If the average of observations x,x +1,x+2,x+3 and x+4 is 12, then what is the average of the last three observations ?

Answer

To solve the problem step by step, we need to find the average of the last three observations given that the average of five observations is 12.

### Step 1: Understand the Observations
The observations are:
- \( x \)
- \( x + 1 \)
- \( x + 2 \)
- \( x + 3 \)
- \( x + 4 \)

### Step 2: Write the Formula for Average
The average of these observations is given by the formula:
\[ \text{Average} = \frac{\text{Sum of observations}}{\text{Number of observations}} \]
In this case, the number of observations is 5.

### Step 3: Set Up the Equation
According to the problem, the average is 12:
\[ 12 = \frac{x + (x + 1) + (x + 2) + (x + 3) + (x + 4)}{5} \]

### Step 4: Simplify the Sum
Now, simplify the sum in the numerator:
\[ x + (x + 1) + (x + 2) + (x + 3) + (x + 4) = 5x + (1 + 2 + 3 + 4) = 5x + 10 \]
So, we can rewrite the equation as:
\[ 12 = \frac{5x + 10}{5} \]

### Step 5: Multiply Both Sides by 5
To eliminate the fraction, multiply both sides by 5:
\[ 12 \times 5 = 5x + 10 \]
This simplifies to:
\[ 60 = 5x + 10 \]

### Step 6: Solve for \( x \)
Now, isolate \( x \):
\[ 60 - 10 = 5x \]
\[ 50 = 5x \]
\[ x = \frac{50}{5} = 10 \]

### Step 7: Identify the Last Three Observations
The last three observations are:
- \( x + 2 \)
- \( x + 3 \)
- \( x + 4 \)

Substituting \( x = 10 \):
- \( x + 2 = 10 + 2 = 12 \)
- \( x + 3 = 10 + 3 = 13 \)
- \( x + 4 = 10 + 4 = 14 \)

### Step 8: Find the Average of the Last Three Observations
Now, calculate the average of these three observations:
\[ \text{Average} = \frac{12 + 13 + 14}{3} \]
Calculating the sum:
\[ 12 + 13 + 14 = 39 \]
So, the average is:
\[ \text{Average} = \frac{39}{3} = 13 \]

### Final Answer
The average of the last three observations is **13**.

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