A student must secure 35% of the maximum marks to pass in a subject. The student obtained 82 marks and failed by 23 marks. What are the maximum marks for the subject?

Introduction / Context:
Examination and result based questions often involve pass marks expressed as a percentage of the maximum marks. The student's actual marks and the margin by which the student failed are used to back calculate the full mark of the paper. This question is a direct application of percentage and basic algebra and appears frequently in aptitude and competitive exams.

Given Data / Assumptions:

• The pass mark is 35% of the maximum marks.
• The student scored 82 marks.
• The student failed by 23 marks, that is, the student needed 23 more marks to just pass.
• We need to find the maximum marks for the subject.

Concept / Approach:
The idea is to express the pass mark in terms of the unknown maximum marks. If we let the maximum marks be M, then the pass mark is 35% of M, which is (35 / 100) * M. The question says that the student scored 82 marks and is short by 23 marks from the pass mark. So, pass mark = 82 + 23. We equate this to 35% of M and solve for M.

Step-by-Step Solution:
Step 1: Let the maximum marks be M. Step 2: The pass mark is 35% of M, so pass mark = (35 / 100) * M = 0.35M. Step 3: The student obtained 82 marks and failed by 23 marks. So the pass mark must be 82 + 23 = 105. Step 4: Set up the equation: 0.35M = 105. Step 5: Solve for M by dividing both sides by 0.35. So M = 105 / 0.35. Step 6: Compute 105 / 0.35 = 300. Step 7: Therefore, the maximum marks for the subject is 300.

Verification / Alternative check:
We can verify the result by checking whether 35% of 300 actually gives a pass mark consistent with the story. 35% of 300 is (35 / 100) * 300 = 105. The student got 82 marks, and indeed 105 minus 82 equals 23 marks, which matches the statement that the student failed by 23 marks. This confirms that our calculation is correct.

Why Other Options Are Wrong:
For a maximum of 250 marks, 35% would be 87.5, and the shortfall from 82 would be only 5.5, not 23. For a maximum of 350 marks, 35% would be 122.5, which is far from the given difference. For 400 marks, 35% would be 140, so the student would fall short by 58 marks. None of these match the failure margin of 23 marks, so they are invalid options.

Common Pitfalls:
Students sometimes treat the 23 marks as 23% or misinterpret the phrase "failed by 23 marks". Another frequent mistake is to directly take 35% of 82 instead of relating 82 plus 23 to the pass mark. Always read carefully whether a number represents a percentage of the maximum or an absolute number of marks. Building the equation patiently usually leads to the correct answer.

Final Answer:
The maximum marks of the subject is 300.