In an army selection process, the ratio of selected to unselected candidates was 3:1. If 60 fewer candidates had applied and 30 fewer 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 is a classic example of ratio and algebra in selection processes. It deals with the relationship between selected and unselected candidates in an exam or recruitment and describes how that relationship would change if the number of applicants and selected candidates changed. Such questions test the ability to translate ratio statements into equations and handle changes carefully.

Given Data / Assumptions:

Initial ratio of selected to unselected candidates = 3 : 1.
If 60 fewer candidates had applied, and 30 fewer candidates had been selected, the new ratio would have become 5 : 1.
All candidates are either selected or unselected, with no other category.
We are asked to find the original total number of candidates who applied.

Concept / Approach:
Let the numbers of selected and unselected candidates be multiples of the given ratio. Use variables to represent these numbers and then apply the changes described in the question. The new ratio after the hypothetical changes provides a second equation. Solving these equations together yields the original numbers. This is a standard algebraic approach for problems involving changing ratios.

Step-by-Step Solution:
Let initially selected candidates = 3x and unselected candidates = x, based on the ratio 3:1. So total candidates who applied initially = 3x + x = 4x. Under the hypothetical situation, 60 fewer candidates applied, so new total = 4x - 60. Also, 30 fewer candidates are selected, so selected becomes 3x - 30. Then unselected under the new condition = (4x - 60) - (3x - 30) = x - 30. Given new ratio of selected to unselected is 5 : 1, so (3x - 30) / (x - 30) = 5 / 1. Cross multiply: 3x - 30 = 5(x - 30). So 3x - 30 = 5x - 150. Rearrange: 120 = 2x, hence x = 60. Original total candidates = 4x = 4 * 60 = 240.

Verification / Alternative check:
With x = 60, initially selected = 180 and unselected = 60, giving ratio 3:1. Under the hypothetical change, total becomes 240 - 60 = 180, selected becomes 180 - 30 = 150, unselected becomes 180 - 150 = 30. The new ratio 150:30 simplifies to 5:1, which matches the problem statement, confirming our answer.

Why Other Options Are Wrong:
Option 480 would make x = 120, which breaks the second ratio condition when the hypothetical changes are applied.
Option 120 would give x = 30, but then unselected candidates after change become zero, which is inconsistent with a 5:1 ratio.
Option 720 is three times the correct total and would not satisfy both ratio conditions simultaneously when checked.

Common Pitfalls:
A common mistake is to subtract 60 and 30 directly from the ratio numbers, instead of from the actual totals and selected counts.
Students sometimes forget that unselected after the change must be recomputed from the new total minus the new selected, not just from a scaled ratio.
Another pitfall is cross multiplying incorrectly or simplifying the equation in a way that leads to negative or non integral values of x.

Final Answer:
The original number of candidates who applied for the process was 240.