In an army selection process, the ratio of selected to unselected candidates was 4:1. If 90 fewer candidates had applied and 20 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 is another ratio based selection question, similar in style to other army or recruitment process problems. It describes initial numbers of selected and unselected candidates and then presents a hypothetical change in which some candidates are removed from the pool and fewer candidates are selected. The question tests algebraic modelling of these changes and correct use of ratios.

Given Data / Assumptions:

Initially, the ratio of selected to unselected candidates is 4 : 1.
If 90 fewer candidates had applied, and 20 fewer candidates had been selected, the new ratio would have been 5 : 1.
There are only two categories of candidates: selected and unselected.
We need to find the original number of candidates who applied.

Concept / Approach:
We represent the initial numbers of selected and unselected candidates using a variable multiple of the ratio 4:1. Then we adjust those numbers according to the hypothetical changes described, and express the resulting new numbers as another ratio equation. Solving this equation will give the multiple and therefore the original total number of candidates. This method is systematic and avoids guesswork.

Step-by-Step Solution:
Let initially selected candidates = 4x and unselected candidates = x. Total candidates who originally applied = 4x + x = 5x. If 90 fewer candidates had applied, new total would have been 5x - 90. If 20 fewer had been selected, new selected would have been 4x - 20. Therefore new unselected = (5x - 90) - (4x - 20) = x - 70. Given new ratio selected : unselected = 5 : 1, so (4x - 20) / (x - 70) = 5 / 1. Cross multiply: 4x - 20 = 5(x - 70). So 4x - 20 = 5x - 350. Rearrange: 330 = x, since 5x - 4x = x and -20 + 350 = 330. Original total candidates = 5x = 5 * 330 = 1,650.

Verification / Alternative check:
With x = 330, initial selected = 1,320 and unselected = 330, giving ratio 4:1. Under the hypothetical change, total = 1,650 - 90 = 1,560, selected = 1,320 - 20 = 1,300, and unselected = 1,560 - 1,300 = 260. The new ratio 1,300:260 simplifies by dividing by 260 to 5:1. This perfectly matches the problem statement, confirming that 1,650 is correct.

Why Other Options Are Wrong:
Option 3,300 corresponds to taking x = 660, which does not satisfy the second ratio when the changes are applied.
Option 825 would give x = 165, leading to non integer or inconsistent unselected counts when the hypothetical changes are considered.
Option 4,950 is three times 1,650 and again fails to satisfy the new ratio 5:1 after reductions are applied.

Common Pitfalls:
Some students subtract 90 and 20 from the ratio terms instead of the actual counts, confusing ratios with real numbers.
Others forget to recompute unselected candidates after the changes and instead attempt to scale the ratio directly, which is incorrect.
A frequent algebraic error is mishandling signs when bringing terms to one side of the equation, which leads to a wrong value for x.

Final Answer:
The original number of candidates who applied for the selection process was 1,650.