Q. A Question is given followed by two Statements I and II. Consider the Question and the Statements.

# Q. If the average marks in a class are 60, then what is the number of students in the class?

Statement-I: The highest marks in the class are 70 and the lowest marks are 50.

Statement-II: Exclusion of highest and lowest marks from the class does not change the average.

Which one of the following is correct in respect of the above Question and the Statements?

a) The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone.

b) The Question can be answered by using either Statement alone.

c) The Question can be answered by both the Statements together, but cannot be answered using either Statement alone.

d) The Question cannot be answered even by using both the Statements together

Correct answer: d) The Question cannot be answered even by using both the Statements together

#### Question from UPSC Prelims 2024 CSAT

**Explanation : **

To determine the number of students in the class based on the given information, let’s analyze both statements individually and then consider them together.

Given:- Average marks = 60

Question:- What is the number of students in the class?

Analysis:

**Statement I:**

– Information Provided:

– Highest marks = 70

– Lowest marks = 50

– Implications:

– Without additional information about the distribution of marks or the total sum of marks, knowing just the highest and lowest marks doesn’t help in determining the number of students.

– Conclusion: Statement I alone is insufficient to determine the number of students.

**Statement II:**

– Information Provided:

– Removing the highest and lowest marks doesn’t change the average.

– Mathematical Implication:

– Let the number of students be n.

– Total sum of marks = Average × Number of students = 60n

– After removing highest and lowest marks:

– New sum = 60n – 70 – 50 = 60n – 120

– New number of students = n – 2

– New average = 60

– Setting up the equation:

60n – 120/n – 2 = 60

60n – 120 = 60(n – 2)

60n – 120 = 60n – 120

– The equation simplifies to 0 = 0, which is always true regardless of the value of n.

– Conclusion: Statement II alone is insufficient to determine the number of students.

## Combining Statements I and II:

– Even when combining both statements:

– Highest Marks: 70

– Lowest Marks: 50

– Average remains 60 after exclusion.

– Attempts to solve for n with combined information still do not yield a unique solution for the number of students.

– Conclusion: Even together, the statements are insufficient to determine the exact number of students.

Final Answer: d) The Question cannot be answered even by using both the Statements together