https://file.aiquickdraw.com/mathbot/file/1730688718005fju0m0mt.jpg

To solve this problem using the Hamilton method, we need to apportion the 46 available booths based on the size of each group (painters, sculptors, and weavers) and their total membership.

### Step 1: Calculate the Total Population
The total number of members is:
\[
90 + 41 + 40 = 171
\]

### Step 2: Calculate Each Group’s Standard Quota
The standard quota for each group is found by multiplying the total number of booths (46) by the proportion of each group relative to the total.

1. **Painters’ Standard Quota**:
   \[
   \frac{90}{171} \times 46 \approx 24.21
   \]

2. **Sculptors’ Standard Quota**:
   \[
   \frac{41}{171} \times 46 \approx 11.03
   \]

3. **Weavers’ Standard Quota**:
   \[
   \frac{40}{171} \times 46 \approx 10.76
   \]

### Step 3: Assign Initial Booths by Rounding Down
We assign each group the integer part of their standard quota:

- Painters: 24 booths
- Sculptors: 11 booths
- Weavers: 10 booths

This gives a total of:
\[
24 + 11 + 10 = 45 \text{ booths}
\]

Since we have 46 booths, there is **1 booth remaining** to distribute.

### Step 4: Distribute Remaining Booth(s) Using Largest Fractional Part
To allocate the remaining booth, we look at the fractional parts of each group’s standard quota:

- Painters: \(0.21\)
- Sculptors: \(0.03\)
- Weavers: \(0.76\)

The largest fractional part is for the **Weavers**. Thus, the extra booth goes to the Weavers.

### Final Apportionment
- **Painters**: 24 booths
- **Sculptors**: 11 booths
- **Weavers**: 11 booths

Let me know if you would like further details or have any questions!

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Here are 5 related questions:
1. How would the solution change if there were 50 booths instead of 46?
2. What would happen if the number of guild members changed?
3. How does the Hamilton method compare to the Jefferson method of apportionment?
4. Why do we assign the remaining booths based on the largest fractional part?
5. What is the purpose of rounding down initially in the Hamilton method?

**Tip:** The Hamilton method is often used in proportional representation systems due to its simplicity and fairness in distributing resources.

The Civic Arts Guild is having a show. There is room for 46 booths. The guild has 90 painters, 41 sculptors, and 40 weavers. Use the Hamilton method to apportion the booths to each of the three groups.

Learn how to use Hamilton's Method to fairly apportion 46 booths among 90 painters, 41 sculptors, and 40 weavers. This solution explains proportional allocation and fractional part analysis, perfect for high school students in Grades 11-12.

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