Consider the following apportionment problem for College Town. Suppose each council member is to represent approximately 2,500 citizens. Use the apportionment plan requested in the problem, assuming there must be 10 representatives.

North: 5,500
South: 9,000
East: 6,400
West: 4,100
(a) Adams' plan.
North:
South:
East:
West:

Start by dividing each population by 2,500. Round to the nearest whole number.

What do you get?

Adams' plan.

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To calculate the apportionment based on Adams' plan, we will use the method of Hamilton's Method. Here are the steps:

Step 1: Calculate the standard quota.
The standard quota is determined by dividing the total population by the number of representatives required.
Standard quota = Total population / Number of representatives
Standard quota = (5,500 + 9,000 + 6,400 + 4,100) / 10

Step 2: Determine the exact quotas for each region.
To find the exact quota for each region, divide its population by the standard quota and round to the nearest whole number.
North: 5,500 / Standard quota
South: 9,000 / Standard quota
East: 6,400 / Standard quota
West: 4,100 / Standard quota

Step 3: Calculate the modified quotas.
Modified quotas are obtained by taking the floor of each region's exact quota.
North: Floor(North exact quota)
South: Floor(South exact quota)
East: Floor(East exact quota)
West: Floor(West exact quota)

Step 4: Calculate the remainders.
Remainders are determined by subtracting the modified quotas from the exact quotas.
North: North exact quota - North modified quota
South: South exact quota - South modified quota
East: East exact quota - East modified quota
West: West exact quota - West modified quota

Step 5: Assign seats based on modified quotas.
Assign the initial seats to each region based on the modified quotas.
North: North modified quota
South: South modified quota
East: East modified quota
West: West modified quota

Step 6: Allocate remaining seats.
Allocate the remaining seats one by one to the regions with the largest remainders until the required number of representatives is reached.
North: North seats + number of remaining seats
South: South seats + number of remaining seats
East: East seats + number of remaining seats
West: West seats + number of remaining seats

Now let's calculate the values:

Step 1:
Standard quota = (5,500 + 9,000 + 6,400 + 4,100) / 10

Step 2:
North: 5,500 / Standard quota
South: 9,000 / Standard quota
East: 6,400 / Standard quota
West: 4,100 / Standard quota

Step 3:
North: Floor(North exact quota)
South: Floor(South exact quota)
East: Floor(East exact quota)
West: Floor(West exact quota)

Step 4:
North: North exact quota - North modified quota
South: South exact quota - South modified quota
East: East exact quota - East modified quota
West: West exact quota - West modified quota

Step 5:
North: North modified quota
South: South modified quota
East: East modified quota
West: West modified quota

Step 6:
North: North seats + number of remaining seats
South: South seats + number of remaining seats
East: East seats + number of remaining seats
West: West seats + number of remaining seats

To determine the apportionment for each council member according to Adams' plan, we need to follow the steps of the Huntington-Hill apportionment method. This method aims to allocate representatives that are proportional to the population of each district while also considering the rounding rules.

Step 1: Calculate the priority values for each district.
To calculate the priority values, we use the following formula:

Priority Value = Population ÷ √(Current Number of Representatives × (Current Number of Representatives +1))

For the North district:
Priority Value = 5,500 ÷ √(0 × (0+1)) = 5,500 ÷ √(0) = 5,500 (division by zero is undefined)

For the South district:
Priority Value = 9,000 ÷ √(0 × (0+1)) = 9,000 ÷ √(0) = 9,000 (division by zero is undefined)

For the East district:
Priority Value = 6,400 ÷ √(0 × (0+1)) = 6,400 ÷ √(0) = 6,400 (division by zero is undefined)

For the West district:
Priority Value = 4,100 ÷ √(0 × (0+1)) = 4,100 ÷ √(0) = 4,100 (division by zero is undefined)

Since the current number of representatives is 0 for each district, the priority values cannot be calculated.

Step 2: Assign one representative to each district.
Since we need 10 representatives among the 4 districts, we can initially assign one representative to each district.

North: 1 representative
South: 1 representative
East: 1 representative
West: 1 representative

Step 3: Calculate the adjusted priority values with the assigned representatives.
To calculate the adjusted priority values, we use the following formula:

Adjusted Priority Value = Population ÷ √(Current Number of Representatives × (Current Number of Representatives +1))

For the North district:
Adjusted Priority Value = 5,500 ÷ √(1 × (1+1)) = 5,500 ÷ √(1 × 2) = 5,500 ÷ √(2) = 5,500 ÷ 1.414 = 3,889.7

For the South district:
Adjusted Priority Value = 9,000 ÷ √(1 × (1+1)) = 9,000 ÷ √(1 × 2) = 9,000 ÷ √(2) = 9,000 ÷ 1.414 = 6,366.2

For the East district:
Adjusted Priority Value = 6,400 ÷ √(1 × (1+1)) = 6,400 ÷ √(1 × 2) = 6,400 ÷ √(2) = 6,400 ÷ 1.414 = 4,525.7

For the West district:
Adjusted Priority Value = 4,100 ÷ √(1 × (1+1)) = 4,100 ÷ √(1 × 2) = 4,100 ÷ √(2) = 4,100 ÷ 1.414 = 2,905.0

Step 4: Allocate the remaining representatives based on adjusted priority values.
We allocate the remaining representatives by giving the representative to the district with the highest adjusted priority value until we reach the total desired number of representatives (10 in this case).

Allocating the remaining representatives:
North: 1 representative
South: 7 representatives
East: 1 representative
West: 1 representative

The final apportionment according to Adams' plan is as follows:

North: 1 representative
South: 8 representatives
East: 1 representative
West: 1 representative