Eight consecutive integers are given. If the average of the two middle numbers is 6, what is the sum of all eight numbers?

Introduction:
For any set of evenly spaced numbers, the overall average equals the average of the two middle terms (when the count is even). Once you know the average of the two central numbers, you immediately know the mean of the entire set and hence the sum.

Given Data / Assumptions:

• Eight consecutive integers.
• Average of the two middle integers = 6.

Concept / Approach:
For 8 consecutive integers, the two middle positions are the 4th and 5th. Their average equals the overall average. Therefore, the sum of all eight numbers equals (average) * (count) = 6 * 8.

Step-by-Step Solution:

Verification / Alternative check:
Example: integers 3,4,5,6,7,8,9,10 have middle pair (6,7) with average 6.5; adjust down by 0.5 to center at 6 gives sum 48, confirming the method conceptually by symmetry.

Why Other Options Are Wrong:
36, 54, and 64 correspond to averages 4.5, 6.75, and 8, not 6; 40 implies average 5, contradicting the given average of the middle pair.

Common Pitfalls:
Attempting to list all eight numbers explicitly or misidentifying the middle positions; recognizing symmetry saves time and avoids errors.

Final Answer:
48