Two students appeared for an examination. One of them secured 26 marks more than the other, and his marks were 75% of the sum of their marks. What are the marks obtained by the two students?

Introduction / Context:
This problem is a simple algebra word problem involving two unknown quantities (the marks of two students) and two equations derived from the statements. It tests your ability to translate verbal conditions into algebraic equations and then solve them systematically.

Given Data / Assumptions:

There are two students; let their marks be x and y.
One student scored 26 marks more than the other.
The higher mark is equal to 75% of the sum of both students' marks.
Marks are measured on the same scale (for example, out of 100).
We must find the actual marks obtained by both students.

Concept / Approach:
Let the higher scoring student have marks x and the other student have marks y. Then x = y + 26. Also, x is 75% of (x + y), so x = 0.75 * (x + y). These two equations in x and y can be solved simultaneously. Solving them gives the unique pair of marks that satisfies both conditions.

Step-by-Step Solution:
Step 1: Express the relationships using equations. From the difference condition: x = y + 26. From the percentage condition: x = 0.75 * (x + y). Step 2: Use fractional form for clarity. Write 0.75 as 3/4: x = (3/4) * (x + y). Multiply both sides by 4: 4x = 3x + 3y, so x = 3y. Step 3: Combine with the first equation. We now have x = 3y and also x = y + 26. So 3y = y + 26, which gives 2y = 26 and hence y = 13. Step 4: Find x. x = 3y = 3 * 13 = 39. Thus, the two students scored 39 marks and 13 marks.

Verification / Alternative Check:
Check the conditions: the difference between their marks is 39 - 13 = 26, which matches the first condition. The sum of their marks is 39 + 13 = 52. Seventy five percent of this sum is 0.75 * 52 = 39, which is exactly the higher mark. Both conditions are satisfied, so the solution is correct.

Why Other Options Are Wrong:
53 and 27 differ by 26, but 53 is not 75% of (53 + 27).
60 and 34 also differ by 26, yet 60 is not 75% of their sum.
78 and 52 keep the same difference, but 78 is not 75% of (78 + 52); the percentage condition fails.

Common Pitfalls:
A common mistake is to mix up which mark is 75% of the sum or to mis-handle the percentage as 75 / 100 in equations. Some learners also forget to convert 0.75 into a fraction before clearing the denominator, which can cause arithmetic slips. Always write both conditions clearly in algebraic form and solve systematically to avoid guesswork.

Final Answer:
The two students obtained marks of 39 and 13 respectively.