We need to apportion 70 representatives among five states with the following populations: - Apprecia: 300,500 - Beauti: 200,000 - Contentia: 50,000 - Dashin: 38,000 - Enjoya: 21,500 Compute the geometric mean (Huntington-Hill) for each state, accurate to two decimal places.

To compute the **Huntington-Hill geometric mean** for each state, we'll follow these steps: 1. **Calculate the Total Population:** \[ \text{Total Population} = 300,500 + 200,000 + 50,000 + 38,000 + 21,500 = 610,000 \] 2. **Determine the Standard Divisor (D):** \[ D = \frac{\text{Total Population}}{\text{Total Seats}} = \frac{610,000}{70} \approx 8,714.29 \] 3. **Compute the Quotas for Each State:** \[ \text{Quota}_i = \frac{\text{Population}_i}{D} \] | State | Population | Quota ≈ | |------------|------------|---------| | Apprecia | 300,500 | 34.48 | | Beauti | 200,000 | 22.98 | | Contentia | 50,000 | 5.74 | | Dashin | 38,000 | 4.36 | | Enjoya | 21,500 | 2.47 | 4. **Determine the Lower and Upper Quotas:** - **Lower Quota (Lᵢ):** The integer part of the Quota. - **Upper Quota (Uᵢ):** The next integer above the Quota. | State | Lower Quota (Lᵢ) | Upper Quota (Uᵢ) | |------------|------------------|------------------| | Apprecia | 34 | 35 | | Beauti | 22 | 23 | | Contentia | 5 | 6 | | Dashin | 4 | 5 | | Enjoya | 2 | 3 | 5. **Calculate the Geometric Mean (√(Lᵢ × Uᵢ)) for Each State:** \[ \text{Geometric Mean}_i = \sqrt{Lᵢ \times Uᵢ} \] | State | Geometric Mean (√(Lᵢ × Uᵢ)) ≈ | |------------|-------------------------------| | Apprecia | √(34 × 35) = √1190 ≈ **34.50** | | Beauti | √(22 × 23) = √506 ≈ **22.49** | | Contentia | √(5 × 6) = √30 ≈ **5.48** | | Dashin | √(4 × 5) = √20 ≈ **4.47** | | Enjoya | √(2 × 3) = √6 ≈ **2.45** | **Final Geometric Means:** - **Apprecia:** 34.50 - **Beauti:** 22.49 - **Contentia:** 5.48 - **Dashin:** 4.47 - **Enjoya:** 2.45 These geometric means are used in the Huntington-Hill method to prioritize seat allocation based on population and current seat count.