The average of six numbers is 3.95. The average of two of these numbers is 3.4, and the average of another two of these numbers is 3.85. What is the average of the remaining two numbers?

Introduction / Context:
This problem is a direct application of the concept of averages and how they relate to total sums. It checks whether you can move comfortably between average and total, and then separate the contribution of different groups of numbers from the whole set.

Given Data / Assumptions:

• There are six numbers.

• Average of all six numbers = 3.95.

• Average of two of the numbers = 3.4.

• Average of another two of the numbers = 3.85.

• We assume all averages are arithmetic means and all numbers are real values.

Concept / Approach:
The key idea is that average = total sum / number of items. From this, total sum = average * number of items. Once we know the total sum of all six numbers, and the sums of two separate pairs of numbers, we can subtract those known sums from the overall total to get the combined sum of the remaining two numbers. Finally, we divide that remaining sum by 2 to get their average.

Step-by-Step Solution:
Total sum of all six numbers = 3.95 * 6 = 23.70. Sum of the first pair with average 3.4 = 3.4 * 2 = 6.80. Sum of the second pair with average 3.85 = 3.85 * 2 = 7.70. Combined sum of these four numbers = 6.80 + 7.70 = 14.50. Sum of the remaining two numbers = 23.70 − 14.50 = 9.20. Average of the remaining two numbers = 9.20 / 2 = 4.60.

Verification / Alternative check:
One good check is to reconstruct the overall average. If two numbers average 3.4, two numbers average 3.85 and two numbers average 4.6, then total sum = 2 * 3.4 + 2 * 3.85 + 2 * 4.6 = 6.8 + 7.7 + 9.2 = 23.7. Dividing this by 6 gives 23.7 / 6 = 3.95, which matches the original average. This confirms that the derived average of 4.6 for the remaining pair is consistent with the given information.

Why Other Options Are Wrong:
An average of 3.6 or 6.5 or 7.3 would change the total sum of all six numbers and would no longer give an overall average of 3.95. If you plug any of those values back and recompute the total, you will not get 23.7 as the total sum. Therefore, those options do not satisfy the relationships between the group averages and the global average.

Common Pitfalls:
A common mistake is to try to average the averages directly without taking into account how many numbers each average represents. Another error is to forget that the remaining two numbers must account for the difference between the known total and the sums of the other four numbers. Always move through sums first, then come back to averages to avoid algebraic confusion.

Final Answer:
The average of the remaining two numbers is 4.6.