In an army recruitment process, the ratio of selected candidates to unselected candidates was 9 : 2. If 80 fewer candidates had applied in total and 20 fewer candidates had been selected, the ratio of selected to unselected would have become 5 : 1. How many candidates originally applied for the process?

Introduction / Context:
This problem uses ratios to relate selected and unselected candidates in an army selection process. We have two scenarios: the original one and a hypothetical one with different totals and selections. By converting the verbal description of the ratios into algebraic equations, we can determine the original total number of applicants. Such ratio based word problems are common in aptitude exams.

Given Data / Assumptions:

• Initially, ratio of selected to unselected candidates = 9 : 2.
• If 80 fewer candidates had applied and 20 fewer candidates had been selected, the new ratio of selected to unselected would be 5 : 1.
• We assume only two categories: selected and unselected candidates.
• We are asked to find the original total number of applicants.

Concept / Approach:
Let the numbers of selected and unselected candidates initially be 9k and 2k respectively, for some positive integer k. Then the total number of applicants is 11k. In the hypothetical situation, total applicants become 11k − 80 and selected candidates become 9k − 20. The new number of unselected candidates is total minus selected. Using the new ratio 5 : 1, we set up an equation and solve for k. Finally we compute 11k to get the original total applicants.

Step-by-Step Solution:
Let initially selected = 9k and unselected = 2k. Total initial applicants = 9k + 2k = 11k. In the second scenario, 80 fewer apply, so total becomes 11k − 80. Also, 20 fewer are selected, so selected become 9k − 20. Hence unselected in the second scenario = (11k − 80) − (9k − 20) = 11k − 80 − 9k + 20 = 2k − 60. Given that the new ratio selected : unselected is 5 : 1, we have: (9k − 20) / (2k − 60) = 5 / 1. So 9k − 20 = 5 * (2k − 60). 9k − 20 = 10k − 300. Bring like terms together: −k = −280, so k = 280. Original total applicants = 11k = 11 * 280 = 3080.

Verification / Alternative check:
Using k = 280, initial selected = 9 * 280 = 2520 and unselected = 2 * 280 = 560. The ratio 2520 : 560 simplifies to 9 : 2. New total after removing 80 is 3080 − 80 = 3000. New selected = 2520 − 20 = 2500, so new unselected = 3000 − 2500 = 500. The new ratio is 2500 : 500 = 5 : 1, which matches the condition. Hence the value 3080 is consistent with both scenarios.

Why Other Options Are Wrong:
Values like 6160, 1540, 9240 or 2200 do not satisfy both ratio conditions when we recompute selected and unselected numbers using the described changes. Only 3080 leads to integer values of selected and unselected candidates that fit both given ratios.

Common Pitfalls:
Students sometimes incorrectly set up the second scenario, especially when determining the new number of unselected candidates. Another common mistake is to forget that both selected and unselected counts must remain positive when solving for k. Careful bookkeeping of each change (fewer applicants, fewer selected) and using total = selected + unselected at every stage avoids these errors.

Final Answer:
The original number of candidates who applied for the process was 3080.