To fairly apportion the 102 representatives among the 4 states using Hamilton's, Jefferson's, and Webster's methods, we need to first calculate the priority values for each method.
Here are the populations of the 4 states:
- State A: 10,000,000
- State B: 8,000,000
- State C: 5,000,000
- State D: 3,000,000
Hamilton's method:
- Step 1: Calculate the total population of all 4 states.
Total population = 10,000,000 + 8,000,000 + 5,000,000 + 3,000,000 = 26,000,000
- Step 2: Divide the total number of representatives by the total population.
Priority value = 102 / 26,000,000 = 0.00392307692
- Step 3: Multiply the priority value by each state's population to get its number of representatives.
State A: 10,000,000 * 0.00392307692 = 39.2307692 representatives (rounded to 39)
State B: 8,000,000 * 0.00392307692 = 31.3846154 representatives (rounded to 31)
State C: 5,000,000 * 0.00392307692 = 19.6153846 representatives (rounded to 20)
State D: 3,000,000 * 0.00392307692 = 11.7692308 representatives (rounded to 12)
Jefferson's method:
- Step 1: Divide the total number of representatives by 2.
Half of the representatives = 102 / 2 = 51
- Step 2:
- State A: sqrt(10,000,000) = 3,162.27766
- State B: sqrt(8,000,000) = 2,828.42712
- State C: sqrt(5,000,000) = 2,236.06798
- State D: sqrt(3,000,000) = 1,732.05081
- Step 3: Divide each state's result from step 2 by the sum of all states' results from step 2 and multiply the result by the total number of representatives.
State A: (3,162.27766 / (3,162.27766 + 2,828.42712 + 2,236.06798 + 1,732.05081)) * 102 = 39 representatives (rounded to 39)
State B: (2,828.42712 / (3,162.27766 + 2,828.42712 + 2,236.06798 + 1,732.05081)) * 102 = 35 representatives (rounded to 35)
State C: (2,236.06798 / (3,162.27766 + 2,828.42712 + 2,236.06798 + 1,732.05081)) * 102 = 22 representatives (rounded to 22)
State D: (1,732.05081 / (3,162.27766 + 2,828.42712 + 2,236.06798 + 1,732.05081)) * 102 = 6 representatives (rounded to 6)
Webster's method:
- Step 1: Calculate the average population of the 4 states.
Average population = (10,000,000 + 8,000,000 + 5,000,000 + 3,000,000) / 4 = 6,500,000
- Step 2: Calculate each state's excess population by subtracting the average population from its population.
State A's excess population = 10,000,000 - 6,500,000 = 3,500,000
State B's excess population = 8,000,000 - 6,500,000 = 1,500,000
State C's excess population = 5,000,000 - 6,500,000 = -1,500,000
State D's excess population = 3,000,000 - 6,500,000 = -3,500,000
- Step 3: Calculate each state's modified excess population by taking the square root of its absolute excess population.
State A's modified excess population = sqrt(3,500,000) = 1,870.82869
State B's modified excess population = sqrt(1,500,000) = 1,224.74487
State C's modified excess population = sqrt(1,500,000) = 1,224.74487
State D's modified excess population = sqrt(3,500,000) = 1,870.82869
- Step 4: Calculate each state's priority value by dividing its modified excess population by the sum of all states' modified excess populations and multiplying the result by the total number of representatives.
State A's priority value = (1,870.82869 / (1,870.82869 + 1,224.74487 + 1,224.74487 + 1,870.82869)) * 102 = 38 representatives (rounded to 38)
State B's priority value = (1,224.74487 / (1,870.82869 + 1,224.74487 + 1,224.74487 + 1,870.82869)) * 102 = 25 representatives (rounded to 25)
State C's priority value = (1,224.74487 / (1,870.82869 + 1,224.74487 + 1,224.74487 + 1,870.82869)) * 102 = 25 representatives (rounded to 25)
State D's priority value = (1,870.82869 / (1,870.82869 + 1,224.74487 + 1,224.74487 + 1,870.82869)) * 102 = 14 representatives (rounded to 14)
Therefore, using Hamilton's method, State A gets 39 representatives, State B gets 31 representatives, State C gets 20 representatives, and State D gets 12 representatives. Using Jefferson's method, State A gets 39 representatives, State B gets 35 representatives, State C gets 22 representatives, and State D gets 6 representatives. Finally, using Webster's method, State A gets 38 representatives, State B gets 25 representatives, State C gets 25 representatives, and State D gets 14 representatives.