UPSC CSE Prelims 2024

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

Question: 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?

EXPLANATION

Correct Option (4)

The question requires determining the number of students in a class, given that the average marks are 60. Let 'N' be the number of students and 'S' be the total marks. The relationship is S/N = 60, which implies S = 60N. To find N, either N itself or the total marks S must be ascertainable.

Evaluation of Statement I: Statement I provides the highest marks (70) and the lowest marks (50). This information defines the range of marks within the class but does not offer any data to calculate the total sum of marks (S) or the total number of students (N). Therefore, Statement I alone is insufficient to answer the question.
Evaluation of Statement II: Statement II states that the exclusion of the highest and lowest marks from the class does not change the average. Let H be the highest marks and L be the lowest marks.
- Original average: S/N = 60.
- New average (after excluding H and L): (S - H - L) / (N - 2) = 60.
- From this, it follows that S - H - L = 60(N - 2).
- Substituting S = 60N into the equation: 60N - H - L = 60N - 120.
- This simplifies to H + L = 120.
Statement II alone only provides the sum of the highest and lowest marks (H + L = 120). It does not provide the total sum of marks (S) or the number of students (N). Therefore, Statement II alone is insufficient to answer the question.
Evaluation of Both Statements Together:
- From Statement I, H = 70 and L = 50.
- From Statement II, H + L = 120.
Substituting the values from Statement I into the relation derived from Statement II (70 + 50 = 120) confirms consistency between the statements. However, this combined information still only pertains to the specific values of the highest and lowest marks. It does not provide any additional data or equations that would allow for the determination of the total sum of marks (S) or the number of students (N).

Therefore, since neither statement alone nor both statements together provide sufficient information to determine the number of students, the question cannot be answered even by using both statements together.

Incorrect Options:

Options 1, 2, and 3 are incorrect because, as established through the individual and combined analysis of the statements, the provided information is insufficient to calculate the number of students in the class. Neither statement independently, nor their combination, yields a unique numerical value for 'N'.