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Thirteen legislative seats are being apportioned among three states, resulting in the standard quotas shown in the following table.

\begin{tabular}{|l|l|}
\hline
State 1 & 2.67 \\
\hline
State 2 & 6.92 \\
\hline
State 3 & 4.17 \\
\hline
\end{tabular}

Using the Hamilton method, what is the final apportionment?

A. [tex]$2, 7, 4$[/tex]
B. [tex]$3, 7, 4$[/tex]
C. [tex]$2, 6, 4$[/tex]
D. [tex]$3, 6, 4$[/tex]



Answer :

GinnyAnswer
To solve this problem using the Hamilton method, we follow these steps:

1. Calculate the Standard Quota:
- State 1: 2.67
- State 2: 6.92
- State 3: 4.17

2. Initial Apportionment: Assign each state the integer part of its standard quota.
- State 1: [tex]\(\lfloor 2.67 \rfloor = 2\)[/tex]
- State 2: [tex]\(\lfloor 6.92 \rfloor = 6\)[/tex]
- State 3: [tex]\(\lfloor 4.17 \rfloor = 4\)[/tex]

3. Total Initial Apportionment:
[tex]\[ 2 + 6 + 4 = 12 \][/tex]
Thirteen seats are available, so one more seat needs to be allocated because [tex]\(13 - 12 = 1\)[/tex].

4. Calculate the Remainders (Decimal Parts) of Each State:
- State 1: [tex]\(2.67 - 2 = 0.67\)[/tex]
- State 2: [tex]\(6.92 - 6 = 0.92\)[/tex]
- State 3: [tex]\(4.17 - 4 = 0.17\)[/tex]

5. Allocate Remaining Seats Based on the Largest Remainders:
We have one remaining seat. We assign this seat to the state with the largest remainder.
- The remainders are:
- State 1: 0.67
- State 2: 0.92
- State 3: 0.17

The largest remainder is 0.92, which belongs to State 2.

6. Final Apportionment:
- State 1: 2 seats
- State 2: 7 seats (initial 6 + 1 additional seat)
- State 3: 4 seats

By following these calculations, the final apportionment according to the Hamilton method is:
[tex]\[ \boxed{(2, 7, 4)} \][/tex]

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