A student has to secure 40 percent marks to pass an examination. He obtains 67 marks and fails by 13 marks. What are the maximum marks for the exam?

Introduction / Context:
This question links passing percentage, actual marks obtained, and the shortfall in marks to determine the maximum marks of the examination. It illustrates a standard technique in percentage problems where the passing score is expressed as a percentage of total marks and also as a specific number of marks.

Given Data / Assumptions:
- Let the maximum marks be M.
- Passing percentage = 40 percent of M.
- Student scores 67 marks.
- He fails by 13 marks, meaning he is 13 marks short of the passing marks.
- Therefore, passing marks = 67 + 13 = 80 marks.

Concept / Approach:
Passing marks can be expressed in two ways: as 40 percent of the maximum marks and as 80 marks from the shortfall information. Equating these two expressions allows us to solve for the maximum marks M. This is a typical direct proportion equation where we find the unknown base corresponding to a known percentage and value.

Step-by-Step Solution:
Step 1: From the statement "fails by 13 marks," we know passing marks = 67 + 13 = 80.Step 2: Passing marks are also 40 percent of the maximum marks M.Step 3: Write the equation: 40 percent of M = 80.Step 4: Convert 40 percent to decimal form: 40 percent = 0.40.Step 5: So 0.40 * M = 80.Step 6: Solve for M: M = 80 / 0.40.Step 7: Compute M = 80 / 0.40 = 200.

Verification / Alternative check:
Check by recomputing the passing marks from the found maximum. If M = 200, then 40 percent of 200 = 0.40 * 200 = 80 marks. The student scored 67 marks, which is indeed 13 marks less than 80. This confirms that maximum marks of 200 are consistent with the information.

Why Other Options Are Wrong:
- 300 and 240 would give passing marks of 120 and 96 respectively at 40 percent, which do not differ from 67 by exactly 13 marks.
- 150 would give passing marks of 60, but the student already scored 67, so he would be passing not failing.

Common Pitfalls:
Some students incorrectly assume 67 is 40 percent of the maximum, ignoring the "fails by 13 marks" detail. Others subtract 13 from 67 instead of adding, misinterpreting the direction of the shortfall. Always carefully translate "fails by" or "passes by" into how far the candidate's score is from the required passing marks, and then set up a correct equation.

Final Answer:
The maximum marks for the examination are 200.