The average of 11 numbers is 10.9. The average of the first six numbers is 10.5 and the average of the last six numbers is 11.4. What is the value of the middle (6th) number?

Introduction / Context:
This question involves overlapping averages and the idea of a middle term in a sequence of numbers. You are given the average of all 11 numbers, the average of the first six, and the average of the last six. From these overlapping pieces of information, you must determine the middle number, which is common to both groups. This type of question is useful for building confidence in handling sums and averages of partially overlapping sets.

Given Data / Assumptions:
- There are 11 numbers in total.
- Average of all 11 numbers = 10.9.
- Average of the first six numbers = 10.5.
- Average of the last six numbers = 11.4.
- The middle number is the 6th number, which belongs to both the first six and the last six.

Concept / Approach:
Let the 11 numbers be a1, a2, ..., a11. The 6th number is a6. The first six numbers are a1 to a6 and the last six numbers are a6 to a11. If we add the sums of the first six and the last six numbers, the middle term a6 is counted twice, while the others are counted once. We can relate this combined sum to the sum of all 11 numbers and then solve for a6.

Step-by-Step Solution:
Step 1: Total sum of all 11 numbers = 11 * 10.9 = 119.9.Step 2: Sum of the first six numbers = 6 * 10.5 = 63.Step 3: Sum of the last six numbers = 6 * 11.4 = 68.4.Step 4: Let the middle number (6th number) be m.Step 5: The sum of first six plus last six equals (sum of all 11 numbers) + m, because a6 is counted twice.Step 6: Therefore, 63 + 68.4 = 119.9 + m.Step 7: Left side = 131.4, so 131.4 = 119.9 + m.Step 8: Hence m = 131.4 - 119.9 = 11.5.

Verification / Alternative Check:
The combined sum of the first six and last six numbers is 63 + 68.4 = 131.4. If the middle number is 11.5, then the sum of all 11 numbers becomes 131.4 - 11.5 = 119.9. Dividing 119.9 by 11 yields 10.9, which matches the given overall average. This consistency confirms that the middle number must be 11.5.

Why Other Options Are Wrong:
Values like 10.5, 12.5, or 13.5 do not make the total sum match 119.9 when you apply the overlapping average logic. Substituting any of these values in place of 11.5 leads to a mismatch between the computed overall average and the given 10.9.

Common Pitfalls:
Some learners mistakenly think the middle number is simply the average of 10.5 and 11.4, or they miscount how many times the middle term appears in the combined sum of the first six and last six numbers. Always remember that the overlapping element is counted twice in such combined sums, and set up the equation based on that fact.

Final Answer:
The middle (6th) number is 11.5.