Two students appeared for an examination. One student secured 21 marks more than the other, and his marks were equal to 80% of the sum of their marks. What were the marks obtained by the two students?

Introduction / Context:
This aptitude question tests linear equation skills and understanding of percentages in word problems. You must translate the relation between two students marks into algebra and then solve for their individual scores.

Given Data / Assumptions:
- Two students appeared for the same examination. - Let the lower score be y marks. - The higher score is 21 marks more, so it is y + 21. - The higher score is 80% of the sum of their marks. - All marks are assumed to be non negative and realistic for a test.

Concept / Approach:
We convert the statement into an equation. The sum of their marks is (y + y + 21). The higher score equals 0.8 times this sum. That equation can be solved for y, after which the higher score is obtained by adding 21. Finally we check which option matches the pair of marks.

Step-by-Step Solution:
Step 1: Let the lower score be y and the higher score be y + 21. Step 2: Sum of marks = y + (y + 21) = 2y + 21. Step 3: Given that higher score = 80% of sum, so y + 21 = 0.8(2y + 21). Step 4: Expand right side: y + 21 = 1.6y + 16.8. Step 5: Rearrange: 21 - 16.8 = 1.6y - y gives 4.2 = 0.6y. Step 6: Hence y = 4.2 / 0.6 = 7. Step 7: Higher score = y + 21 = 7 + 21 = 28. The marks are 28 and 7.

Verification / Alternative check:
With marks 28 and 7, the sum is 35. Eighty percent of 35 is 0.8 * 35 = 28, which matches the higher score. The difference between the scores is 28 - 7 = 21, which also matches the given condition. So the pair 28 and 7 satisfies both conditions exactly.

Why Other Options Are Wrong:
- 88 and 67 differ by 21 but 80% of their sum is not 88. - 89 and 68 also differ by 21 but do not match the 80% relation. - 98 and 77 differ by 21 but again fail the 80% condition. - None of these is a generic distractor and is not required since one option already fits perfectly.

Common Pitfalls:
Many learners misuse percentage and set 80% of the larger mark equal to the sum instead of 80% of the sum equal to the larger mark. Others forget to represent both conditions simultaneously and try to guess values. Writing a clear variable equation avoids confusion and makes the solution systematic.

Final Answer:
The two students obtained 28 and 7 marks respectively.