Introduction / Context:
This is a classic ratio and algebra problem in the context of recruitment. You are given an initial ratio of selected to unselected candidates, and a hypothetical scenario that changes both the total applicants and the number selected. From this changed scenario you must work backwards to find the original total number of applicants. This type of problem appears frequently in aptitude tests, especially in government and banking exams.
Given Data / Assumptions:
- Initial ratio of selected candidates (S) to unselected candidates (U) = 14 : 25.
- If 35 fewer candidates had applied, and 10 fewer were selected, the new ratio S : U would be 3 : 5.
- The total number of candidates is S + U.
- We assume all numbers are positive integers.
Concept / Approach:
First, express the initial numbers of selected and unselected candidates in terms of a single variable using the ratio 14 : 25. Then represent the new scenario: the total applicants decrease by 35, selected candidates decrease by 10, and therefore unselected candidates change accordingly. Use the new ratio 3 : 5 to form a second equation. Solving this equation yields the scaling factor and the original numbers of selected and unselected candidates, from which we compute the total applicants.
Step-by-Step Solution:
1) Let initial selected candidates be S and unselected candidates be U with S : U = 14 : 25.
2) So S = 14k and U = 25k for some positive integer k.
3) Total initial applicants = S + U = 14k + 25k = 39k.
4) In the hypothetical scenario, 35 fewer candidates apply, so new total = 39k - 35.
5) Also, 10 fewer candidates are selected, so new selected = S - 10 = 14k - 10.
6) New unselected = (new total) - (new selected) = (39k - 35) - (14k - 10) = 25k - 25.
7) The new ratio of selected to unselected is given as 3 : 5, so (14k - 10) : (25k - 25) = 3 : 5.
8) Write as a fraction: (14k - 10) / (25k - 25) = 3 / 5.
9) Cross-multiply: 5(14k - 10) = 3(25k - 25).
10) Expand: 70k - 50 = 75k - 75.
11) Rearrange: 75k - 70k = -50 + 75, so 5k = 25, hence k = 5.
12) Therefore S = 14k = 70 and U = 25k = 125.
13) Total number of candidates who applied = S + U = 70 + 125 = 195.
Verification / Alternative check:
Check the hypothetical scenario with these values. New total applicants = 195 - 35 = 160. New selected = 70 - 10 = 60. New unselected = 160 - 60 = 100. The new ratio is 60 : 100, which simplifies to 3 : 5 when divided by 20. This perfectly matches the condition given in the hypothetical scenario, confirming that the original total of 195 applicants is correct.
Why Other Options Are Wrong:
Other totals like 205, 185, 175 or 165 do not satisfy both the original and the hypothetical ratio conditions when you try to split them into selected and unselected candidates with the given changes. Only 195 allows the numbers S and U to fit the ratio 14 : 25, while still satisfying the new 3 : 5 ratio after reducing the number of applicants and selected candidates as specified.
Common Pitfalls:
A usual mistake is to incorrectly express the new unselected candidates or to assume that the decrease of 35 applicants affects only unselected candidates. Another pitfall is to form the wrong ratio equation, such as mixing up the order of selected and unselected. Always define variables clearly, write down expressions for both scenarios and then apply the ratios carefully to avoid such errors.
Final Answer:
The total number of candidates who had applied for the recruitment is
195.
Discussion & Comments