Rakesh is 17th from the right and Ankit is 15th from the left in a line of students. If they interchange their places, the position of Ankit becomes 19th from the left. How many students are there in the line?
35
Understanding the Student Seating Arrangement Problem
This problem involves two students, Rakesh and Ankit, sitting in a line. We are given their initial positions from opposite ends of the line. When they swap places, we are given Ankit's new position from one end. Our goal is to find the total number of students in the line using this information.
Let's break down the given information:
- Rakesh's initial position: 17th from the right end.
- Ankit's initial position: 15th from the left end.
- After interchanging places: Ankit is now at the spot where Rakesh was initially sitting.
- Ankit's new position (which is Rakesh's original spot): 19th from the left end.
Step-by-Step Solution for Finding Total Students
When Rakesh and Ankit interchange their places, Ankit moves to Rakesh's original position, and Rakesh moves to Ankit's original position. The key piece of information is Ankit's new position, which tells us about the position of the spot Rakesh originally occupied.
- Identify the position of a specific spot from both ends: Ankit's new position is 19th from the left. Since Ankit moved into Rakesh's original spot, this means Rakesh's original spot is 19th from the left.
- We are also given Rakesh's original position from the other end: Rakesh was 17th from the right. So, Rakesh's original spot is 17th from the right.
- Now we have the position of one specific spot (Rakesh's original spot) from both the left and the right ends of the line: 19th from the left and 17th from the right.
- To find the total number of students in a line, if you know the position of one person or spot from both ends, you use the formula:
Total Students = (Position from Left) + (Position from Right) - 1
The reason we subtract 1 is that the position from the left and the position from the right both count the spot itself, so we would be counting it twice if we just added the two positions.
Calculation of Total Students
Using the information that Rakesh's original spot is 19th from the left and 17th from the right, we can apply the formula:
Total Students = 19 (\text{from left}) + 17 (\text{from right}) - 1
Total Students = 36 - 1
Total Students = 35
Therefore, there are 35 students in the line.
Total Students in the Line
Based on the calculation, the total number of students in the line is 35.
| Student | Initial Position (Left) | Initial Position (Right) | Position After Swap (Left) | Position After Swap (Right) |
|---|---|---|---|---|
| Rakesh | 15th (Ankit's original) | 17th | N/A (New spot) | N/A (New spot) |
| Ankit | 15th | 17th (Rakesh's original) | 19th | N/A (Need to calculate) |
The crucial point is that Ankit's new position (19th from left) tells us about Rakesh's original position. This spot is 19th from the left and 17th from the right, allowing us to find the total number of students.
Revision Table: Key Concepts
| Concept | Explanation | Formula (where applicable) |
|---|---|---|
| Position from an end | The count of items from a specific side up to the item in question, including the item itself. | - |
| Interchanging Positions | Two individuals swap spots. The new position of one individual gives information about the original position of the other. | - |
| Total items in a line | The total count of items. Can be calculated if one item's position from both ends is known. | Total = Position from Left + Position from Right - 1 |
Additional Information: Seating Arrangement Problems
Seating arrangement problems often involve relative positions in a line or circle. Line arrangements are common, where positions are given from the left, right, top, or bottom. Key variations include:
- Finding the total number of people given two positions from opposite ends.
- Finding the position of a person from the other end given their position from one end and the total number of people.
- Finding the number of people between two people given their positions.
- Problems involving interchanging positions, like the one solved here, where the new position of one person helps identify the properties of the other person's original spot.
Understanding the formula "Total = Position from Left + Position from Right - 1" is fundamental for solving many linear arrangement problems. The key is often to identify a single point in the line whose position is known from both directions, either directly or indirectly through an interchange.
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