The average of 13 results is 68. The average of the first 7 results is 63 and that of the last 7 results is 70. What is the seventh (overlapping) result?

Introduction:
This average puzzle uses overlapping groups. The “first seven” and “last seven” sets overlap in exactly one element (the seventh result). Sums from the two groups can be combined and compared with the total sum to isolate the overlapped value.

Given Data / Assumptions:

• Total results = 13; overall average = 68 → total sum = 68 * 13 = 884
• First 7 average = 63 → sum = 63 * 7 = 441
• Last 7 average = 70 → sum = 70 * 7 = 490

Concept / Approach:
The two seven-element blocks together count every element once except the central (7th) element, which is counted twice. Hence: (sum first 7 + sum last 7) = total sum + seventh result. Solve for the seventh result.

Step-by-Step Solution:

Verification / Alternative check:
Visualize the 13 results in order. The first seven include positions 1–7; the last seven include 7–13. Position 7 is double-counted; subtract the total once to isolate it.

Why Other Options Are Wrong:
65.5, 73.5, 94, and 52 do not satisfy the overlap arithmetic when plugged back into the sums.

Common Pitfalls:
Adding averages instead of sums or forgetting that the seventh item is counted twice in the two partial sums.

Final Answer:
47