To solve the problem of which number can be removed from the set \( S = \{0, 2, 4, 5, 9\} \) without changing the average, we can follow these steps:
### Step 1: Calculate the Average of the Set \( S \)
The average (mean) of a set is calculated using the formula:
\[
\text{Average} = \frac{\text{Sum of all elements}}{\text{Total number of elements}}
\]
First, we find the sum of all elements in the set \( S \):
\[
0 + 2 + 4 + 5 + 9 = 20
\]
Next, we count the total number of elements in the set \( S \):
\[
\text{Total number of elements} = 5
\]
Now, we can calculate the average:
\[
\text{Average} = \frac{20}{5} = 4
\]
### Step 2: Identify the Condition for Removing a Number
We need to determine which number can be removed from the set without changing the average. A key rule is that if we remove an element that is equal to the current average, the average of the new set will remain the same.
### Step 3: Check Each Option
The options we have for removal are \( 0, 2, 4, \) and \( 5 \). Since the average is \( 4 \), we check if any of these numbers is equal to \( 4 \):
- **Removing \( 0 \)**: New set \( = \{2, 4, 5, 9\} \)
- **Removing \( 2 \)**: New set \( = \{0, 4, 5, 9\} \)
- **Removing \( 4 \)**: New set \( = \{0, 2, 5, 9\} \)
- **Removing \( 5 \)**: New set \( = \{0, 2, 4, 9\} \)
### Step 4: Calculate the New Average for Each Case
1. **Removing \( 0 \)**:
\[
\text{New Average} = \frac{2 + 4 + 5 + 9}{4} = \frac{20}{4} = 5
\]
2. **Removing \( 2 \)**:
\[
\text{New Average} = \frac{0 + 4 + 5 + 9}{4} = \frac{18}{4} = 4.5
\]
3. **Removing \( 4 \)**:
\[
\text{New Average} = \frac{0 + 2 + 5 + 9}{4} = \frac{16}{4} = 4
\]
4. **Removing \( 5 \)**:
\[
\text{New Average} = \frac{0 + 2 + 4 + 9}{4} = \frac{15}{4} = 3.75
\]
### Step 5: Conclusion
From the calculations, we see that removing \( 4 \) keeps the average the same at \( 4 \). Therefore, the number that can be removed from the set \( S \) without changing the average is:
\[
\boxed{4}
\]
