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.