How many job applicants had applied if the ratio of selected to unselected candidates was 19 : 17, and if 1,200 less had applied and 800 less were selected, the ratio of selected to unselected would have been 1 : 1?

Introduction / Context:
This question involves ratios and a hypothetical change in total applicants and selected candidates. We are given the original ratio of selected to unselected applicants, along with a scenario where fewer applicants apply and fewer are selected, leading to a new ratio. The task is to find the original total number of applicants. This type of problem is common in exam settings and requires setting up algebraic equations from ratio information.

Given Data / Assumptions:
• Original ratio of selected to unselected candidates = 19 : 17. • Suppose originally selected = 19k and unselected = 17k for some k. • Therefore original total applicants = 19k + 17k = 36k. • If 1,200 fewer had applied, total applicants would be 36k − 1,200. • If 800 fewer had been selected, new selected = 19k − 800. • In that scenario, the ratio selected : unselected becomes 1 : 1, meaning selected = unselected.

Concept / Approach:
We represent the original numbers in terms of k using the ratio, then apply the hypothetical changes and use the condition that selected equals unselected. This gives a linear equation in k. Solving for k allows us to find the original total number of applicants 36k. Working with symbols first keeps the reasoning clear and prevents confusion when inserting the numerical adjustments later.

Step-by-Step Solution:
Step 1: Let originally selected = 19k and unselected = 17k. Step 2: Then original total applicants = 36k. Step 3: In the hypothetical scenario, total applicants decrease by 1,200, so new total = 36k − 1,200. Step 4: Selected candidates decrease by 800, so new selected = 19k − 800. Step 5: Since the new ratio selected : unselected is 1 : 1, we must have new selected = new unselected. Step 6: New unselected = new total − new selected = (36k − 1,200) − (19k − 800). Step 7: Simplify new unselected: 36k − 1,200 − 19k + 800 = 17k − 400. Step 8: Set new selected equal to new unselected: 19k − 800 = 17k − 400. Step 9: Rearrange: 19k − 17k = -400 + 800. Step 10: So 2k = 400, giving k = 200. Step 11: Original total applicants = 36k = 36 * 200 = 7,200.

Verification / Alternative Check:
Compute the actual numbers with k = 200. Original selected = 19 * 200 = 3,800. Original unselected = 17 * 200 = 3,400. Original total = 3,800 + 3,400 = 7,200, which matches. Now apply the hypothetical changes: new total = 7,200 − 1,200 = 6,000. New selected = 3,800 − 800 = 3,000. New unselected = 6,000 − 3,000 = 3,000. With selected and unselected both 3,000, the ratio is 1 : 1 as required. This verifies that the answer 7,200 is correct.

Why Other Options Are Wrong:
• 6,000, 8,400 and 4,800 do not satisfy the given conditions when we attempt to reconstruct selected and unselected counts using the ratio and adjustments. • Only 7,200 allows the hypothetical scenario to achieve equal selected and unselected counts.

Common Pitfalls:
Common mistakes include assuming the total number of applicants remains unchanged when applying the hypothetical adjustment or misplacing the 800 reduction (for example, subtracting it from unselected instead of selected). Clearly labeling original and new quantities and carefully applying the conditions for the new ratio helps avoid such errors.

Final Answer:
The original number of job applicants who had applied is 7,200.