In a competitive exam, the number of students who passed was four times the number of students who failed. If instead 35 fewer students had appeared and 9 more had failed, the ratio of passed to failed students would have been 2 : 1. How many students appeared for the exam in total?

Introduction / Context:
This is a word problem involving ratios and changes in the numbers of passed and failed students. Initially, the number of passed students is four times the number of failed students. Then, under a hypothetical scenario where fewer students appear and slightly more fail, a new ratio of passed to failed students is given. Our task is to find the original total number of students who appeared for the exam.

Given Data / Assumptions:
• Let the number of failed students initially be F. • Then the number of passed students initially is 4F. • Total students initially = 5F. • Hypothetical change: 35 fewer students appear, so new total = 5F − 35. • In this scenario, the number of failed students becomes F + 9. • The new ratio of passed to failed students is 2 : 1.

Concept / Approach:
We express the initial condition in terms of F and use it to describe the new situation. After the hypothetical changes, the number of passed students is total minus failed. We then form an equation representing the ratio 2 : 1 and solve for F. Once F is known, the initial total students is 5F. This requires careful bookkeeping of how each quantity changes.

Step-by-Step Solution:
Step 1: Initially, passed = 4F, failed = F, total = 5F. Step 2: In the new scenario, total students = 5F − 35. Step 3: New failed students = F + 9. Step 4: New passed students = (new total) − (new failed) = (5F − 35) − (F + 9) = 4F − 44. Step 5: Given that new ratio of passed to failed is 2 : 1, we have (4F − 44) : (F + 9) = 2 : 1. Step 6: Convert to equation: (4F − 44) / (F + 9) = 2 / 1. Step 7: Cross-multiply: 4F − 44 = 2(F + 9). Step 8: Expand right side: 4F − 44 = 2F + 18. Step 9: Rearrange: 4F − 44 − 2F − 18 = 0 ⇒ 2F − 62 = 0. Step 10: Solve for F: 2F = 62 ⇒ F = 31. Step 11: Initial total students = 5F = 5 * 31 = 155.

Verification / Alternative check:
Initially: failed = 31, passed = 4 * 31 = 124, total = 155. New scenario: total = 155 − 35 = 120, failed = 31 + 9 = 40, passed = 120 − 40 = 80. Check new ratio: 80 : 40 = 2 : 1 as required. This confirms that F = 31 and total 155 are consistent with all conditions.

Why Other Options Are Wrong:
• 145, 165 and 175 do not satisfy the derived equations when back-substituted into the scenario; they fail either the initial 4 : 1 ratio or the new 2 : 1 ratio after adjustments.

Common Pitfalls:
A common mistake is to confuse the new total with the old or to forget to adjust both passed and failed counts correctly. Another error is to misinterpret the phrase "9 more had failed" as subtracting 9 instead of adding. Carefully tracking all changes with clear algebra prevents such errors.

Final Answer:
The total number of students who appeared for the exam is 155.