The average of 11 numbers is 10.9. If the average of the first six numbers is 10.5 and the average of the last six numbers is 11.4, then what is the value of the middle number in this list?

Introduction / Context:
This question checks a very common aptitude concept involving averages, overlapping groups, and how a single middle value affects two different partial averages. Such problems are popular in competitive exams because they require candidates to combine the average formula with a little logical thinking about how sums are counted more than once.

Given Data / Assumptions:

• There are 11 numbers in total.
• The average of all 11 numbers is 10.9.
• The average of the first 6 numbers is 10.5.
• The average of the last 6 numbers is 11.4.
• The middle number is counted in both the first 6 and the last 6 numbers because of overlap.

Concept / Approach:
The key concepts are: average = sum / number of items and the idea that when two overlapping groups share a common element, that element is counted twice if we add the sums of those groups. We can use this to form a simple equation and solve for the middle number. This technique is widely used in problems about consecutive terms, marks of students, and daily temperature lists.

Step-by-Step Solution:
Let the 11 numbers be N1, N2, ..., N11 with N6 as the middle number.Total sum of all 11 numbers = average * count = 10.9 * 11 = 119.9.Sum of first 6 numbers = 10.5 * 6 = 63.Sum of last 6 numbers = 11.4 * 6 = 68.4.When we add the sums of the first 6 and the last 6, the middle number N6 is included twice.So, 63 + 68.4 = (sum of all 11 numbers) + N6.Therefore, 131.4 = 119.9 + N6.N6 = 131.4 - 119.9 = 11.5.

Verification / Alternative check:
We can quickly check: suppose the total sum is 119.9 and the middle number is 11.5. Then the sum of the other 10 numbers is 119.9 - 11.5 = 108.4. The first 5 and last 5 plus the middle should still match the given averages. The calculations we did are consistent and follow the standard overlapping averages method, so the answer is reliable.

Why Other Options Are Wrong:
Values like 9.5 or 10.5 would be too small and would reduce the combined overlap sum, making the derived averages incorrect. Numbers like 12.5 and 13.5 are too large and would make the sums of the first six or last six inconsistent with 10.5 and 11.4 respectively. Only 11.5 keeps all three averages valid at the same time.

Common Pitfalls:
A frequent mistake is to treat the two sets of six numbers as disjoint and forget that the middle number is common to both sets. Another common error is to average the two averages directly, which does not respect the overlap structure. Careless multiplication or subtraction can also lead to wrong answers even with the correct method. Writing the equation for sums clearly usually avoids these errors.

Final Answer:
The value of the middle number is 11.5.