An engineering student has to secure 25% of the maximum marks in an examination in order to pass. He obtains 45 marks and still falls short of the pass mark by 45 marks. What are the maximum marks for the examination?

Introduction / Context:
This question is a straightforward application of percentage and pass mark calculations. It tests your ability to convert a pass percentage requirement into an actual mark and then use the failure information to deduce the maximum marks of the exam.

Given Data / Assumptions:
- Pass percentage required = 25% of maximum marks. - The student scores 45 marks. - He is short of the pass mark by 45 marks. - Let the maximum marks of the exam be M. - All marks are assumed to be non negative and expressed on the same scale.

Concept / Approach:
The pass mark is 25% of the maximum marks, which equals (25 / 100) * M = M / 4. The student scored 45 marks and still needed 45 more marks to reach the pass mark, so the pass mark must be 45 + 45. Once we find the pass mark, we can set it equal to M / 4 and solve for M. This leads directly to the maximum marks of the exam.

Step-by-Step Solution:
Step 1: Let the maximum marks of the exam be M. Step 2: Pass mark = 25% of M = (25 / 100) * M = M / 4. Step 3: The student scores 45 marks and fails by 45 marks. Step 4: Therefore, pass mark = 45 (scored) + 45 (short) = 90 marks. Step 5: Set pass mark equal to M / 4: M / 4 = 90. Step 6: Multiply both sides by 4: M = 90 * 4 = 360. Step 7: Hence, the maximum marks for the exam are 360.

Verification / Alternative check:
Check if 25% of 360 is indeed 90. Compute 25% of 360 as 360 * 25 / 100 = 360 / 4 = 90. The student has 45 marks, which is exactly 45 less than 90, so he fails by 45 marks. This matches the problem statement and confirms that the maximum marks are correctly found as 360.

Why Other Options Are Wrong:
Option A (385): 25% of 385 is 96.25, which is not 45 + 45. Option B (410): 25% of 410 is 102.5, inconsistent with the given deficit. Option C (435): 25% of 435 is 108.75, again not matching 90. Option E (300): 25% of 300 is 75, so the student would not be short by 45 marks here.

Common Pitfalls:
Students sometimes confuse the shortfall with the percentage and incorrectly set up the equation, for example using 45 as 25% directly. Another common mistake is to forget that pass marks equal the student's marks plus the shortfall. Writing a clear equation that equates the pass mark to both expressions avoids such mistakes. Always identify the pass mark and then relate it to the required percentage of the maximum marks.

Final Answer:
The maximum marks for the examination are 360.