The average of 6 consecutive natural numbers is K. If the next two natural numbers are also included, how much more than K will the average of these 8 numbers be?

Options

A. 1.3
B. 1
C. 2
D. 1.8

Answer

To solve the problem step by step, we will first define the six consecutive natural numbers and then calculate the average of these numbers. After that, we will include the next two natural numbers and find the new average. Finally, we will determine how much more this new average is than the original average \( K \).

### Step 1: Define the six consecutive natural numbers
Let the six consecutive natural numbers be:
- \( n, n+1, n+2, n+3, n+4, n+5 \)

### Step 2: Calculate the sum of these six numbers
The sum of these numbers can be calculated as follows:
\[ \text{Sum} = n + (n+1) + (n+2) + (n+3) + (n+4) + (n+5) = 6n + 15 \]

### Step 3: Calculate the average of the six numbers
The average \( K \) of these six numbers is given by:
\[ K = \frac{\text{Sum}}{\text{Number of terms}} = \frac{6n + 15}{6} = n + 2.5 \]

### Step 4: Include the next two natural numbers
The next two natural numbers after the six are:
- \( n+6 \) and \( n+7 \)

### Step 5: Calculate the new sum including these two numbers
The new sum becomes:
\[ \text{New Sum} = (6n + 15) + (n+6) + (n+7) = 6n + 15 + n + 6 + n + 7 = 8n + 28 \]

### Step 6: Calculate the new average of the eight numbers
The new average of these eight numbers is:
\[ \text{New Average} = \frac{\text{New Sum}}{\text{Number of terms}} = \frac{8n + 28}{8} = n + 3.5 \]

### Step 7: Determine how much more the new average is than \( K \)
Now, we find the difference between the new average and the original average \( K \):
\[ \text{Difference} = \text{New Average} - K = (n + 3.5) - (n + 2.5) = 3.5 - 2.5 = 1 \]

### Final Answer
The average of the eight numbers is \( 1 \) more than the average \( K \) of the six consecutive natural numbers.

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