Out of 135 applicants for a post, 60 are graduates and 80 have work experience. What is the ratio of the minimum possible number of graduate applicants with experience to the maximum possible number of graduate applicants with experience?

Introduction / Context:
This is a classic set theory and counting problem based on overlapping groups. We are given the number of graduates, the number of applicants with experience and the total number of applicants. The question asks for the possible extremes of overlap between the two groups and then requests the ratio of minimum to maximum overlap. This is a common style of aptitude question about unions and intersections of sets.

Given Data / Assumptions:

• Total number of applicants = 135.
• Number of graduates (set G) = 60.
• Number of applicants with experience (set E) = 80.
• We assume all applicants are counted once in the total.
• We must find the minimum and maximum possible number of applicants who are both graduate and experienced, that is the size of G intersection E.

Concept / Approach:
The maximum intersection occurs when one smaller group is entirely contained within the other group. The minimum intersection is found by spreading the groups apart as much as possible without exceeding the total number of applicants. The standard formula using inclusion exclusion is helpful: size of union = size of G + size of E minus size of intersection. By using the fact that the union cannot exceed the total of 135, we can derive the minimum intersection. The maximum intersection is simply the size of the smaller group.

Step-by-Step Solution:
Let x be the number of applicants who are both graduates and experienced. The union G ∪ E contains G + E − x applicants. So G + E − x ≤ total applicants. 60 + 80 − x ≤ 135. 140 − x ≤ 135, so x ≥ 5. Therefore, the minimum possible intersection size is 5. The maximum intersection occurs when all graduates are experienced, so maximum x = 60 (the smaller of 60 and 80). Thus the ratio of minimum to maximum intersection is 5 : 60. Simplify 5 : 60 by dividing both terms by 5 to get 1 : 12.

Verification / Alternative check:
We can check the extreme cases. For the minimum overlap of 5, the union has 60 + 80 − 5 = 135 applicants, exactly matching the total. For the maximal overlap of 60, the union has 60 + 80 − 60 = 80 applicants, which is allowed because the union can be less than or equal to the total. This confirms that 5 and 60 are valid values and that the ratio 1 : 12 is consistent.

Why Other Options Are Wrong:
A ratio such as 5 : 1 or 1 : 5 reverses or miscalculates the comparison. The option 12 : 1 swaps the order of minimum and maximum. Ratios like 2 : 5 do not arise from any reasonable interpretation of the intersection values. Only 1 : 12 correctly reflects minimum 5 and maximum 60.

Common Pitfalls:
A common error is to think the minimum intersection is 0, ignoring the total limit on union size. Another mistake is to assume maximum intersection is the larger of the two group sizes rather than the smaller. Remembering the inclusion exclusion relationship and that the union cannot exceed the total avoids these mistakes.

Final Answer:
The required ratio of minimum to maximum number of graduates with experience is 1 : 12.