The average (arithmetic mean) of 8 numbers is 65. If one of the numbers, 65, removed, what is the average of the remaining 7 numbers?

Answer

To solve the problem step by step, we will follow the logical sequence based on the information given.

### Step-by-Step Solution:

1. **Understand the Average Formula**:
The average (arithmetic mean) of a set of numbers is calculated using the formula:
\[ \text{Average} = \frac{\text{Sum of all values}}{\text{Total number of values}} \]

2. **Calculate the Sum of the 8 Numbers**:
Given that the average of 8 numbers is 65, we can find the total sum of these numbers. Let the sum of the 8 numbers be \( S \).
\[ S = \text{Average} \times \text{Total number of values} = 65 \times 8 = 520 \]

3. **Identify the Number to be Removed**:
One of the numbers is 65, which we will remove from the total sum.

4. **Calculate the New Sum After Removal**:
After removing the number 65, the new sum \( S' \) of the remaining 7 numbers will be:
\[ S' = S - 65 = 520 - 65 = 455 \]

5. **Calculate the New Average**:
Now, we need to find the average of the remaining 7 numbers. Using the average formula again:
\[ \text{New Average} = \frac{S'}{\text{Total number of remaining values}} = \frac{455}{7} \]

6. **Perform the Division**:
Now, we divide 455 by 7:
\[ \text{New Average} = 65 \]

### Final Answer:
The average of the remaining 7 numbers is **65**.