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 need to find the difference between the average of 8 consecutive natural numbers and the average of the first 6 consecutive natural numbers, which is denoted as K.

### Step 1: Define the 6 consecutive natural numbers
Let the first of the 6 consecutive natural numbers be \( Y \). Therefore, the 6 consecutive natural numbers are:
- \( Y \)
- \( Y + 1 \)
- \( Y + 2 \)
- \( Y + 3 \)
- \( Y + 4 \)
- \( Y + 5 \)

### Step 2: Calculate the sum of the 6 consecutive natural numbers
The sum of these numbers can be calculated as follows:
\[ \text{Sum} = Y + (Y + 1) + (Y + 2) + (Y + 3) + (Y + 4) + (Y + 5) \]
\[ = 6Y + (0 + 1 + 2 + 3 + 4 + 5) = 6Y + 15 \]

### Step 3: Calculate the average of the 6 consecutive natural numbers
The average \( K \) of these 6 numbers is:
\[ K = \frac{\text{Sum}}{6} = \frac{6Y + 15}{6} = Y + \frac{15}{6} = Y + 2.5 \]

### Step 4: Define the next two natural numbers
The next two consecutive natural numbers after the first 6 are:
- \( Y + 6 \)
- \( Y + 7 \)

### Step 5: Calculate the sum of the 8 consecutive natural numbers
Now, we need to find the sum of all 8 consecutive natural numbers:
\[ \text{Sum of 8 numbers} = Y + (Y + 1) + (Y + 2) + (Y + 3) + (Y + 4) + (Y + 5) + (Y + 6) + (Y + 7) \]
\[ = 8Y + (0 + 1 + 2 + 3 + 4 + 5 + 6 + 7) = 8Y + 28 \]

### Step 6: Calculate the average of the 8 consecutive natural numbers
The average of these 8 numbers is:
\[ \text{Average of 8 numbers} = \frac{\text{Sum}}{8} = \frac{8Y + 28}{8} = Y + \frac{28}{8} = Y + 3.5 \]

### Step 7: Find the difference between the averages
Now, we need to find how much more the average of the 8 numbers is than \( K \):
\[ \text{Difference} = \left(Y + 3.5\right) - \left(Y + 2.5\right) \]
\[ = 3.5 - 2.5 = 1 \]

### Final Answer
The average of the 8 numbers is 1 more than \( K \).