Finding Class Size After Marks Correction

This problem involves calculating the total number of students in a class using information about the average marks before and after correcting a single student's score. We need to determine the class size (number of students) when we know the initial average, the incorrect score, the correct score, and the final average.

Understanding Average Marks Calculation

The average marks are calculated by dividing the sum of all marks by the total number of students. The formula is:

$ \text{Average} = \frac{\text{Sum of Marks}}{\text{Number of Students}} $

If we know the average and the number of students, we can find the sum of marks:

$ \text{Sum of Marks} = \text{Average} \times \text{Number of Students} $

When a single student's mark is corrected, the sum of marks changes, which in turn affects the average.

Step-by-Step Calculation

Let's denote the number of students in the class as '$N$'.

1. Initial Information:
   • Initial average marks = 30.8
   • Incorrect mark entered = 24
   • Correct mark should be = 42
   • New average marks after correction = 31.4
2. Calculate the change in total marks:

   The difference between the correct mark and the incorrect mark is:

   $ \text{Change in Marks} = \text{Correct Mark} - \text{Incorrect Mark} $

   $ \text{Change in Marks} = 42 - 24 = 18 $

   So, the total sum of marks for the class increased by 18 after the correction.

3. Set up equations based on the average:

   Let the initial sum of marks be '$S_{initial}$'.

   $ S_{initial} = \text{Initial Average} \times N = 30.8 \times N $

   After correcting the mark, the new sum of marks, '$S_{new}$', is:

   $ S_{new} = S_{initial} + \text{Change in Marks} $

   $ S_{new} = (30.8 \times N) + 18 $

   The new average is given by:

   $ \text{New Average} = \frac{S_{new}}{N} $

   $ 31.4 = \frac{(30.8 \times N) + 18}{N} $

4. Solve for the number of students ($N$):

   Multiply both sides by '$N$':

   $ 31.4 \times N = (30.8 \times N) + 18 $

   Subtract '$30.8 \times N$' from both sides:

   $ (31.4 \times N) - (30.8 \times N) = 18 $

   $ (31.4 - 30.8) \times N = 18 $

   $ 0.6 \times N = 18 $

   Divide by 0.6 to find '$N$':

   $ N = \frac{18}{0.6} $

   To simplify the division, multiply the numerator and denominator by 10:

   $ N = \frac{180}{6} $

   $ N = 30 $

Conclusion

The calculation shows that there are 30 students in the class. The correction of one student's marks from 24 to 42 increased the total marks by 18, leading to an increase in the average marks from 30.8 to 31.4.