The class average x in a test increases by 4 when the score of a student is rectified, whose corrected score is 100 instead of 0. Later, the score of another student was found to have been recorded as 81 in place of 56. If there are no other corrections and the final corrected average is y, then $y-x$ is

The correct option is (b) - 3.

[as per provisional answerkey]

Solution

Step 1: Determine the number of students (n).
Let the number of students in the class be n.
The first correction involves changing a score from 0 to 100. This is an increase of 100 marks.
We are told this increase causes the class average x to increase by 4.
Formula for change in average: Total Change / Number of Students = Increase in Average
$100/n=4$
$n=100/4=25$ students.

Step 2: Calculate the effect of the second correction.
The second correction involves changing a score from 81 to 56.
This is a decrease in the total marks.
Change in marks = $56-81=-25$ marks.

Step 3: Find the final average y and the difference y - x.
Let the initial total marks be T. Then $x=T/25$.
After the first correction (0 to 100), the new average is $x+4$.
After the second correction (-25 marks), the final total marks become: (Initial Total + 100 - 25).
Final Average y = (Initial Total + 75) / 25
y = (Initial Total / 25) + (75 / 25)
$y=x+3$
Therefore, $y-x=3$.

Why the other options are incorrect

• Option (a) - 2: This would be the result if the second correction resulted in a decrease of 50 marks instead of 25, or if the number of students was calculated incorrectly.
• Option (c) - 5: This value might be reached if a student incorrectly added the second change ($81-56=25$) to the average instead of subtracting it from the total, or miscalculated the class size.
• Option (d) - 6: This value does not align with the calculated class size of 25; it would require a different net change in marks or a different number of students.

Key Concept

The relationship between the total sum, the number of observations, and the average, specifically how a net change in the sum affects the average across a fixed number of entities.