(Mandatory) (5 points) Calculate the Huntington-Hill number. City A: population 4500, 8 representatives 281,251.000 281,250.000 281,260.000 281,250.100
The Correct Answer and Explanation is:
Answer:
To calculate the Huntington-Hill number for City A with a population of 4,500 and 8 representatives, we use the Huntington-Hill method formula:Huntington-Hill number=Populationn(n+1)\text{Huntington-Hill number} = \frac{\text{Population}}{\sqrt{n(n+1)}}Huntington-Hill number=n(n+1)Population
Where:
- Population = 4,500
- n=8n = 8n=8 (current number of representatives)
Step-by-step Calculation:Huntington-Hill number=45008×9=450072\text{Huntington-Hill number} = \frac{4500}{\sqrt{8 \times 9}} = \frac{4500}{\sqrt{72}}Huntington-Hill number=8×94500=72450072≈8.485\sqrt{72} \approx 8.48572≈8.48545008.485≈530.18\frac{4500}{8.485} \approx 530.188.4854500≈530.18
Final Answer:
530.18\boxed{530.18}530.18
Textbook-Style Explanation
The Huntington-Hill method, also known as the method of equal proportions, is used to fairly allocate a fixed number of representatives (such as seats in a legislative body) among various regions or states based on population. It is designed to minimize the relative differences in the ratio of population to representation among all entities.
To determine whether a city or state should receive an additional representative, the Huntington-Hill number is calculated. This number represents the average number of people per representative, adjusted using a geometric mean to balance proportional fairness.
The formula is:Huntington-Hill number=Pn(n+1)\text{Huntington-Hill number} = \frac{P}{\sqrt{n(n+1)}}Huntington-Hill number=n(n+1)P
Where:
- PPP is the population,
- nnn is the current number of representatives,
- n(n+1)\sqrt{n(n+1)}n(n+1) is the geometric mean of the current and next representative counts.
For City A, which has a population of 4,500 and currently 8 representatives, we compute:8×9=72≈8.485\sqrt{8 \times 9} = \sqrt{72} \approx 8.4858×9=72≈8.485
Then divide the population by this geometric mean:45008.485≈530.18\frac{4500}{8.485} \approx 530.188.4854500≈530.18
This Huntington-Hill number (530.18) can be compared to priority values of other cities or states. If it is higher than another city’s value, City A would have a stronger claim for an additional representative. This method ensures representation grows fairly with population while avoiding over-representation of small regions or under-representation of large ones. It is used in real-world contexts such as U.S. Congressional apportionment after each census.
