Sriram conducts an experiment and records 11 observations whose average is 92. The average of the first 5 observations is 89, and the average of the last 5 observations is 86. What is the value of the 6th observation?

Introduction / Context:
This problem is about using averages of overlapping groups of observations to find a specific missing value. It shows how average information about different parts of a data set can be combined to reconstruct individual observations. This is a standard type of aptitude question that tests your fluency with sums and averages.

Given Data / Assumptions:

• Total number of observations = 11.
• Average of all 11 observations = 92.
• Average of the first 5 observations = 89.
• Average of the last 5 observations = 86.
• We must find the 6th observation, which is not included entirely in either the first or last 5, but is common when we consider the whole data set.

Concept / Approach:
Average is total divided by count. From the overall average, we can compute the total sum of all 11 observations. From the averages of the first 5 and last 5 observations, we can compute the sums of these subsets. The key idea is that the sum of all 11 observations equals the sum of the first 5 plus the sum of the last 5 plus the 6th observation counted once, but actually the 6th observation is not included in either of the first 5 or last 5 sums. So we simply subtract the two partial sums from the overall sum to isolate the 6th observation.

Step-by-Step Solution:
Step 1: Compute the total of all 11 observations using the overall average. Total (all 11) = 11 * 92 = 1012. Step 2: Compute the total of the first 5 observations. Total (first 5) = 5 * 89 = 445. Step 3: Compute the total of the last 5 observations. Total (last 5) = 5 * 86 = 430. Step 4: Let the 6th observation be x. Step 5: Note that the sum of all 11 observations is the sum of the first 5, plus the 6th, plus the last 5. Step 6: Therefore, 1012 = 445 + x + 430. Step 7: Add the two known sums: 445 + 430 = 875. Step 8: So 1012 = 875 + x, so x = 1012 - 875 = 137. Step 9: Therefore, the 6th observation is 137.

Verification / Alternative check:
We can verify by constructing a hypothetical data set consistent with these sums, but the algebra itself is already a direct check. The combined sum of first 5 and last 5 observations is 875. Subtracting this from the total 1012 leaves 137, which must be the 6th observation. There is no other place for this missing amount to come from, so the logic is sound.

Why Other Options Are Wrong:
Values like 134, 139, 141 or 131 would give a different total sum for the 11 observations and would not maintain the given averages for the first 5 and last 5. For example, if the 6th observation were 134, the total would be 875 + 134 = 1009, leading to an overall average less than 92. Only 137 ensures the total stays 1012 and all averages are satisfied.

Common Pitfalls:
A frequent error is to think that the 6th observation is somehow the average of 89 and 86 or to attempt to average the averages. Another mistake is double counting or omitting some observations when adding partial sums. Clearly identifying the total sum and subtracting the known parts is the most reliable method.

Final Answer:
The 6th observation has a value of 137.