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To determine how to allocate the 23 emergency physicians across the three clinics (A, B, and C) proportionally based on the number of patients each clinic serves, we can use the following steps:

1. **Calculate the proportion of patients in each clinic** relative to the total number of patients.
2. **Multiply these proportions by the total number of physicians** (23) to find the number of physicians for each clinic.
3. **Round the results** to the nearest whole numbers since the number of physicians must be whole numbers.

### Step-by-Step Solution

1. **Total Patients Across All Clinics**:
   - Clinic A: 214 patients
   - Clinic B: 65 patients
   - Clinic C: 301 patients
   - Total = \( 214 + 65 + 301 = 580 \)

2. **Calculate Each Clinic's Proportion of the Total Patients**:
   - For Clinic A: \( \frac{214}{580} \approx 0.369 \)
   - For Clinic B: \( \frac{65}{580} \approx 0.112 \)
   - For Clinic C: \( \frac{301}{580} \approx 0.519 \)

3. **Assign Physicians Proportionally**:
   - Clinic A: \( 0.369 \times 23 \approx 8.487 \), rounded to 8 physicians.
   - Clinic B: \( 0.112 \times 23 \approx 2.576 \), rounded to 3 physicians.
   - Clinic C: \( 0.519 \times 23 \approx 11.937 \), rounded to 12 physicians.

4. **Final Allocation of Physicians**:
   - Clinic A: 8 physicians
   - Clinic B: 3 physicians
   - Clinic C: 12 physicians

This allocation ensures that the 23 physicians are distributed proportionally to the patient load in each clinic.

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Would you like more details on these calculations, or do you have any questions?

### Related Questions:
1. How would the allocation change if the total number of physicians were different, say 25?
2. What if Clinic C experienced an increase in patient load—how would that affect the allocation?
3. How can we ensure that each clinic has at least one physician, even if one clinic has very few patients?
4. What method would be used if the number of physicians needed to be allocated in strict proportion, without rounding?
5. How does rounding affect the final allocation of resources in this type of proportional distribution?

#### Tip:
In proportional distributions, always check the rounded values to ensure they add up to the total. Adjustments may be needed if they do not.

If there are 23 emergency physicians available, how should they be appointed across three clinics with patient counts of 214, 65, and 301?

Determine the proportional allocation of 23 emergency physicians across three clinics with patient counts of 214, 65, and 301. This problem involves calculating ratios and rounding to distribute resources effectively. Suitable for high school students.

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