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The table below shows the number of computers that are assigned to four different schools and the number of students in those schools.




School		Number of ComputersNumber of Students

Rose		20686

Lincoln		29500

Midway		24663

Valley		25706






Calculate the Huntington-Hill number for each of the schools. In this case, the population is the number of students and the number of representatives is the number of computers. (Round your answers to the nearest integer.)



Rose: [ ]

Lincoln: [ ]

Midway: [ ]

Valley: [ ]



Use the Huntington-Hill apportionment principle to determine to which school a new computer should be assigned.


ARose
BLincoln
CMidway
DValley

Verified Answer :

  • We calculate the Huntington-Hill (HH) numbers using the formula \(HH = \frac{p}{\sqrt{n(n+1)}}\).
  • The calculated values are: Rose \(\approx 33\), Lincoln \(\approx 17\), Midway \(\approx 27\), and Valley \(\approx 117\).
  • Valley has the highest HH number, therefore it is the school that receives the new computer.
  • Final Answer: Rose: 33; Lincoln: 17; Midway: 27; Valley: 117; Assignment: \(\boxed{Valley}\).

Explanation

  1. Problem Analysis and Formula Setup

The Huntington-Hill (HH) method is an apportionment principle used to assign resources based on population. The priority value for assigning an additional resource is calculated using the formula:
\[HH = \frac{p}{\sqrt{n(n+1)}}\]
where \(p\) is the population (number of students) and \(n\) is the current number of units (number of computers). The school with the highest calculated HH value is prioritized to receive the next unit. We have the following data for the schools:

  • Rose: \(p = 686\), \(n = 20\)
  • Lincoln: \(p = 500\), \(n = 29\)
  • Midway: \(p = 663\), \(n = 24\)
  • Valley: \(p = 406\), \(n = 3\)
  1. Calculating Huntington-Hill Numbers

Using the formula \(HH = \frac{p}{\sqrt{n(n+1)}}\), we calculate the Huntington-Hill number for each school:

  • Rose: \(HH = \frac{686}{\sqrt{20 \times 21}} = \frac{686}{\sqrt{420}} \approx 33.473\), which rounds to \(\boxed{33}\).
  • Lincoln: \(HH = \frac{500}{\sqrt{29 \times 30}} = \frac{500}{\sqrt{870}} \approx 16.952\), which rounds to \(\boxed{17}\).
  • Midway: \(HH = \frac{663}{\sqrt{24 \times 25}} = \frac{663}{\sqrt{600}} \approx 27.067\), which rounds to \(\boxed{27}\).
  • Valley: \(HH = \frac{406}{\sqrt{3 \times 4}} = \frac{406}{\sqrt{12}} \approx 117.202\), which rounds to \(\boxed{117}\).
  1. Determining the Assignment

Comparing the calculated Huntington-Hill numbers:

  • Rose: 33.473
  • Lincoln: 16.952
  • Midway: 27.067
  • Valley: 117.202

Since the school with the highest Huntington-Hill number receives the additional resource, the school that should receive the new computer is Valley, as it has the highest value of \(117.202\). The final choice is Valley.

Examples

The Huntington-Hill method is commonly used in political science to apportion seats in a legislature, such as the U.S. House of Representatives. By calculating priority values for each state, it ensures that the distribution of seats is as close as possible to a proportional representation based on population. This same principle can be applied to many organizational scenarios, such as distributing computers to classrooms, medical equipment to hospital wards, or budgets to different regional offices, ensuring that resources are allocated in a fair and mathematically sound manner.

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