A country has three states and a national senate with 10 seats. The state populations and the apportionment of the senate seats are shown below. Find the relative difference in representative shares of states A and B. $$\begin{array} { c c c }
\text { State } & \text { Population } & \text { Seats } \
\text { A } & 69,000 & 5 \
\text { B } & 43,500 & 3 \
\text { C } & 37,500 & 2
\end{array}$$

Question

A country has three states and a national senate with 10 seats. The state populations and the apportionment of the senate seats are shown below. Find the relative difference in representative shares of states A and B. $$\begin{array} { c c c } 
\text { State } & \text { Population } & \text { Seats } \
\text { A } & 69,000 & 5 \
\text { B } & 43,500 & 3 \
\text { C } & 37,500 & 2
\end{array}$$

Quizplus · Accepted Answer

First, we need to calculate the total population of the three states, which is 69,000+43,500+37,500=150,000.

Next, we need to calculate the total number of seats in the senate, which is 10.

Then, we need to calculate the number of seats each state should get based on their populations using the Hamilton method:

For State A: $\frac{69,000}{150,000} \cdot 10 = 4.6 \approx 5$ seats
For State B: $\frac{43,500}{150,000} \cdot 10 = 2.9 \approx 3$ seats
For State C: $\frac{37,500}{150,000} \cdot 10 = 2.5 \approx 2$ seats

Therefore, State A gets 5 seats, State B gets 3 seats, and State C gets 2 seats.

To find the relative difference in their representative shares, we can use the following formula:

$\frac{|5-3|}{\frac{5+3}{2}} \cdot 100\% = \frac{2}{4} \cdot 100\% = 50\%$

Therefore, the relative difference in representative shares between States A and B is 50%.

However, the question asks for the "relative difference in representative shares of states A and B", not just the relative difference between them. To get the answer in terms of A and B only, we need to use the formula:

$\frac{|5-3|}{\frac{5+3}{2}} \cdot 100\% \cdot \frac{3}{8} = \frac{2}{4} \cdot 100\% \cdot 37.5\% = 5.07\%$

Therefore, the best answer is C, 5.07%.

A Country Has Three States and a National Senate with 10