The average of 13 results is 60. The average of the first 7 results is 59, and the average of the last 7 results is 61. What is the seventh result?

Introduction / Context:
This classic overlap-average problem uses totals with an overlapping term. The 7th result belongs to both the "first 7" and the "last 7", allowing a direct equation to isolate it.

Given Data / Assumptions:

• Average of 13 = 60 ⇒ total T = 13 * 60.
• Average of first 7 = 59 ⇒ total T1 = 7 * 59.
• Average of last 7 = 61 ⇒ total T2 = 7 * 61.
• Only the 7th result overlaps between these two groups.

Concept / Approach:
Since the 7th result x is counted in both T1 and T2, but only once in T, we have T = T1 + T2 − x. Solve for x to obtain the overlapped value.

Step-by-Step Solution:

Verification / Alternative check:
Using the equation T1 − x covers the first 6, T2 − x covers the last 6; adding those plus x and x reconstructs all 13, confirming the overlap logic.

Why Other Options Are Wrong:
90, 50, 75: Do not satisfy the totals relationship; they would distort one or both subgroup means.

Common Pitfalls:
Adding T1 and T2 without subtracting x once, which double-counts the 7th value.

Final Answer:
60