A student, by mistake, wrote 64 instead of 46 as one number while finding the average of 10 given numbers and obtained the average as 50. What is the correct average of the 10 numbers?

Introduction / Context:
In this problem, a student makes a mistake while computing the average of 10 numbers by entering 64 instead of 46. The reported average is therefore wrong. We must determine the correct average when the error is corrected.

Given Data / Assumptions:

• Number of given numbers = 10.
• Reported average based on the wrong value is 50.
• One number should have been 46 but was recorded as 64.
• We need the correct average of all 10 numbers.

Concept / Approach:
The wrong average is based on a wrong total. First we recover this total by multiplying the average 50 by 10. Then we correct the total by subtracting the wrong value and adding the correct value. The final correct average is the corrected total divided by 10. Because the wrong value is larger than the correct one, the corrected average will be smaller than 50.

Step-by-Step Solution:
Step 1: Incorrect total sum = 50 * 10 = 500. Step 2: Wrong value used = 64; correct value = 46. Step 3: The error in the total is 64 - 46 = 18 extra units. Step 4: Corrected total = 500 - 18 = 482. Step 5: Correct average = corrected total / number of values = 482 / 10 = 48.2.

Verification / Alternative check:
The mean decreased from 50 to 48.2, a drop of 1.8. This equals the error 18 divided by 10, which matches our understanding that each point of total error spread equally over 10 numbers reduces the mean by 0.1. Thus the new average 48.2 is reasonable and consistent with the arithmetic.

Why Other Options Are Wrong:
Option 48 or 48.1 would both correspond to totals different from the corrected sum of 482. For instance, an average of 48 would require a total of 480, implying a correction of 20, not 18. Option 49 corresponds to a total of 490, which is too high when we know the original total had to be reduced. Only 48.2 aligns exactly with the net correction.

Common Pitfalls:
Students may mistakenly add 18 instead of subtracting it or might divide by 9 thinking that only one number changed. However, the correction affects the total of all 10 numbers, and the average should always be recomputed using the full count of observations. Careful attention to whether the wrong value is higher or lower avoids sign errors.

Final Answer:
The correct average of the 10 numbers is 48.2.