Finding the Minimum Number of Students in a Ranking Problem

This problem involves calculating the total number of students in a group, considering rankings from the top and bottom within a subset of passed students and also accounting for failed students. The key phrase "minimum students" suggests we must consider the possibility of overlapping ranks within the passed list.

Understanding the Given Information

• Rohan's rank from the top in the list of passed students: 13th
• Raman's rank from the bottom in the list of passed students: 36th
• Number of students between Rohan and Raman in the passed list: 5
• Number of failed students: 9

Calculating Passed Students: Two Possible Cases

When dealing with ranks from opposite ends (top and bottom) for two different individuals and a specified number of students between them, there are two scenarios for the arrangement of students in the list of passed students:

1. Non-overlapping Case: The students are arranged linearly without their ranks crossing over from the perspective of a single direction (e.g., viewing from the top, Rohan is above Raman, and Raman is ranked from the bottom).
2. Overlapping Case: The ranks overlap. From the perspective of viewing from the top, the person ranked from the bottom appears higher up in the list than the person ranked from the top. This scenario usually results in a lower total number of students.

Case 1: Non-Overlapping Ranks

In this case, the total number of passed students is simply the sum of Rohan's rank from the top, the number of students between them, and Raman's rank from the bottom.

Total Passed (Non-overlapping) = (Rohan's rank from top) + (Number of students between) + (Raman's rank from bottom)

\( \text{Total Passed} = 13 + 5 + 36 \)

\( \text{Total Passed} = 54 \)

Case 2: Overlapping Ranks

In this case, Rohan's rank from the top and Raman's rank from the bottom "overlap". From the top, Raman would appear before Rohan in the list. The formula for the total number of people in the list in this overlapping scenario is:

Total Passed (Overlapping) = (Rohan's rank from top) + (Raman's rank from bottom) - (Number of students between) - 2 (subtracting 2 accounts for Rohan and Raman being included in both ranks and also counted within the 'between' count relative to the ranks).

\( \text{Total Passed} = 13 + 36 - 5 - 2 \)

\( \text{Total Passed} = 49 - 7 \)

\( \text{Total Passed} = 42 \)

Determining the Minimum Number of Passed Students

The minimum number of passed students is obtained from the overlapping case, as it results in a smaller total for the ranked list.

• Non-overlapping passed students: 54
• Overlapping passed students: 42

The minimum number of passed students is 42.

Calculating the Minimum Total Students

The total number of students in the class includes both the students who passed and the students who failed.

Total Students = Passed Students + Failed Students

To find the minimum total students, we use the minimum number of passed students calculated above.

Minimum Total Students = (Minimum Passed Students) + (Failed Students)

\( \text{Minimum Total Students} = 42 + 9 \)

\( \text{Minimum Total Students} = 51 \)

Therefore, the minimum number of students in the class is 51.

Revision Table: Key Concepts

Additional Information on Ranking Problems

Ranking and order problems are common in logical reasoning sections of competitive exams. They test your ability to understand relative positions based on given information.

• Single Person Rank: If a person's rank is given from both ends (e.g., 10th from top and 15th from bottom), the total number of people is Rank from top + Rank from bottom - 1. We subtract 1 because the person is counted twice.
• Finding Rank from Other End: If total people and a person's rank from one end are given, their rank from the other end is (Total People - Rank from one end) + 1.
• Students Between Two People:
  • In a non-overlapping scenario: Total - (Rank1 + Rank2) = Number between.
  • In an overlapping scenario: (Rank1 + Rank2) - Total - 2 = Number between. Conversely, if Number between is known, Total = Rank1 + Rank2 - Number between - 2.
• Conditions for Overlapping: Overlapping is only possible if the sum of the ranks from opposite ends is greater than the number of people between them plus two (Rank1 + Rank2 > Number between + 2). In our case, 13 + 36 = 49, and Number between + 2 = 5 + 2 = 7. Since 49 > 7, overlapping is possible. Also, Rank1 + Rank2 must be greater than the total number of people in the list for non-overlapping, and less than the total for overlapping (this check is iterative). The formula Total = R1 + R2 - N - 2 directly calculates the total for overlapping if possible.