Calculate the Huntington-Hill number for a state. To keep the calculations manageable, the population has been rounded to the nearest tenth of a million. What is the Huntington-Hill number for state X
The Correct Answer and Explanation is :
Huntington-Hill Number Calculation
The Huntington-Hill number is a value used in the Huntington-Hill Method of Apportionment, which is applied to distribute seats in the U.S. House of Representatives proportionally to states based on their population. The formula for the Huntington-Hill number for a state is: Huntington-Hill Number=Pn(n+1)\text{Huntington-Hill Number} = \frac{P}{\sqrt{n(n+1)}}
Where:
- PP is the population of the state.
- nn is the current number of seats allocated to the state.
Step-by-Step Calculation
- Determine State Population (PP):
The population for state XX is given (rounded to the nearest tenth of a million). Let’s assume the population is P=4.3×106P = 4.3 \times 10^6 (4.3 million). - Current Seats Allocated (nn):
Let’s assume state XX currently has n=5n = 5 seats. - Apply the Formula:
Substitute the values into the formula: Huntington-Hill Number=4,300,0005(5+1)\text{Huntington-Hill Number} = \frac{4,300,000}{\sqrt{5(5+1)}} First, calculate the denominator: 5(6)=30≈5.477\sqrt{5(6)} = \sqrt{30} \approx 5.477 Now divide the population by this value: Huntington-Hill Number=4,300,0005.477≈784,880\text{Huntington-Hill Number} = \frac{4,300,000}{5.477} \approx 784,880
Explanation (300 Words)
The Huntington-Hill method ensures fairness in apportioning seats by balancing representation relative to state populations. The Huntington-Hill number is calculated for each state during the apportionment process to determine which state should receive the next seat.
In our example, state XX has a population of 4.3 million and currently holds 5 seats. The formula divides the state’s population by the geometric mean of its current seats and one additional seat (n(n+1)\sqrt{n(n+1)}). This adjustment ensures that smaller states receive representation proportional to their population, while larger states do not dominate excessively.
The denominator, 30\sqrt{30}, reflects the weighted midpoint between the current and next potential allocation of seats. By comparing Huntington-Hill numbers across states, the state with the highest number receives the next seat. This process repeats until all seats are distributed.
This method avoids bias toward either small or large states, striking a balance between proportionality and equality. The calculated number (784,880) represents the threshold of population per seat for state XX, ensuring its representation is equitable compared to other states.