The average of six numbers is 3.95. The average of two of these numbers is 3.40, and the average of another two of the numbers is 3.85. What is the average of the remaining two numbers?

Introduction:
This question is about breaking an overall average into parts. The average of all six numbers is known, and the averages of two separate pairs among them are given. You must find the average of the remaining two numbers.

Given Data / Assumptions:
Average of six numbers = 3.95. Average of one pair of numbers = 3.40. Average of another pair of numbers = 3.85. We need the average of the remaining two numbers.

Concept / Approach:
The idea is to translate averages into totals. For any group: Total = Average * Number of terms. We calculate: 1. Total of all six numbers. 2. Total of the first two numbers. 3. Total of the second pair of two numbers. Then subtract these from the total to get the sum of the remaining two numbers, and finally divide by 2 to get their average.

Step-by-Step Solution:
Step 1: Total of all six numbers = 6 * 3.95. 6 * 3.95 = 23.70. Step 2: Total of the first pair = 2 * 3.40 = 6.80. Step 3: Total of the second pair = 2 * 3.85 = 7.70. Step 4: Sum of the remaining two numbers = 23.70 - 6.80 - 7.70. Sum of remaining two numbers = 23.70 - 14.50 = 9.20. Step 5: Average of the remaining two numbers = 9.20 / 2 = 4.60.

Verification / Alternative Check:
You can quickly check: if two numbers average 3.40, their total is slightly less than 7; another two average 3.85, total slightly less than 8. The total of all six numbers is about 24, so the remaining pair must total about 9, giving an average a bit above 4, which matches 4.60.

Why Other Options Are Wrong:
3.60: This would make the total of the last two too small, and the overall average would no longer be 3.95. 6.50 and 7.30: These are too high and would push the overall average far above 3.95. 5.10: This does not match the exact calculated sum of 9.20.

Common Pitfalls:
A common error is to average the three averages directly: (3.95 + 3.40 + 3.85) / 3, which is incorrect because the groups have different sizes. Always convert averages into totals, manipulate the totals, and then convert back to averages.

Final Answer:
The average of the remaining two numbers is 4.60.