Two students appear for an examination. One scores 24 marks more than the other, and his marks are 65 percent of the sum of their marks. What are the marks obtained by the two students?

Introduction / Context:
This is a classic algebra and percentage question involving two unknowns linked by both a difference and a percentage condition. It checks the ability to set up and solve simultaneous equations based on a word problem statement.

Given Data / Assumptions:

• Let the higher scoring student have marks x and the other have marks y.
• x is 24 more than y.
• x is 65 percent of the sum (x + y).
• We must find x and y.

Concept / Approach:
We translate the verbal conditions into two equations: x = y + 24 and x = 0.65(x + y). Solving this system gives unique values for x and y that satisfy both relationships.

Step-by-Step Solution:
Let the higher marks = x and the lower marks = y.From the problem, x = y + 24.Also, x = 65 percent of (x + y) = 0.65(x + y).Write 0.65 as 65 / 100 = 13 / 20.So x = (13 / 20)(x + y).Multiply by 20: 20x = 13x + 13y.So 7x = 13y and x = (13/7)y.But x = y + 24, so y + 24 = (13/7)y.Multiply both sides by 7: 7y + 168 = 13y.Thus 168 = 6y and y = 28.Then x = y + 24 = 28 + 24 = 52.

Verification / Alternative check:
Check the percentage condition: sum = 52 + 28 = 80. Sixty five percent of 80 is 0.65 * 80 = 52, which indeed equals the marks of the higher student, confirming the solution pair (52, 28).

Why Other Options Are Wrong:
Pairs such as 78 and 54 or 85 and 61 do not satisfy the condition that the higher score is both 24 more than the lower score and 65 percent of the total. Substituting them breaks one or both equations.

Common Pitfalls:
Misidentifying which student has higher marks or misplacing the 65 percent on the wrong side of the equation are common mistakes. It is important to define variables clearly and ensure that each textual condition is precisely captured in algebraic form.

Final Answer:
The two students score 52 marks and 28 marks respectively.