As part of a "green" initiative, California wants to apportion 150
 new electric vehicles to its university system campuses. Given the following table, use the Hamilton method to determine how many of the vehicles should be apportioned to the Los Angeles campus based on the number of students.

Student Enrollment in the University of California System
Campus	Enrollment	Full Time Equivalent (FTE)
Los Angeles	37,221
28,292
San Diego	25,938
18,274
Santa Cruz	15,012
4597

The Hamilton method is a method of apportionment that involves the following steps:

1. **Calculate the standard divisor:** This is the total population divided by the total number of items to be apportioned.

2. **Calculate the standard quota for each group:** Divide each group's population by the standard divisor.

3. **Assign the lower quota (integer part) to each group.**

4. **Distribute the remaining items:** After assigning the lower quota, there will usually be some items left over. These are distributed to the groups with the largest fractional parts of their standard quotas.

Now, let's apply these steps to the given problem.

### Step 1: Calculate the Standard Divisor

The total number of Full-Time Equivalent (FTE) students is:

\[
\text{Total FTE} = 28,292 + 18,274 + 4,597 = 51,163
\]

The standard divisor is:

\[
\text{Standard Divisor} = \frac{\text{Total FTE}}{\text{Total Vehicles}} = \frac{51,163}{150} \approx 341.09
\]

### Step 2: Calculate the Standard Quota for Each Campus

The standard quota for each campus is calculated by dividing the FTE of each campus by the standard divisor.

- **Los Angeles:** 
\[
\frac{28,292}{341.09} \approx 82.98
\]

- **San Diego:** 
\[
\frac{18,274}{341.09} \approx 53.59
\]

- **Santa Cruz:** 
\[
\frac{4,597}{341.09} \approx 13.48
\]

### Step 3: Assign the Lower Quota

The lower quota (integer part) for each campus is:

- **Los Angeles:** 82
- **San Diego:** 53
- **Santa Cruz:** 13

Adding these lower quotas gives:

\[
82 + 53 + 13 = 148
\]

This means 148 vehicles have been assigned, leaving 2 vehicles to be distributed.

### Step 4: Distribute the Remaining Vehicles

We distribute the remaining vehicles to the campuses with the largest fractional parts of their quotas. The fractional parts are:

- **Los Angeles:** 0.98
- **San Diego:** 0.59
- **Santa Cruz:** 0.48

The two largest fractional parts are from Los Angeles (0.98) and San Diego (0.59). Therefore, we assign one additional vehicle to Los Angeles and one additional vehicle to San Diego.

### Final Apportionment:

- **Los Angeles:** 82 + 1 = **83 vehicles**
- **San Diego:** 53 + 1 = **54 vehicles**
- **Santa Cruz:** 13 vehicles

### Conclusion:
The Los Angeles campus should be apportioned **83 vehicles** based on the Hamilton method.

---

Would you like more details or have any questions? Here are some related questions to explore:

1. How does the Hamilton method compare to other apportionment methods?
2. What are the potential issues or biases with the Hamilton method?
3. How would the apportionment change if the total number of vehicles was increased?
4. What is the significance of the fractional parts in apportionment?
5. How does the standard divisor change if we exclude one campus?

**Tip:** Understanding how the fractional parts impact final distributions can help when dealing with similar allocation problems.

As part of a 'green' initiative, California wants to apportion 150 new electric vehicles to its university system campuses. Given the following table, use the Hamilton method to determine how many of the vehicles should be apportioned to the Los Angeles campus based on the number of students.

Explore how to apportion 150 electric vehicles among California university campuses using the Hamilton method. This problem involves calculating the standard divisor, determining standard quotas, and distributing remaining vehicles based on fractional parts. A detailed solution for students in Grades 10-12.

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