Calculating Class Size From Average Marks Discrepancy
A student's marks were recorded incorrectly. The incorrect entry was 84, but the correct mark should have been 94. This error caused the class average marks to decrease by \(\frac{5}{4}\). The question asks for the total number of students in the class.
Step-by-Step Solution
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Calculate the difference in marks:
The difference between the correct mark and the incorrectly entered mark is:
Correct Mark - Incorrect Mark = \(94 - 84 = 10\) marks.
This difference of 10 marks represents the total deficit in the sum of marks for the class due to the error.
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Relate mark difference to average change:
The total difference in marks (10) when spread across all students affects the class average. Let \(N\) be the number of students in the class.
The change in the average mark is calculated as the total difference in marks divided by the number of students (\(N\)).
Change in Average = \(\frac{\text{Total Difference in Marks}}{N}\)
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Set up the equation:
The problem states that the average marks decreased by \(\frac{5}{4}\). Since the entered mark (84) is less than the correct mark (94), the total sum of marks is lower, leading to a lower average. Therefore, the change in average is negative, \(-\frac{5}{4}\).
So, we have the equation:
\(-\frac{10}{N} = -\frac{5}{4}\)
This simplifies to:
\(\frac{10}{N} = \frac{5}{4}\)
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Solve for the number of students (N):
To find \(N\), we can cross-multiply:
\(10 \times 4 = 5 \times N\)
\(40 = 5N\)
Now, divide by 5:
\(N = \frac{40}{5}\)
\(N = 8\)
Thus, there are 8 students in the class.
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