In an experiment Sriram records 11 observations whose average is 92. The average of the first five observations is 89 and the average of the last five observations is 86. What is the value of the sixth observation?

Introduction:
This question deals with averages of overlapping groups of observations. We know the average of all 11 observations, the average of the first five and the average of the last five. Using these values, we can find the value of the middle or sixth observation that is not part of either group.

Given Data / Assumptions:
- Total number of observations = 11. - Average of all 11 observations = 92. - Average of the first five observations = 89. - Average of the last five observations = 86. - We must find the sixth observation.

Concept / Approach:
The average of a set of numbers gives us their total sum. So, from the given averages we can compute three sums: the sum of all 11 observations, the sum of the first five observations and the sum of the last five observations. The sixth observation is included in the total sum but not in either of the sums of the first or last five observations. Therefore, we can find it by subtracting these partial sums from the total.

Step-by-Step Solution:
Step 1: Let the observations be O1, O2, O3, O4, O5, O6, O7, O8, O9, O10 and O11. Step 2: Total sum of all 11 observations = 11 * 92. Step 3: Compute 11 * 92 = 1012, so the total sum S = 1012. Step 4: Sum of the first five observations S1 = 5 * 89 = 445. Step 5: Sum of the last five observations S2 = 5 * 86 = 430. Step 6: The total sum S can be written as S1 + O6 + S2. Step 7: Therefore, 1012 = 445 + O6 + 430. Step 8: Combine 445 and 430 to get 875, so 1012 = 875 + O6. Step 9: Solve for O6: O6 = 1012 - 875 = 137.

Verification / Alternative Check:
Check that the sums are consistent. S1 = 445, S2 = 430 and O6 = 137. Adding them gives 445 + 430 + 137 = 1012, which matches the total sum of all 11 observations. The averages of the first and last five observations computed from these sums also remain 89 and 86 respectively, confirming the result.

Why Other Options Are Wrong:
Options 134, 139, 141 and 143 would give total sums different from 1012 when combined with 445 and 430, contradicting the given overall average of 92.

Common Pitfalls:
Some learners mistakenly divide the overall average by 2 or attempt to average 89 and 86 directly. Others may forget that the sixth observation is not part of either group of five, and incorrectly count some observations twice. The correct approach is to work with total sums and subtract the known partial sums from the overall sum.

Final Answer:
The sixth observation recorded by Sriram is 137.