If the mean of `x_(1),x_(2)` is 7.5 and the mean of `x_(1),x_(2),x_(3)` is 8, then the value of `x_(3)` is

Options

A. A. 9
B. B. 8
C. C. 7.5
D. D. 6

Answer

To solve the problem step by step, we will use the information given about the means of the sets of numbers.

### Step 1: Understand the Mean of Two Numbers
We know that the mean of \( x_1 \) and \( x_2 \) is given as 7.5. The formula for the mean is:
\[ \text{Mean} = \frac{x_1 + x_2}{2} \]
Setting this equal to 7.5, we have:
\[ \frac{x_1 + x_2}{2} = 7.5 \]

### Step 2: Calculate the Sum of \( x_1 \) and \( x_2 \)
To find the sum \( x_1 + x_2 \), we multiply both sides of the equation by 2:
\[ x_1 + x_2 = 7.5 \times 2 = 15 \]
This gives us our first equation:
\[ x_1 + x_2 = 15 \quad \text{(Equation 1)} \]

### Step 3: Understand the Mean of Three Numbers
Next, we know that the mean of \( x_1, x_2, \) and \( x_3 \) is 8. Using the mean formula again, we have:
\[ \text{Mean} = \frac{x_1 + x_2 + x_3}{3} \]
Setting this equal to 8, we have:
\[ \frac{x_1 + x_2 + x_3}{3} = 8 \]

### Step 4: Calculate the Sum of \( x_1, x_2, \) and \( x_3 \)
To find the sum \( x_1 + x_2 + x_3 \), we multiply both sides of the equation by 3:
\[ x_1 + x_2 + x_3 = 8 \times 3 = 24 \]
This gives us our second equation:
\[ x_1 + x_2 + x_3 = 24 \quad \text{(Equation 2)} \]

### Step 5: Substitute the Value of \( x_1 + x_2 \)
Now we can substitute the value of \( x_1 + x_2 \) from Equation 1 into Equation 2:
\[ 15 + x_3 = 24 \]

### Step 6: Solve for \( x_3 \)
To find \( x_3 \), we subtract 15 from both sides:
\[ x_3 = 24 - 15 \]
Calculating this gives:
\[ x_3 = 9 \]

### Final Answer
Thus, the value of \( x_3 \) is:
\[ \boxed{9} \]