should either be the nearest upper or lower whole number of its standard quota.

13. Jefferson’s Method
Find a modifier divisor, d, such that each group’s modified quota is rounded down to the nearest whole number, the sum of the quotas is the number of items to be apportioned.
The modified quotients that are rounded down are called modified lower quotas.
Apportion to each group its modified lower quota.

14. Example 3: Using Jefferson’s Method
A rapid transit service operates 130 buses along six routes, A, B, C, D, E, and F. The number of buses assigned to each route is based on the average number of daily passengers per route. Use Jefferson’s method with d = 486 to apportion the buses.
Route A B C D E F
Total
Average Number of Passengers
4360
5130
7080
10,245
15,535
22650
65,000

15. Example 3: Using Jefferson’s Method
Solution: Using d = 486, the table below illustrates Jefferson’s method.

16. Adam’s Method
Find a modifier divisor, d, such that when each group’s modified quota is rounded up to the nearest whole number, the sum of the quotas is the number of items to be apportioned.
The modified quotients that are rounded up are called modified upper quotas.
Apportion to each group its modified upper quota.

17. Example 4: Using Adam’s Method
A rapid transit service operates 130 buses along six routes, A, B, C, D, E, and F. The number of buses assigned to each route is based on the average number of daily passengers per route. Use Adam’s method to apportion the buses.
Route A B C D E F
Total
Average Number of Passengers
4360
5130
7080
10,245
15,535
22,650
65,000

18. Example 4: Using Adam’s Method
Solution: We begin by guessing at a possible modified divisor, d, that we hope will work. We guess d = 512.

19. Example 4: Using Adam’s Method
Because the sum of modified upper quotas is too high, we need to lower the modified quotas.
In order to do this, we must try a higher divisor. If we let d = 513, we obtain 129 for the sum of modified upper quotas.
Because this sum is too low, we decide to increase the divisor just a bit to d = instead.

20. Webster’s Method
Find a modifier divisor, d, such that when each group’s modified quota is rounded to the nearest whole number, the sum of the quotas is the number of items to be apportioned.
The modified quotients that are rounded down are called modified rounded quotas.
Apportion to each group its modified rounded quota.

21. Example 5: Using Webster’s Method
A rapid transit service operates 130 buses along six routes, A, B, C, D, E, and F. The number of buses assigned to each route is based on the average number of daily passengers per route. Use Webster’s method to apportion the buses.
Route A B C D E F
Total
Average Number of Passengers
4360
5130
7080
10,245
15,535
22650
65,000

22. Example 5: Using Webster’s Method
Solution: We begin by guessing at a possible modified divisor, d, that we hope will work. We guess d = 502, a divisor greater than the standard divisor, 500.
We would obtain 129 for the sum of modified rounded quotas.
Because 129 is too low, this suggests that d is too large. So, we guess a lower d than 500. Say, d = 498.

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