A college offers tutoring in Math, English, Chemistry, and Biology. The number of students enrolled in each subject is listed below. The college can only afford to hire 18 tutors.  Using Huntington-Hill's method determine the initial geometric means and the initial allocations with the standard divisor.  You do not need to finish the apportionment.
Subject	Students Enrolled	Initial Geometric Mean	Initial Allocation
Math	305	Correct7.4833147735479	Incorrect
English	220	Correct5.4772255750517	Incorrect
Chemistry	140	Correct3.4641016151378	Incorrect
Biology	80	Correct1.4142135623731	Incorrect
Give decimal answers accurate to at least 3 decimal places

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?

---

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.

A college offers tutoring in Math, English, Chemistry, and Biology. The number of students enrolled in each subject is listed below. The college can only afford to hire 18 tutors. Using Huntington-Hill's method, determine the initial geometric means and the initial allocations with the standard divisor. You do not need to finish the apportionment.

This problem involves using the Huntington-Hill method to allocate 18 tutors across four subjects based on student enrollment. The solution covers the calculation of standard divisor, initial allocations, and comparison with geometric means for Math, English, Chemistry, and Biology. Suitable for college-level mathematics students.

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