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?
- (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 using 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 ✓ UPSC's answer
Why the answer is (d)
• Let the number of students be $n$, the sum of marks be $S$, and the average be 60, so $S = 60n$.
• Statement I provides the maximum mark (70) and minimum mark (50), but does not provide information about the distribution of the remaining $n-2$ marks or the total sum, making it insufficient to determine $n$.
• Statement II states that removing the highest (70) and lowest (50) marks does not change the average, implying the average of the remaining $n-2$ students is also 60.
• This gives the equation: $\frac{S - 70 - 50}{n - 2} = 60$, which simplifies to $S - 120 = 60n - 120$, or $S = 60n$.
• This equation is an identity that holds true for any $n$ (where $n > 2$) and does not yield a specific value for $n$.
• Since neither statement alone nor both together provide a unique value for the number of students, the question cannot be answered.
Why the other options are wrong
- (a) The Question can be answered by using one of the Statements alone, but cannot be answered…
- Neither statement alone provides sufficient information to determine the number of students.
- (b) The Question can be answered by using either Statement alone
- Neither statement alone is sufficient to determine the number of students.
- (c) The Question can be answered by using both the Statements together, but cannot be…
- Using both statements together results in an identity ($S=60n$) that does not solve for a specific number of students.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2024, held on 16 June 2024. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.