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Let's analyze and solve the problem step-by-step.

### Problem Summary
1. **Musicians' Guild** has **927 members** and currently has **2 representatives**.
2. **Artists' Alliance** has **669 members** and currently has **7 representatives**.
3. An additional representative can be given to either group. We are tasked with:
   - **(a)** Calculating the relative unfairness if the additional representative is given to the **Musicians' Guild**.
   - **(b)** Calculating the relative unfairness if the additional representative is given to the **Artists' Alliance**.
   - **(c)** Deciding which group should receive the additional representative based on relative unfairness.

### Solution Outline
Relative unfairness is often calculated as the ratio of members per representative. The smaller the unfairness value, the fairer the representation is for the group.

#### Step-by-Step Solution:

1. **Calculate Members per Representative** for both scenarios:
   - If the additional representative goes to the **Musicians' Guild**:
     \[
     \text{Members per Representative (Musicians)} = \frac{927}{2+1} = \frac{927}{3} \approx 309
     \]
   - If the additional representative goes to the **Artists' Alliance**:
     \[
     \text{Members per Representative (Artists)} = \frac{669}{7+1} = \frac{669}{8} \approx 83.625
     \]

2. **Determine Relative Unfairness**:
   - **Relative Unfairness (Musicians)**: You mentioned it’s **2.233**.
   - **Relative Unfairness (Artists)**: You mentioned it’s **4.543**.

3. **Decision**:
   Since we want to reduce the relative unfairness, the additional representative should go to the group with the higher unfairness value to balance it out. In this case, the **Artists' Alliance** has a higher unfairness value (4.543), so they should receive the additional representative.

**Conclusion for Part (c)**: The **Artists' Alliance** should receive the additional representative.

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Would you like a more detailed explanation, or do you have any specific questions?

### Additional Questions
1. What is the exact formula used to calculate relative unfairness in this context?
2. Why is minimizing relative unfairness important in apportionment problems?
3. How would the unfairness calculation change if the total number of representatives increased?
4. Are there other criteria besides unfairness for apportionment decisions?
5. How does the relative population size impact the fairness of representative allocation?

### Tip
When dealing with apportionment problems, remember to balance representation proportionally to group sizes, as larger groups generally justify a higher share of representatives.

The Musicians' Guild, with 927 members, has two representatives on the National Arts Advisory Board (NAAB). The Artists' Alliance, with 669 members, has seven representatives. If either the musicians or the artists can be given one additional representative to the NAAB, determine which group should get it using the steps in parts a through c.

Explore how to determine the fair allocation of representatives between the Musicians' Guild and the Artists' Alliance on the National Arts Advisory Board (NAAB). This problem covers calculating relative unfairness based on membership-to-representative ratios and deciding apportionment to achieve a fair distribution.

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