The average of five consecutive even number (in increasing order ) is k. If the next four consecutive even numbers are included , then the average of all the numbers is

Options

A. 2 k - 1
B. k + 5
C. k + 4
D. k + 6

Answer

To solve the problem step by step, we will first define the five consecutive even numbers and then calculate their average. After that, we will include the next four consecutive even numbers and find the new average.

### Step 1: Define the five consecutive even numbers
Let the first even number be \( x \). Therefore, the five consecutive even numbers can be expressed as:
- First number: \( x \)
- Second number: \( x + 2 \)
- Third number: \( x + 4 \)
- Fourth number: \( x + 6 \)
- Fifth number: \( x + 8 \)

### Step 2: Calculate the average of the five numbers
The average \( k \) of these five numbers is calculated as follows:
\[ k = \frac{x + (x + 2) + (x + 4) + (x + 6) + (x + 8)}{5} \]
Simplifying the numerator:
\[ k = \frac{5x + (2 + 4 + 6 + 8)}{5} = \frac{5x + 20}{5} = x + 4 \]

### Step 3: Define the next four consecutive even numbers
The next four consecutive even numbers after the first five are:
- Sixth number: \( x + 10 \)
- Seventh number: \( x + 12 \)
- Eighth number: \( x + 14 \)
- Ninth number: \( x + 16 \)

### Step 4: Calculate the total sum of all nine numbers
Now, we will calculate the total sum of all nine consecutive even numbers:
\[ \text{Total Sum} = (x + (x + 2) + (x + 4) + (x + 6) + (x + 8)) + (x + 10 + x + 12 + x + 14 + x + 16) \]
This simplifies to:
\[ \text{Total Sum} = (5x + 20) + (4x + 52) = 9x + 72 \]

### Step 5: Calculate the new average of all nine numbers
The average of all nine numbers is:
\[ \text{New Average} = \frac{\text{Total Sum}}{9} = \frac{9x + 72}{9} = x + 8 \]

### Step 6: Relate the new average to \( k \)
From Step 2, we know \( k = x + 4 \). Therefore, we can express the new average in terms of \( k \):
\[ \text{New Average} = x + 8 = (k - 4) + 8 = k + 4 \]

### Conclusion
Thus, the average of all the numbers when the next four consecutive even numbers are included is:
\[ \boxed{k + 4} \]