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 will follow these instructions:

### Step 1: Write down the observations
The observations given are:
- \( x \)
- \( x + 3 \)
- \( x + 6 \)
- \( x + 9 \)
- \( x + 12 \)

### Step 2: Calculate the mean of the observations
The mean of these five observations is given by the formula:
\[ \text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}} \]
So, we can write:
\[ \text{Mean} = \frac{x + (x + 3) + (x + 6) + (x + 9) + (x + 12)}{5} \]

### Step 3: Simplify the sum of observations
Now, let's simplify the sum:
\[ x + (x + 3) + (x + 6) + (x + 9) + (x + 12) = x + x + 3 + x + 6 + x + 9 + x + 12 \]
This simplifies to:
\[ 5x + (3 + 6 + 9 + 12) = 5x + 30 \]

### Step 4: Set the mean equal to 15
According to the problem, the mean is equal to 15:
\[ \frac{5x + 30}{5} = 15 \]

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

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

### Step 7: Calculate the mean of the first three observations
Now, we can find the mean of these three observations:
\[ \text{Mean} = \frac{9 + 12 + 15}{3} \]
Calculating the sum:
\[ 9 + 12 + 15 = 36 \]
Now, divide by 3:
\[ \text{Mean} = \frac{36}{3} = 12 \]

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

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