Question

Mahesh and Ramesh are studying in the same class. How many students are there in their class if the difference in their ranks is 8?
Statement 1: Mahesh has equal number of students, who are above as well as below, in terms of rank
Statement 2: The number of students above Ramesh’s rank is equal to the number of students between Mahesh and Ramesh’s ranks.

• A. statement (1) alone is sufficient to answer the question
• B. statement (2) alone is sufficient to answer the question
• C. both the statements together are needed to answer the question
• D. statement (1) alone or statement (2) alone is sufficient to answer the question
• E. neither statement (1) nor statement (2) suffices to answer the question

Answer 1

The Correct Option is C

The problem requires us to determine the number of students in Mahesh and Ramesh's class using the given statements.

Given:

• Mahesh and Ramesh are students in the same class. 
• The difference in their ranks is 8.

Statement 1: Mahesh has an equal number of students above him and below him.

Let's analyze this statement:

• This means Mahesh's rank is exactly in the middle of the total number of students.
• If the total number of students is \(n\), then Mahesh's rank can be expressed as \(\left(\frac{n + 1}{2}\right)\), given \(n\) is odd.

This statement alone does not give us the exact number of students, as we do not know the total number of students in the class yet.

Statement 2: The number of students above Ramesh’s rank is equal to the number of students between Mahesh and Ramesh’s ranks.

Let's analyze this statement:

• If the number of students between Mahesh and Ramesh is equal to the number of students above Ramesh, we need to express these in terms of rank.
• If Ramesh's rank is \(r_R\), the number of students above him is \((r_R - 1)\).
• Students between Mahesh and Ramesh = |Mahesh's rank - Ramesh's rank| - 1 = 8 - 1 = 7.

From this statement, we know that there are 7 students above Ramesh, but without the exact rank of either Mahesh or the total students, we can't determine the precise number of students.

Using Both Statements Together:

• From Statement 1, Mahesh’s rank is the middle rank of the class.
• Using both statements: If Ramesh's rank is 8 places away from Mahesh's, there are exactly 7 students between them (from Statement 2).
• Let's calculate:
  • Since Mahesh’s rank divides the class into two equal parts, and the students between Mahesh and Ramesh are equal to students above Ramesh, then there are 7 students above and 7 students below Mahesh.
  • This implies Mahesh’s rank is 8, so \(\frac{n + 1}{2} = 8 \Rightarrow n + 1 = 16 \Rightarrow n = 15\).

Therefore, both statements together are needed to calculate the total number of students in the class, which is 15. Thus, the correct answer is: both the statements together are needed to answer the question.