If the mean of the observation `x``,` `x+3``,` `x+5``,` `x+7` and `x+10` is `9``,` then mean of the last three observations is

Options

A. `101/3`
B. `102/3`
C. `11``1/3`
D. `112/3`

Answer

To solve the problem step by step, we need to find the mean of the last three observations given that the mean of the five observations is 9.

### Step 1: List the Observations
The observations given are:
- \( x \)
- \( x + 3 \)
- \( x + 5 \)
- \( x + 7 \)
- \( x + 10 \)

### Step 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.

### Step 3: Set Up the Equation
According to the problem, the mean is 9. Thus, we can set up the equation:
\[ 9 = \frac{x + (x + 3) + (x + 5) + (x + 7) + (x + 10)}{5} \]

### Step 4: Simplify the Equation
Now, let's simplify the sum in the numerator:
\[ x + (x + 3) + (x + 5) + (x + 7) + (x + 10) = 5x + (3 + 5 + 7 + 10) = 5x + 25 \]
So, we can rewrite the equation as:
\[ 9 = \frac{5x + 25}{5} \]

### Step 5: Multiply Both Sides by 5
To eliminate the fraction, multiply both sides by 5:
\[ 9 \times 5 = 5x + 25 \]
This simplifies to:
\[ 45 = 5x + 25 \]

### Step 6: Solve for \( x \)
Now, isolate \( x \):
\[ 45 - 25 = 5x \]
\[ 20 = 5x \]
\[ x = \frac{20}{5} = 4 \]

### Step 7: Find the Last Three Observations
Now that we have \( x = 4 \), we can find the last three observations:
- \( x + 5 = 4 + 5 = 9 \)
- \( x + 7 = 4 + 7 = 11 \)
- \( x + 10 = 4 + 10 = 14 \)

So, the last three observations are 9, 11, and 14.

### Step 8: Calculate the Mean of the Last Three Observations
Now, we need to find the mean of these last three observations:
\[ \text{Mean} = \frac{9 + 11 + 14}{3} \]
Calculating the sum:
\[ 9 + 11 + 14 = 34 \]
So, the mean is:
\[ \text{Mean} = \frac{34}{3} \]

### Step 9: Final Answer
The mean of the last three observations is:
\[ \frac{34}{3} = 11 \frac{1}{3} \]