Questions: Ten years, the populations are recounted. Reapportion the 50 seats. County Population Seats Jefferson 120,000 Clay 62,400 Madison 144,800 Jackson 163,200 Franklin 236,800

Ten years, the populations are recounted. Reapportion the 50 seats.

County Population Seats
Jefferson 120,000
Clay 62,400
Madison 144,800
Jackson 163,200
Franklin 236,800

Transcript text: Ten years, the populations are recounted. Reapportion the 50 seats. \begin{tabular}{|l|l|l|} \hline County & Population & \multicolumn{1}{c|}{ Seats } \\ \hline Jefferson & 120,000 & \\ \hline Clay & 62,400 & \\ \hline Madison & 144,800 & \\ \hline Jackson & 163,200 & \\ \hline Franklin & 236,800 & \\ \hline \end{tabular}

Solution

Solution Steps

To reapportion the 50 seats among the counties based on their populations, we can use the method of equal proportions. This method involves calculating a priority value for each seat allocation and distributing the seats one by one to the counties with the highest priority values until all seats are allocated.

Step 1: Initial Population Data

We start with the population data for each county:

Jefferson: 120,000
Clay: 62,400
Madison: 144,800
Jackson: 163,200
Franklin: 236,800

Step 2: Initial Seat Allocation

Each county is initially allocated 1 seat. Therefore, the initial seat allocation is:

Jefferson: 1
Clay: 1
Madison: 1
Jackson: 1
Franklin: 1

Step 3: Calculate Priority Values

We use the method of equal proportions to calculate the priority value for each county. The priority value for a county with population $P$ and $s$ seats is given by: $\text{Priority Value} = \frac{P}{\sqrt{s(s+1)}}$

Step 4: Allocate Remaining Seats

We allocate the remaining 45 seats (since 5 seats are already allocated) one by one to the county with the highest priority value at each step. The final seat allocation after distributing all 50 seats is:

Jefferson: 8 seats
Clay: 4 seats
Madison: 10 seats
Jackson: 11 seats
Franklin: 17 seats