The average of 25 results is 18. The average of the first 12 is 14 and the average of the last 12 is 17. What is the 13th (middle) result?

Introduction:
Here, two disjoint groups (first twelve and last twelve) exclude the central 13th result. By comparing their combined sum with the total sum of all 25 results, we can directly extract the middle value without reconstructing the full list.

Given Data / Assumptions:

• Total count = 25; overall average = 18 → total sum = 18 * 25 = 450
• First 12 average = 14 → sum = 14 * 12 = 168
• Last 12 average = 17 → sum = 17 * 12 = 204

Concept / Approach:
The first 12 and last 12 are disjoint and together contain 24 results (excluding the 13th). Therefore, (sum first 12 + sum last 12) = total sum − 13th. Rearranging yields the 13th value.

Step-by-Step Solution:

Verification / Alternative check:
Summing all 25 as (first 12 + 13th + last 12) reproduces the total 450, confirming that the 13th must be 78.

Why Other Options Are Wrong:
28, 72, 85, and 62 do not reconcile the three averages simultaneously when plugged back into the partition sums.

Common Pitfalls:
Mistaking the setup for an overlapping case (like 7-and-7 over 13). Here, the groups are disjoint; no item is double-counted.

Final Answer:
78