If the mean of five observations x, x + 4, x + 8, x + 12 and x + 16 is 15, then the value of x is

Options

A. 5
B. 6
C. 7
D. 8

Answer

To find the value of \( x \) given that the mean of the observations \( x, x + 4, x + 8, x + 12, \) and \( x + 16 \) is 15, we can follow these steps:

### Step 1: Write down the observations
The observations are:
- \( x \)
- \( x + 4 \)
- \( x + 8 \)
- \( x + 12 \)
- \( x + 16 \)

### Step 2: Calculate the sum of the observations
The sum of these observations can be calculated as follows:
\[ \text{Sum} = x + (x + 4) + (x + 8) + (x + 12) + (x + 16) \]

### Step 3: Simplify the sum
Combining like terms:
\[ \text{Sum} = x + x + 4 + x + 8 + x + 12 + x + 16 = 5x + (4 + 8 + 12 + 16) \]
Calculating the constant terms:
\[ 4 + 8 + 12 + 16 = 40 \]
Thus, the sum becomes:
\[ \text{Sum} = 5x + 40 \]

### Step 4: Set up the equation for the mean
The mean is given by the formula:
\[ \text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}} \]
Here, the mean is 15 and the number of observations is 5:
\[ 15 = \frac{5x + 40}{5} \]

### Step 5: Multiply both sides by 5 to eliminate the fraction
\[ 15 \times 5 = 5x + 40 \]
\[ 75 = 5x + 40 \]

### Step 6: Solve for \( x \)
Subtract 40 from both sides:
\[ 75 - 40 = 5x \]
\[ 35 = 5x \]
Now, divide both sides by 5:
\[ x = \frac{35}{5} = 7 \]

### Conclusion
The value of \( x \) is \( 7 \).