The average marks of 100 students are given to be 40. It was found later that marks of one student were 53 which were misread as 83. The corrected mean marks are

The correct option is 39.7.

To find the corrected mean, we first need to understand the relationship between the number of observations, the mean, and the sum of all observations.
The formula for the mean (average) is:
$$\text{Mean}=\frac{\text{Sum of all observations}}{\text{Total number of observations}}$$

From the given problem, we have:
Number of students ($n$) = 100
Incorrect mean = 40

Using these, we can calculate the incorrect sum of marks:
$$\text{Incorrect Sum of Marks}=\text{Incorrect Mean}\times n$$
$$\text{Incorrect Sum of Marks}=40\times 100=4000$$

Now, we are told that the marks of one student were misread as 83 instead of the correct value of 53.
To find the correct sum of marks, we subtract the incorrect value (83) and add the correct value (53) to our incorrect sum:
$$\text{Correct Sum of Marks}=\text{Incorrect Sum of Marks}-\text{Incorrect Value}+\text{Correct Value}$$
$$\text{Correct Sum of Marks}=4000-83+53$$
$$\text{Correct Sum of Marks}=4000-30=3970$$

Finally, we calculate the corrected mean marks by dividing the correct sum of marks by the total number of students:
$$\text{Corrected Mean}=\frac{\text{Correct Sum of Marks}}{n}$$
$$\text{Corrected Mean}=\frac{3970}{100}=39.7$$