Solving the Class Rank Sum Problem

Problem Analysis

The problem involves determining the sum of ranks for three students: Amar, Akbar, and Anthony. We are given the total number of students (60) and relationships between Amar's rank and the ranks of the other three students (Akbar, Amita, Anthony). Key constraints are that all ranks must be distinct positive integers and no rank can exceed the total number of students.

Solution Steps

1. Define variables for the ranks:
   • Let $R_A$ be the rank of Amar.
   • Let $R_{Ak}$ be the rank of Akbar.
   • Let $R_{Am}$ be the rank of Amita.
   • Let $R_{An}$ be the rank of Anthony.
2. Establish the relationships based on the question:
   • $R_A = 2 \times R_{Ak}$
   • $R_A = 3 \times R_{Am}$
   • $R_A = 7 \times R_{An}$
   This implies:
   • $R_{Ak} = R_A / 2$
   • $R_{Am} = R_A / 3$
   • $R_{An} = R_A / 7$
3. Determine a valid rank for Amar ($R_A$). Since $R_{Ak}$, $R_{Am}$, and $R_{An}$ must be integers, $R_A$ must be divisible by 2, 3, and 7. The smallest common multiple (LCM) of 2, 3, and 7 is $2 \times 3 \times 7 = 42$. Let's assume $R_A = 42$.
4. Calculate the other ranks using $R_A = 42$:
   • $R_{Ak} = 42 / 2 = 21$
   • $R_{Am} = 42 / 3 = 14$
   • $R_{An} = 42 / 7 = 6$
   The ranks are 42, 21, 14, and 6. These are all distinct positive integers and less than or equal to 60.
5. Calculate the sum of the ranks of Amar, Akbar, and Anthony: Sum = $R_A + R_{Ak} + R_{An}$ Sum = $42 + 21 + 6$ Sum = $69$

Conclusion

The sum of the ranks of Amar, Akbar, and Anthony is 69.