The average of 9 observations was found to be 35. Later it was discovered that an observation 81 had been misread as 18. What is the correct average of the 9 observations?

Introduction / Context:
This question is about correcting an average after discovering that one observation was misread. The reported average is based on the wrong value, so we must adjust the total sum by replacing the incorrect observation with the correct one and then find the new average.

Given Data / Assumptions:

• Number of observations = 9.
• Reported average based on a misread value = 35.
• One observation was read as 18 instead of the correct value 81.
• We need the correct average for the 9 observations.

Concept / Approach:
Average equals total sum divided by number of observations. First we find the incorrect total that produced the average 35. Then we correct this total by removing the wrong value and adding the correct one. Finally, dividing the corrected total by 9 gives the correct average. The error in this case is quite large, so the corrected average changes significantly.

Step-by-Step Solution:
Step 1: Incorrect total sum = reported average * number of observations = 35 * 9 = 315. Step 2: The wrong value used was 18; the correct value should be 81. Step 3: To correct the total, subtract the wrong value and add the correct one: corrected total = 315 - 18 + 81. Step 4: Compute 315 - 18 = 297. Step 5: Now add 81: 297 + 81 = 378. Step 6: Correct average = corrected total / number of observations = 378 / 9 = 42.

Verification / Alternative check:
The difference between the correct and incorrect values is 81 - 18 = 63. Since this 63 was missing from the total, the correct total must be 315 + 63 = 378. Dividing by 9 again gives 42, which confirms our corrected average by a shorter method.

Why Other Options Are Wrong:
Option 35 is the original incorrect average based on the misread value. Option 28 is too low and would correspond to removing rather than adding the missing marks. Option 45 is higher than what the single correction can justify. Only 42 matches the revised total and respects the size of the correction.

Common Pitfalls:
Learners sometimes subtract the difference 63 from the total instead of adding it or forget to multiply the average by the number of observations to recover the total. Always reconstruct the total, apply the correction carefully and then recompute the average with the correct sum.

Final Answer:
The correct average of the 9 observations is 42.