To determine the initial allocations using Huntington-Hill's method, the process involves the following key steps:

Steps for Huntington-Hill's Apportionment Method:

1. Standard Divisor Calculation:

   • First, calculate the standard divisor (SD) by dividing the total number of students by the number of available tutors.

   $\text{Standard Divisor (SD)} = \frac{\text{Total Students Enrolled}}{\text{Total Tutors}}$

2. Initial Allocation:

   • To get the initial allocation for each subject, divide the number of students enrolled in each subject by the standard divisor.

   $\text{Initial Allocation} = \frac{\text{Students Enrolled}}{\text{Standard Divisor}}$

3. Geometric Mean Calculation:

   • The geometric mean for an allocation $n$ is calculated as:

   $\text{Geometric Mean} = \sqrt{n(n+1)}$ This is used to determine whether to round the allocation up or down. The initial allocation is compared with the geometric mean. If the initial allocation is greater than the geometric mean, round up. Otherwise, round down.

Step 1: Calculate the Standard Divisor

The total number of students enrolled is:

$305 \, (\text{Math}) + 220 \, (\text{English}) + 140 \, (\text{Chemistry}) + 80 \, (\text{Biology}) = 745$

The total number of tutors available is 18. Therefore, the standard divisor (SD) is:

$\text{SD} = \frac{745}{18} \approx 41.389$

Step 2: Calculate the Initial Allocations

• Math:
  $\frac{305}{41.389} \approx 7.371$

• English:
  $\frac{220}{41.389} \approx 5.315$

• Chemistry:
  $\frac{140}{41.389} \approx 3.383$

• Biology:
  $\frac{80}{41.389} \approx 1.933$

Step 3: Compare with Geometric Means

We are given the geometric means for each subject:

• Math: 7.483
• English: 5.477
• Chemistry: 3.464
• Biology: 1.414

Initial Allocations vs. Geometric Means:

• Math:
  The initial allocation for Math is 7.371, which is less than the geometric mean of 7.483, so we round down.

• English:
  The initial allocation for English is 5.315, which is less than the geometric mean of 5.477, so we round down.

• Chemistry:
  The initial allocation for Chemistry is 3.383, which is less than the geometric mean of 3.464, so we round down.

• Biology:
  The initial allocation for Biology is 1.933, which is greater than the geometric mean of 1.414, so we round up.

Summary:

• Math: Round down to 7
• English: Round down to 5
• Chemistry: Round down to 3
• Biology: Round up to 2

These are the initial allocations before further adjustments are made in the Huntington-Hill method.

Would you like more details or have any questions?

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Here are 5 related questions you can explore:

1. How does the Huntington-Hill method compare to other apportionment methods?
2. What are other applications of the Huntington-Hill method outside of tutoring allocation?
3. How does rounding affect the final distribution of resources?
4. Can the Huntington-Hill method handle fractional allocations effectively?
5. What happens if the total number of students changes after an initial allocation?

Tip: Huntington-Hill's method is widely used in political apportionment, like distributing seats in legislatures based on population.