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

Options

A. `12(1)/3`
B. `12(2)/3`
C. `11(1)/3`
D. `13(2)/3`

Answer

To solve the problem step by step, we will follow the method outlined in the video transcript.

### Step-by-Step Solution

1. **Identify the Observations**:
The observations given are:
\[ x, \quad x + 3, \quad x + 5, \quad x + 7, \quad x + 10 \]

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

3. **Set Up the Equation**:
We know that the mean is 9, so we can set up the equation:
\[ 9 = \frac{x + (x + 3) + (x + 5) + (x + 7) + (x + 10)}{5} \]

4. **Simplify the Sum**:
Now, let's simplify the sum in the numerator:
\[ x + (x + 3) + (x + 5) + (x + 7) + (x + 10) = 5x + (3 + 5 + 7 + 10) \]
Calculating the constants:
\[ 3 + 5 + 7 + 10 = 25 \]
So, the sum becomes:
\[ 5x + 25 \]

5. **Substitute Back into the Equation**:
Now substitute this back into the equation:
\[ 9 = \frac{5x + 25}{5} \]

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

7. **Isolate \(x\)**:
Subtract 25 from both sides:
\[ 45 - 25 = 5x \]
This gives:
\[ 20 = 5x \]
Now divide both sides by 5:
\[ x = \frac{20}{5} = 4 \]

8. **Find the Observations**:
Now substitute \(x = 4\) back into the observations:
\[ 4, \quad 4 + 3 = 7, \quad 4 + 5 = 9, \quad 4 + 7 = 11, \quad 4 + 10 = 14 \]
So the observations are:
\[ 4, \quad 7, \quad 9, \quad 11, \quad 14 \]

9. **Calculate the Mean of the Last Three Observations**:
The last three observations are 9, 11, and 14. To find their mean:
\[ \text{Mean} = \frac{9 + 11 + 14}{3} \]
Calculate the sum:
\[ 9 + 11 + 14 = 34 \]
Now divide by 3:
\[ \text{Mean} = \frac{34}{3} = 11 \frac{1}{3} \]

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