The average of 100 items was found to be 30. Later it was noticed that two items had been wrongly recorded as 32 and 12 instead of the correct values 23 and 11. What is the correct average of the 100 items after rectifying these two errors?

Introduction / Context:
This is another example of correcting an average when mistakes in recorded values are discovered. The original average of 100 items was computed using two incorrect values. Once we know which values were wrong and what their correct values should be, we can adjust the total and recompute the average. The question reinforces how averages depend on the total sum of all items.

Given Data / Assumptions:

• Number of items = 100.
• Original average (with errors) = 30.
• Two items were recorded as 32 and 12.
• Their correct values are 23 and 11.
• We must find the corrected average of the 100 items.

Concept / Approach:
Average is defined as total sum divided by number of items. The original total was computed using the wrong values 32 and 12. To correct the total, we subtract these wrong values and then add the correct ones 23 and 11. This gives the true total sum. Dividing this corrected total by 100 yields the correct average. It is important to handle the arithmetic carefully and interpret whether the total increases or decreases after correction.

Step-by-Step Solution:
Step 1: Compute the original total based on the incorrect average. Original total = 100 * 30 = 3000. Step 2: Determine the wrong and correct totals for the two items. Wrong contribution = 32 + 12 = 44. Correct contribution = 23 + 11 = 34. Step 3: Find the net change in the total when correcting the two items. The original total used 44, but it should have used 34. So we must reduce the total by 44 − 34 = 10. Step 4: Calculate the corrected total. Corrected total = original total − 10 = 3000 − 10 = 2990. Step 5: Compute the corrected average. Corrected average = 2990 / 100 = 29.9.
Verification / Alternative check:
We can directly adjust the original total. Corrected total = 3000 − (32 + 12) + (23 + 11). Corrected total = 3000 − 44 + 34 = 3000 − 10 = 2990. Dividing by 100 again gives 29.9, confirming our earlier result.
Why Other Options Are Wrong:
An average of 29.8 or 29.5 would imply a greater reduction in the total than the 10 units actually required. An average of 29 or 30.1 would correspond to total sums that do not match the corrected values (they would be 2900 and 3010 respectively).
Common Pitfalls:
One frequent error is to add instead of subtracting the net difference, mistakenly increasing the total when it should decrease. Another mistake is to replace only one of the wrong values or to miscalculate the difference between 44 and 34.
Final Answer:
The correct average of the 100 items is 29.9.