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The question in your image involves apportioning 280 nurses among five units (Emergency Care, Intensive Care, Maternity, Pediatrics, and Surgery) using Hamilton's method, based on the number of beds in each unit.

Here are the steps to solve this using Hamilton's method:

1. **Total number of beds**: First, sum up the total number of beds in all units:
   - ECU: 21 beds
   - ICU: 18 beds
   - MU: 35 beds
   - PU: 32 beds
   - SU: 24 beds
   
   Total = \( 21 + 18 + 35 + 32 + 24 = 130 \) beds.

2. **Calculate the standard quota**: For each unit, calculate how many nurses they should get based on their proportion of the total beds. The formula is:
   
   \[
   \text{Standard quota for each unit} = \left( \frac{\text{Beds in the unit}}{\text{Total beds}} \right) \times \text{Total nurses} 
   \]

   Using 280 nurses:

   - ECU: \( \frac{21}{130} \times 280 = 45.23 \)
   - ICU: \( \frac{18}{130} \times 280 = 38.77 \)
   - MU: \( \frac{35}{130} \times 280 = 75.38 \)
   - PU: \( \frac{32}{130} \times 280 = 68.92 \)
   - SU: \( \frac{24}{130} \times 280 = 51.69 \)

3. **Assign whole numbers**: Start by assigning each unit the whole number part of their standard quota:

   - ECU: 45 nurses
   - ICU: 38 nurses
   - MU: 75 nurses
   - PU: 68 nurses
   - SU: 51 nurses

   Total = \( 45 + 38 + 75 + 68 + 51 = 277 \) nurses.

   We still need to assign \( 280 - 277 = 3 \) more nurses.

4. **Distribute remaining nurses**: The remaining nurses are given to the units with the largest fractional remainders from their standard quotas. The fractional parts are:

   - ECU: \( 0.23 \)
   - ICU: \( 0.77 \)
   - MU: \( 0.38 \)
   - PU: \( 0.92 \)
   - SU: \( 0.69 \)

   The three largest fractional parts are PU (0.92), ICU (0.77), and SU (0.69). So, we assign 1 additional nurse to each of these units.

5. **Final distribution**:
   - ECU: 45 nurses
   - ICU: 39 nurses
   - MU: 75 nurses
   - PU: 69 nurses
   - SU: 52 nurses

This gives us a total of 280 nurses.

Would you like further explanation or have any questions?

Here are some related questions:
1. How does Hamilton's method differ from Jefferson's method?
2. What happens if two units have the same fractional part when distributing remaining resources?
3. How would the method change if there were 300 nurses instead of 280?
4. Can Hamilton's method ever lead to paradoxes? If so, which ones?
5. How does apportionment apply to real-world situations like political representation?

**Tip**: In problems involving proportional distribution, always check that your final allocation matches the total available resources to avoid rounding errors.

A certain hospital has five major units: Emergency Care (ECU), Intensive Care (ICU), Maternity (MU), Pediatrics (PU), and Surgery (SU). There are 280 nurses working in these five units and they are apportioned to the units based on the number of beds in each unit using Hamilton's method. Find the distribution of nurses.

Learn how to apportion 280 nurses among five hospital units based on their bed counts using Hamilton's method. This step-by-step solution covers proportional allocation and rounding with the method, making it perfect for high school students studying mathematics or healthcare management.

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