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 will follow the process of finding the mean and then calculating the mean of the last three observations.

### Step 1: Understand the Observations
We have the observations:
- \( x \)
- \( x + 3 \)
- \( x + 5 \)
- \( x + 7 \)
- \( x + 10 \)

### Step 2: Calculate the Mean
The mean of these observations is given as 9. The formula for the mean is:
\[ \text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}} \]
Here, the number of observations is 5.

### Step 3: Set Up the Equation
We can express the mean as:
\[ \frac{x + (x + 3) + (x + 5) + (x + 7) + (x + 10)}{5} = 9 \]

### Step 4: Simplify the Equation
Now, simplify the numerator:
\[ x + (x + 3) + (x + 5) + (x + 7) + (x + 10) = 5x + (3 + 5 + 7 + 10) = 5x + 25 \]
So, we have:
\[ \frac{5x + 25}{5} = 9 \]

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

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

### 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 calculate the mean of the last three observations:
\[ \text{Mean} = \frac{9 + 11 + 14}{3} \]
Calculating the sum:
\[ 9 + 11 + 14 = 34 \]
Now, divide by 3:
\[ \text{Mean} = \frac{34}{3} \]

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

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