In a competitive examination, the average mark for all applicants is 60. If 20% of the applicants scored 85 marks and 25% scored 95 marks, what is the average mark of the remaining applicants?

Introduction / Context:
This is a weighted average question. We are told the overall average mark of all candidates and the marks scored by two specific groups. We need to find the average for the remaining group of candidates. Such problems are very common in exam analysis, data interpretation and statistics, where we break a population into subgroups and relate their averages to the overall mean.

Given Data / Assumptions:

Overall average for all candidates = 60 marks.
20% of candidates scored 85 marks each.
25% of candidates scored 95 marks each.
The remaining 55% scored some common average that we need to find.
All candidates are counted once and marks are in the same subject or test.

Concept / Approach:
Let the total number of candidates be N. Then 20% of N is 0.20N, and 25% of N is 0.25N. The remaining candidates are 0.55N. We use the formula for the overall average as total marks divided by total candidates, and express total marks as the sum of contributions from each group. Then we set this equal to 60N and solve for the unknown average of the remaining 55% group.

Step-by-Step Solution:
Step 1: Let the total number of candidates be N. Candidates scoring 85 marks: 0.20N. Candidates scoring 95 marks: 0.25N. Remaining candidates: N - 0.20N - 0.25N = 0.55N. Let the average mark of the remaining group be x. Step 2: Compute the total marks from each group. Group 1: total marks = 0.20N * 85 = 17N. Group 2: total marks = 0.25N * 95 = 23.75N. Remaining group: total marks = 0.55N * x. Step 3: Use the overall average condition. Overall total marks = average * total candidates = 60 * N. So, 17N + 23.75N + 0.55N * x = 60N. Step 4: Combine like terms. 17N + 23.75N = 40.75N. So, 40.75N + 0.55N * x = 60N. Step 5: Subtract 40.75N from both sides. 0.55N * x = 60N - 40.75N = 19.25N. Step 6: Divide both sides by 0.55N. x = 19.25 / 0.55. x = 35. Thus, the average of the remaining applicants is 35 marks.

Verification / Alternative check:
As a simple check, suppose N = 100 for convenience. Then 20 candidates scored 85, 25 scored 95, and 55 scored x. Total marks from first two groups are 20 * 85 = 1,700 and 25 * 95 = 2,375, giving 4,075. If x is 35, the remaining group contributes 55 * 35 = 1,925 marks. Total marks = 4,075 + 1,925 = 6,000. Average = 6,000 / 100 = 60, which matches the given overall average.

Why Other Options Are Wrong:
60 cannot be the remaining group average because that would keep the overall average at 60 only if the other groups also averaged 60, which they do not.
52 and 45 produce total marks that do not match the required overall average of 60.
55 also leads to a higher total average than 60 when combined with 85 and 95 on the given proportions.

Common Pitfalls:
One common mistake is to compute the simple average of 85 and 95 and ignore the overall average and proportions. Another error is to treat the percentages as additive weights without carefully converting them into the number of candidates. Some may set up the equation but forget to multiply the averages by the respective number of candidates, thereby mixing averages and totals incorrectly. Always convert averages to total marks using number of candidates and then work with those totals.

Final Answer:
The average marks of the remaining applicants is 35.