if the mean of the observations `x, x+ 4, x + 5, x + 7, x + 9` is 9, then th mean of the last three observations is :

Options

A. 7
B. 10
C. 11
D. 8

Answer

To solve the problem step by step, we will follow the given observations and calculate the required mean.

### Step-by-Step Solution:

1. **Identify the Observations**: The observations given are:
- \( x \)
- \( x + 4 \)
- \( x + 5 \)
- \( x + 7 \)
- \( x + 9 \)

2. **Write the Mean Formula**: The mean of these observations is given by the formula:
\[ \text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}} \]
Here, the number of observations is 5.

3. **Set Up the Equation**: According to the problem, the mean is equal to 9:
\[ 9 = \frac{x + (x + 4) + (x + 5) + (x + 7) + (x + 9)}{5} \]

4. **Simplify the Sum**: Combine the terms in the numerator:
\[ x + (x + 4) + (x + 5) + (x + 7) + (x + 9) = 5x + (4 + 5 + 7 + 9) = 5x + 25 \]

5. **Rewrite the Equation**: Substitute the simplified sum back into the equation:
\[ 9 = \frac{5x + 25}{5} \]

6. **Eliminate the Denominator**: Multiply both sides by 5:
\[ 9 \times 5 = 5x + 25 \quad \Rightarrow \quad 45 = 5x + 25 \]

7. **Isolate \( x \)**: Subtract 25 from both sides:
\[ 45 - 25 = 5x \quad \Rightarrow \quad 20 = 5x \]

8. **Solve for \( x \)**: Divide both sides by 5:
\[ x = \frac{20}{5} = 4 \]

9. **Find the Last Three Observations**: Substitute \( x = 4 \) into the last three observations:
- \( x + 5 = 4 + 5 = 9 \)
- \( x + 7 = 4 + 7 = 11 \)
- \( x + 9 = 4 + 9 = 13 \)

10. **Calculate the Mean of the Last Three Observations**: The last three observations are 9, 11, and 13. The mean is calculated as:
\[ \text{Mean} = \frac{9 + 11 + 13}{3} \]

11. **Sum the Observations**: Calculate the sum:
\[ 9 + 11 = 20 \quad \Rightarrow \quad 20 + 13 = 33 \]

12. **Final Mean Calculation**: Divide the sum by 3:
\[ \text{Mean} = \frac{33}{3} = 11 \]

### Conclusion:
The mean of the last three observations is **11**.