If the mean of five observations `x,x+3,x+6,x+9 and x+12` is 15, then find the mean of first three observations

Options

A. 11
B. 12
C. 15.5
D. 11.2

Answer

To solve the problem step by step, we need to find the mean of the first three observations given that the mean of five observations \(x, x+3, x+6, x+9, x+12\) is 15.

### Step 1: Write down the observations
The five observations are:
1. \(x\)
2. \(x + 3\)
3. \(x + 6\)
4. \(x + 9\)
5. \(x + 12\)

### Step 2: Calculate the sum of the observations
To find the mean, we first need to calculate the sum of these observations:
\[ \text{Sum} = x + (x + 3) + (x + 6) + (x + 9) + (x + 12) \]
\[ = x + x + 3 + x + 6 + x + 9 + x + 12 \]
\[ = 5x + (3 + 6 + 9 + 12) \]
\[ = 5x + 30 \]

### Step 3: Set up the equation for the mean
The mean of these five observations is given as 15. The formula for the mean is:
\[ \text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}} \]
Thus, we have:
\[ 15 = \frac{5x + 30}{5} \]

### Step 4: Solve for \(x\)
To eliminate the fraction, multiply both sides by 5:
\[ 15 \times 5 = 5x + 30 \]
\[ 75 = 5x + 30 \]
Now, subtract 30 from both sides:
\[ 75 - 30 = 5x \]
\[ 45 = 5x \]
Now, divide both sides by 5:
\[ x = \frac{45}{5} = 9 \]

### Step 5: Find the first three observations
Now that we have \(x\), we can find the first three observations:
1. \(x = 9\)
2. \(x + 3 = 9 + 3 = 12\)
3. \(x + 6 = 9 + 6 = 15\)

### Step 6: Calculate the mean of the first three observations
Now, we find the mean of the first three observations \(9, 12, 15\):
\[ \text{Mean} = \frac{9 + 12 + 15}{3} \]
\[ = \frac{36}{3} = 12 \]

### Final Answer
The mean of the first three observations is **12**.

---