Two students appeared for an examination. One student scored 20 marks more than the other, and his marks were 55 percent of the sum of their marks together. What are the marks obtained by the two students?

Introduction / Context:
This question examines understanding of percentages and linear equations in a word problem involving two unknown quantities. The relationship between the marks of the two students is given in two different forms: a fixed difference and a percentage of their combined total. Solving such questions quickly is important in quantitative aptitude sections.

Given Data / Assumptions:
Let marks of the first student be A. Let marks of the second student be B. A is 20 marks more than B, so A - B = 20. A is 55 percent of the sum of their marks, so A = 55 percent of (A + B). All marks are assumed to be non negative integers.

Concept / Approach:
We convert the percentage relationship into an algebraic equation. Since A is 55 percent of the total A + B, we write A = 55/100 * (A + B). Combined with the difference equation A - B = 20, we can solve the system for A and B using basic algebra. This method is typical when there are two unknowns and two independent equations.

Step-by-Step Solution:
From the first condition: A - B = 20. From the second condition: A = 55/100 * (A + B) = 0.55 * (A + B). Rewrite: A = 0.55A + 0.55B. Bring like terms together: A - 0.55A = 0.55B. This gives 0.45A = 0.55B. So A / B = 0.55 / 0.45 = 55 / 45 = 11 / 9. Thus A : B = 11 : 9. Difference in ratio units = 11 - 9 = 2 parts correspond to actual difference 20 marks. So 2 parts = 20 marks which implies 1 part = 10 marks. Therefore A = 11 * 10 = 110 and B = 9 * 10 = 90.

Verification / Alternative check:
Check the conditions. Difference: 110 minus 90 equals 20 which matches the first condition. Sum: 110 plus 90 equals 200. Now 55 percent of 200 is 0.55 * 200 = 110, which matches A. Both conditions are satisfied, so the pair 110 and 90 is correct.

Why Other Options Are Wrong:
For 92 and 72, the difference is 20 but 55 percent of their sum 164 is 90.2, not 92. For 83 and 63, 55 percent of 146 is 80.3, not 83. For 64 and 44, 55 percent of 108 is 59.4. For 102 and 82, 55 percent of 184 is 101.2, not 102. Only the pair 110 and 90 satisfies both the difference and percentage conditions.

Common Pitfalls:
A common error is to treat 55 percent as relating only to one student rather than to the sum of their marks. Another mistake is to attempt random trial and error with answer options without setting up the equations, which is slow and error prone. Working systematically with ratios and equations leads to the correct solution more reliably.

Final Answer:
The two students obtained 110 and 90 marks respectively.