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
To determine the apportionment using different methods, we'll go through each method step by step.
a. Hamilton Method:
1. Calculate the quota for each person: The quota is determined by dividing each person's contribution by the total contributions.
Sue's quota = (Sue's contribution / Total contributions) = (320 / (320 + 300 + 220 + 160)) = 0.355
John's quota = (300 / (320 + 300 + 220 + 160)) = 0.331
Mike's quota = (220 / (320 + 300 + 220 + 160)) = 0.243
Betty's quota = (160 / (320 + 300 + 220 + 160)) = 0.071
2. Assign initial shares: Each person starts with their integer part of their quota as their initial share.
Sue's initial share = 0 (integer part of 0.355)
John's initial share = 0 (integer part of 0.331)
Mike's initial share = 0 (integer part of 0.243)
Betty's initial share = 0 (integer part of 0.071)
3. Calculate adjusted shares: Calculate the adjusted shares by subtracting the initial shares from the quotas.
Sue's adjusted share = 0.355 - 0 = 0.355
John's adjusted share = 0.331 - 0 = 0.331
Mike's adjusted share = 0.243 - 0 = 0.243
Betty's adjusted share = 0.071 - 0 = 0.071
4. Repeat the following steps until all the coins are allocated or no change occurs:
- Find the person with the highest adjusted share (if there are ties, choose the person with the highest quota).
- Assign one coin to that person.
- Update the adjusted shares by subtracting one from the person's adjusted share.
b. Jefferson Method:
1. Calculate the divisor: The divisor is determined by dividing the total contributions by the number of coins.
Divisor = (Total contributions / Number of coins) = ((320 + 300 + 220 + 160) / 56) ≈ 10.714
2. Calculate the adjusted quotas: Adjusted quotas are obtained by dividing each person's contribution by the divisor.
Sue's adjusted quota = (320 / 10.714) ≈ 29.854
John's adjusted quota = (300 / 10.714) ≈ 27.977
Mike's adjusted quota = (220 / 10.714) ≈ 20.491
Betty's adjusted quota = (160 / 10.714) ≈ 14.924
3. Assign shares: Each person receives their integer part of the adjusted quota as their share.
Sue's share = 29
John's share = 27
Mike's share = 20
Betty's share = 14
4. Calculate the remainders: Remainders are calculated by subtracting each person's share from their adjusted quota.
Sue's remainder = 29.854 - 29 ≈ 0.854
John's remainder = 27.977 - 27 ≈ 0.977
Mike's remainder = 20.491 - 20 ≈ 0.491
Betty's remainder = 14.924 - 14 ≈ 0.924
5. Determine the priority of the shares based on the remainders: Sort the persons in descending order of their remainders, and assign the remaining coins in this order. If there's a tie, the person with the higher adjusted quota gets priority.
c. Webster Method:
1. Calculate the divisor: Same as the Jefferson Method, and we obtain a divisor of approximately 10.714.
2. Calculate the adjusted quotas: Same as the Jefferson Method, and we get the following adjusted quotas:
Sue's adjusted quota = (320 / 10.714) ≈ 29.854
John's adjusted quota = (300 / 10.714) ≈ 27.977
Mike's adjusted quota = (220 / 10.714) ≈ 20.491
Betty's adjusted quota = (160 / 10.714) ≈ 14.924
3. Assign initial shares: Each person receives their integer part of the adjusted quota as their initial share.
Sue's initial share = 29
John's initial share = 27
Mike's initial share = 20
Betty's initial share = 14
4. Calculate the modified divisor: The modified divisor is determined by dividing the total contributions by the sum of the initial shares.
Modified divisor = (Total contributions / Sum of initial shares) = ((320 + 300 + 220 + 160) / (29 + 27 + 20 + 14)) ≈ 4.348
5. Calculate the modified quotas: The modified quotas are obtained by dividing each person's contribution by the modified divisor.
Sue's modified quota = (320 / 4.348) ≈ 73.571
John's modified quota = (300 / 4.348) ≈ 69.005
Mike's modified quota = (220 / 4.348) ≈ 50.575
Betty's modified quota = (160 / 4.348) ≈ 36.850
6. Calculate the adjusted shares: Calculate the adjusted shares by subtracting the initial shares from the modified quotas.
Sue's adjusted share = 73.571 - 29 ≈ 44.571
John's adjusted share = 69.005 - 27 ≈ 42.005
Mike's adjusted share = 50.575 - 20 ≈ 30.575
Betty's adjusted share = 36.850 - 14 ≈ 22.850
7. Repeat the process according to the Hamilton method until all coins are allocated or no change occurs.