Two students appeared for an examination. One scored 13 marks more than the other and his marks were 76% of the sum of their marks. What were the marks obtained by each student?

Introduction / Context:
This problem combines basic algebra with percentages. We are given a relationship between the marks of two students in terms of their difference and the fact that the higher mark is a certain percentage of the sum of both marks. The aim is to set up equations and solve for the individual marks. Such questions are standard in aptitude tests and help practice translating verbal conditions into algebraic expressions.

Given Data / Assumptions:
There are two students; let their marks be x and y, with x greater than y.
The higher scoring student scored 13 marks more than the other, so x = y + 13.
The higher mark x is equal to 76 percent of the sum of both marks, that is 76 percent of (x + y).
We must find the pair of marks that satisfies these conditions.

Concept / Approach:
We translate the information into two equations. First, x = y + 13 expresses the difference in marks. Second, x = 0.76 * (x + y) captures the percentage relation. Solving these two equations simultaneously will yield the values of x and y. We can express x in terms of y from the first equation and substitute into the second equation, reducing everything to a single variable. After solving for y, we find x and then check whether the values satisfy the conditions and match one of the option pairs.

Step-by-Step Solution:
Let the marks of the higher scoring student be x and the other student be y. Given: x = y + 13. Given: x is 76 percent of the sum of their marks, so x = 0.76 * (x + y). Substitute x = y + 13 into x = 0.76 * (x + y). So y + 13 = 0.76 * ((y + 13) + y) = 0.76 * (2y + 13). Expand the right side: 0.76 * (2y + 13) = 1.52y + 9.88. Set up the equation: y + 13 = 1.52y + 9.88. Bring terms with y to one side: y - 1.52y = 9.88 - 13. Compute: -0.52y = -3.12, so y = -3.12 / -0.52 = 6. Now find x: x = y + 13 = 6 + 13 = 19. Therefore, the two students scored 19 and 6 marks respectively.

Verification / Alternative check:
Check both conditions with x = 19 and y = 6. Difference: 19 - 6 = 13, which satisfies the first condition. Sum of marks: 19 + 6 = 25. Now compute 76 percent of 25: 0.76 * 25 = 19. This equals the higher mark, so the second condition is also satisfied. Thus the pair (19, 6) is fully consistent with the problem statement.

Why Other Options Are Wrong:
For 34 and 21, difference is 13 but 76 percent of their sum 55 equals 41.8, not 34. For 102 and 89, difference is 13 but 76 percent of 191 is 145.16, which does not match 102. For 92 and 79, difference is also 13, but 76 percent of 171 is 129.96, not 92. Only the pair 19 and 6 satisfies both the difference condition and the 76 percent sum condition.

Common Pitfalls:
A common mistake is to interpret 76 percent of the sum incorrectly, such as using only one of the marks in the sum or writing 0.76 * x instead of 0.76 * (x + y). Others may fail to use both equations together and try to guess answers from options without verification. To avoid errors, always write both conditions clearly as equations, substitute one into the other, and solve systematically. Checking the final values against the original conditions is an essential final step.

Final Answer:
The two students obtained marks of 19 and 6 respectively.