The average of 21 numbers is 12. The average of the first 11 numbers is 9 and the average of the last 11 numbers is 15. What is the value of the middle (11th) number in this ordered list?

Introduction / Context:
This problem uses the idea of overlapping groups and averages. We know the average of all 21 numbers, the average of the first 11 numbers, and the average of the last 11 numbers. Since there is overlap at the middle number, we can use totals and subtraction to find its exact value.

Given Data / Assumptions:

• Total count of numbers = 21.
• Average of all 21 numbers = 12.
• Average of the first 11 numbers = 9.
• Average of the last 11 numbers = 15.
• The middle number is the 11th number when they are ordered in sequence.

Concept / Approach:
Let us denote:

• Sum of all 21 numbers as S.
• Sum of first 11 numbers as S1.
• Sum of last 11 numbers as S2.

The middle number is included in both S1 and S2. If we add S1 and S2, the middle number is counted twice. Using this, we can form a simple equation involving S, S1, S2 and the middle number x and then solve for x.

Step-by-Step Solution:
Step 1: Compute sum of all 21 numbers. S = 12 * 21 = 252. Step 2: Compute sum of the first 11 numbers. S1 = 9 * 11 = 99. Step 3: Compute sum of the last 11 numbers. S2 = 15 * 11 = 165. Step 4: Express relationship including the middle number x. When we add S1 and S2, all numbers except the middle one are counted once, but the middle number x is counted twice. Therefore S1 + S2 = S + x. Step 5: Substitute and solve for x. S1 + S2 = 99 + 165 = 264. We also have S = 252, so 264 = 252 + x. Thus x = 264 - 252 = 12.

Verification / Alternative check:
We can check by reconstructing the totals. If the middle number is 12, and S1 = 99, then the sum of the other 10 numbers in the first group is 99 - 12 = 87. In the last 11 numbers, S2 = 165, so the sum of the other 10 numbers in the last group is 165 - 12 = 153. Total sum of all 21 numbers is 87 + 12 + 153 = 252, and the average is 252 / 21 = 12, as given. So the answer is consistent.

Why Other Options Are Wrong:
If the middle number were 10, then S1 + S2 would equal S + 10, giving 99 + 165 = 252 + 10 and thus 264 = 262, which is impossible. Similar contradictions occur for 11 and 13. Only when the middle number is 12 does the relation S1 + S2 = S + x hold true.

Common Pitfalls:
Many students forget that the middle number is counted twice in S1 + S2 and may attempt to average 9 and 15 directly. Others confuse the count of numbers and mistakenly think there are only 20 numbers. The reliable method is to always convert given averages to sums and then carefully account for overlaps when groups share common elements.

Final Answer:
The value of the middle number is 12.