The average of 15 numbers is 7. The average of the first 8 numbers is 6.5, and the average of the last 8 numbers is 8.5. What is the middle (8th) number?

Introduction: When two overlapping blocks of numbers share only the middle element, the sums of the blocks and the total allow you to isolate the shared middle value. This is a standard technique for consecutive-list average problems.

Given Data / Assumptions: Avg of all 15 = 7 ⇒ total T = 15 * 7 = 105. Avg of first 8 ⇒ sum F = 8 * 6.5 = 52. Avg of last 8 ⇒ sum L = 8 * 8.5 = 68. The middle number is counted once in T, but in both F and L.

Concept / Approach: Because the 8th number is included in both block sums, F + L double-counts it relative to T. Therefore, middle = F + L − T.

Step-by-Step Solution: Middle = 52 + 68 − 105 = 15.

Verification / Alternative check: Let the middle be m. Then T = (sum of first 7) + m + (sum of last 7). F = (sum of first 7) + m; L = (sum of last 7) + m. Therefore F + L − T = m, which matches the computation.

Why Other Options Are Wrong: 10, 13, 9, 23: None satisfy the identity F + L − T with the given averages.

Common Pitfalls: Averaging 6.5 and 8.5 directly or assuming symmetry without overlap arithmetic. Use sums with the overlap formula.

Final Answer: The middle number is 15.