Four friends decide to buy a bag of 56 gold coins..Each person contribute the following
Sue $320
John $300
Mike $220
Betty$ 160
Please determine the apportionment using
a.Hamilton method
b.Jefferson method
C.Webster method
The various aportionment methods are described here:
http://www.jdawiseman.com/papers/electsys/apportionment.html
To determine the apportionment using different methods, you need to understand the concepts behind each method. Let's go through each method one by one:
a. Hamilton Method:
The Hamilton method, also known as the "largest remainder method," is a proportional method of apportionment. Here's how you can use it to determine the apportionment:
1. Calculate the quota: To determine the quota for each person, divide their contribution by the total contribution of all friends.
- Sue: $320 / ($320 + $300 + $220 + $160) = 0.320
- John: $300 / ($320 + $300 + $220 + $160) = 0.300
- Mike: $220 / ($320 + $300 + $220 + $160) = 0.220
- Betty: $160 / ($320 + $300 + $220 + $160) = 0.160
2. Calculate the initial allocation: Multiply the quota by the total number of coins to get the initial allocation for each person.
- Sue: 0.320 * 56 = 17.92 (round to 18)
- John: 0.300 * 56 = 16.8 (round to 17)
- Mike: 0.220 * 56 = 12.32 (round to 12)
- Betty: 0.160 * 56 = 8.96 (round to 9)
3. Calculate the remainders: Subtract the initial allocation from the quota for each person.
- Sue: 0.320 - 0.320 = 0 (no remainder)
- John: 0.300 - 0.300 = 0 (no remainder)
- Mike: 0.220 - 0.220 = 0 (no remainder)
- Betty: 0.160 - 0.160 = 0 (no remainder)
4. Distribute remaining coins: If there are any remaining coins after the initial allocation, they are distributed to the individuals with the largest remainders. Since there are no remainders in this case, no further distribution is required. The final apportionment is:
- Sue: 18 coins
- John: 17 coins
- Mike: 12 coins
- Betty: 9 coins
b. Jefferson Method:
The Jefferson method, also known as the "divisor method," is another proportional method of apportionment. Here's how you can use it:
1. Calculate the divisor: The divisor is found by dividing the total number of coins by the total number of people.
Divisor = 56 / 4 = 14
2. Calculate the adjusted shares: Divide each person's contribution by the divisor to get their adjusted share.
- Sue: $320 / 14 = 22.86 (round to 23)
- John: $300 / 14 = 21.43 (round to 21)
- Mike: $220 / 14 = 15.71 (round to 16)
- Betty: $160 / 14 = 11.43 (round to 11)
3. Calculate the initial allocation: Assign each person the whole number part of their adjusted share.
- Sue: 23 coins
- John: 21 coins
- Mike: 16 coins
- Betty: 11 coins
4. Distribute remaining coins: If there are any coins remaining after the initial allocation, they are distributed among the individuals with the highest fractional parts of their adjusted shares. In this case, there are no remaining coins, so no further distribution is required. The final apportionment is:
- Sue: 23 coins
- John: 21 coins
- Mike: 16 coins
- Betty: 11 coins
c. Webster Method:
The Webster method, also known as the "major fractions method," is another proportional method of apportionment. Here's how you can use it:
1. Calculate the adjusted fractions: Divide each person's contribution by the divisor (which is the average of the upper and lower quota) to get their adjusted fraction.
- Sue: $320 / 15 = 21.33
- John: $300 / 15 = 20.00
- Mike: $220 / 15 = 14.67
- Betty: $160 / 15 = 10.67
2. Calculate the fractions of remaining coins: Assign each person the whole number part of their adjusted fraction.
- Sue: 21 coins
- John: 20 coins
- Mike: 14 coins
- Betty: 10 coins
3. Calculate the average fractions: Calculate the average fraction by adding the fractional parts together and dividing by the remaining number of coins.
Average fraction = (0.33 + 0.00 + 0.67) / 7 = 0.143
4. Determine the additional coins: Multiply the average fraction by the remaining number of coins and assign the whole number part to each person.
- Sue: 0.143 * 7 = 1.00 (round to 1)
- John: 0.143 * 7 = 1.00 (round to 1)
- Mike: 0.143 * 7 = 1.00 (round to 1)
5. Distribute the additional coins: Distribute the additional coins to the individuals who have the highest fractional part of their adjusted fraction until all additional coins are assigned. In this case, the remaining coins are distributed equally among Sue, John, and Mike, each receiving 1 additional coin. The final apportionment is:
- Sue: 22 coins
- John: 21 coins
- Mike: 15 coins
- Betty: 10 coins
By using these methods, you can determine the apportionment of the 56 gold coins among the four friends.