The average of 8 numbers is 14. The average of 6 of these numbers is 16. What is the average of the remaining two numbers?

Introduction: This is a sum-partition average problem. Converting averages to totals lets you subtract the known subset and then average what remains. It is a standard technique for decomposing composite averages.

Given Data / Assumptions:

• Overall: 8 numbers, average 14 ⇒ total = 8 * 14 = 112.
• Subset: 6 numbers, average 16 ⇒ subset total = 6 * 16 = 96.

Concept / Approach: The remaining two have total = overall total − subset total. Divide by 2 to get their average. This avoids hunting individual values and relies only on aggregate information.

Step-by-Step Solution:

Verification / Alternative check: If you construct a concrete set matching these sums, the two leftover values will always average to 8, regardless of their individual spread, since the aggregate constraints are fixed.

Why Other Options Are Wrong: 12, 6, 10, and 14 conflict with the fixed totals (112 overall and 96 for the six). Only 8 divides the remainder 16 evenly for two numbers.

Common Pitfalls: Averaging 14 and 16 directly (which would be meaningless here) or dividing the difference of averages by 2 without using totals.

Final Answer: 8