In a class, there are 40 students. Some of them passed the examination and others failed. Raman’s rank among the student who have passed is 13 th from top and 17th from bottom. How many students have failed?
11
Understanding the Student Ranking Problem
This problem involves calculating the number of students who failed an examination, given the total number of students and the ranking of a specific student (Raman) among only those who passed.
To solve this, we first need to determine the total number of students who passed the examination using the information about Raman's rank among them. Once we know the number of passed students, we can subtract this from the total number of students in the class to find the number of students who failed.
Calculating the Number of Passed Students
Raman's rank is given among the students who passed. His rank is 13th from the top and 17th from the bottom within this specific group (the passed students). The total number of items (in this case, students) in a list when you know an item's rank from both the top and the bottom is given by the formula:
Total items = (Rank from Top) + (Rank from Bottom) - 1
Applying this formula to the group of students who passed:
Total number of passed students = Raman's rank from top (among passed) + Raman's rank from bottom (among passed) - 1
Total number of passed students = \(13 + 17 - 1\)
Total number of passed students = \(30 - 1\)
Total number of passed students = \(29\)
So, there are 29 students who passed the examination.
Calculating the Number of Failed Students
We are given the total number of students in the class, which is 40. The total number of students is the sum of the students who passed and the students who failed.
Total students = Number of passed students + Number of failed students
We can rearrange this formula to find the number of failed students:
Number of failed students = Total students - Number of passed students
Substituting the values we know:
Number of failed students = \(40 - 29\)
Number of failed students = \(11\)
Therefore, 11 students have failed the examination.
Let's summarize the steps:
- Identify the group being ranked (students who passed).
- Use the ranks from top and bottom within that group to find the total size of the group (number of passed students).
- Subtract the number of passed students from the total number of students in the class to find the number of failed students.
Summary of Results
| Description | Value |
|---|---|
| Total Students in Class | 40 |
| Raman's Rank from Top (among passed) | 13 |
| Raman's Rank from Bottom (among passed) | 17 |
| Total Students Passed | 29 |
| Total Students Failed | 11 |
Revision Table: Key Concepts in Ranking Problems
| Concept | Explanation | Formula |
|---|---|---|
| Total items in a line/list | If you know an item's rank from both ends. | Rank from Top + Rank from Bottom - 1 |
| Finding rank from other end | If you know total items and rank from one end. | Total items - Rank from Known End + 1 |
| Finding items between two people | If you know their ranks. | Difference between ranks - 1 (if they are in the same line) |
Additional Information: Applying Ranking Logic
Ranking problems are common in competitive exams and aptitude tests. They test your ability to understand positional relationships within a set. The key is to correctly identify the group being ranked for each piece of information given. In this specific problem, Raman's ranks (13th and 17th) apply only to the group of students who passed, not the entire class of 40 students. Failing to recognize this distinction would lead to an incorrect calculation of the number of passed students.
Understanding that the '+ 1' or '- 1' in ranking formulas accounts for the individual being counted from both ends is crucial. If you count from top and bottom, the individual is counted twice, so you subtract 1 to get the correct total count of individuals.
Practice with different types of ranking problems, such as those involving multiple people, finding the number of people between two individuals, or scenarios where some people leave or join the line, will help solidify these concepts.
Similar Questions
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Important Questions from Ordering and Ranking
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