The average of nine consecutive numbers is n. If the next two numbers are also included the new average will

Options

A. increase by 2
B. remain the same
C. increase by 1.5
D. increase by 1

Answer

To solve the problem step by step, we need to understand the concept of averages and how they change when we add more numbers to the set.

### Step 1: Understand the average of nine consecutive numbers
Let’s denote the nine consecutive numbers as:
- \( x, x+1, x+2, x+3, x+4, x+5, x+6, x+7, x+8 \)

The average of these numbers can be calculated as:
\[ \text{Average} = \frac{\text{Sum of the numbers}}{\text{Total numbers}} = \frac{x + (x+1) + (x+2) + (x+3) + (x+4) + (x+5) + (x+6) + (x+7) + (x+8)}{9} \]

### Step 2: Calculate the sum of the nine numbers
The sum of the nine numbers is:
\[ 9x + (0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8) = 9x + 36 \]
So, the average becomes:
\[ \text{Average} = \frac{9x + 36}{9} = x + 4 \]
Given that this average is \( n \), we have:
\[ n = x + 4 \implies x = n - 4 \]

### Step 3: Include the next two consecutive numbers
The next two consecutive numbers after the nine are:
- \( x + 9 \) and \( x + 10 \)

### Step 4: Calculate the new average
Now we have a total of 11 numbers:
\[ x, x+1, x+2, x+3, x+4, x+5, x+6, x+7, x+8, x+9, x+10 \]
The sum of these 11 numbers is:
\[ (9x + 36) + (x + 9) + (x + 10) = 9x + 36 + x + 9 + x + 10 = 11x + 55 \]
The new average is:
\[ \text{New Average} = \frac{11x + 55}{11} = x + 5 \]

### Step 5: Substitute for \( x \)
Since we know \( x = n - 4 \):
\[ \text{New Average} = (n - 4) + 5 = n + 1 \]

### Conclusion
Thus, the new average when the next two numbers are included will be:
\[ \text{New Average} = n + 1 \]

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