In a row of people all facing north, Prince is 5th from the right end. Amit is 15th from the right end. Amit is exactly between Prince and Aditya. If Aditya is sixth from the left end of the line, how many people are there in the row?
30
Understanding Row Arrangement Problems
This problem involves determining the total number of people in a straight line based on their positions from the left and right ends. We are given the positions of three individuals: Prince, Amit, and Aditya, and a crucial piece of information about Amit being exactly between Prince and Aditya.
Analyzing the Given Information
- All people are facing North. This simplifies the directions (left/right).
- Prince is 5th from the right end.
- Amit is 15th from the right end.
- Amit is exactly between Prince and Aditya.
- Aditya is 6th from the left end.
Step-by-Step Solution to Find Total People
Step 1: Find the number of people between Prince and Amit.
Both Prince and Amit's positions are given from the right end. Prince is 5th from the right, and Amit is 15th from the right. Since Amit is further from the right end than Prince, Amit is to the left of Prince in the row.
The number of people between two individuals in a row, when their positions are known from the same end, is given by:
\(\text{Number of people between} = |\text{Position of Person 1} - \text{Position of Person 2}| - 1\)
Using this formula:
Number of people between Amit and Prince = \(|15 - 5| - 1 = 10 - 1 = 9\)
There are 9 people sitting between Amit and Prince.
Step 2: Determine the position of Aditya relative to Amit.
The problem states that Amit is exactly between Prince and Aditya. This means the number of people between Amit and Prince is the same as the number of people between Amit and Aditya.
Number of people between Amit and Aditya = Number of people between Amit and Prince = 9.
Since Amit is to the left of Prince (from the right perspective), and Amit is exactly between them, Aditya must be to the left of Amit.
Step 3: Calculate Aditya's position from the right end.
Amit is 15th from the right end. There are 9 people between Amit and Aditya, and Aditya is to the left of Amit. To find Aditya's position from the right end, we move from Amit towards the left, crossing the 9 people and then Aditya himself.
Aditya's position from the right = (Amit's position from the right) + (Number of people between Amit and Aditya) + 1 (for Aditya himself)
Aditya's position from the right = \(15 + 9 + 1 = 25\)
So, Aditya is 25th from the right end of the row.
Step 4: Use Aditya's positions from both ends to find the total number of people.
We now know:
- Aditya is 6th from the left end.
- Aditya is 25th from the right end.
When the position of a person is known from both ends of a row, the total number of people in the row can be calculated using the formula:
\(\text{Total people} = (\text{Position from Left End}) + (\text{Position from Right End}) - 1\)
Using this formula with Aditya's positions:
\(\text{Total people} = 6 + 25 - 1 = 31 - 1 = 30\)
Therefore, there are 30 people in the row.
Let's verify this with Aditya's position definition:
- Aditya is 6th from the left, meaning there are 5 people to his left.
- Aditya is 25th from the right, meaning there are 24 people to his right.
Total people = (People to the left of Aditya) + (Aditya) + (People to the right of Aditya)
Total people = \(5 + 1 + 24 = 30\)
The result matches.
Here is a summary of the key positions:
| Person | Position from Right | Position from Left |
|---|---|---|
| Prince | 5th | Not directly given |
| Amit | 15th | Not directly given |
| Aditya | 25th (Calculated) | 6th (Given) |
Conclusion
By using the given positions and the information that Amit is exactly between Prince and Aditya, we determined Aditya's position from the right end. With Aditya's positions from both the left and right ends, we successfully calculated the total number of people in the row.
Revision Table: Key Formulas for Row Arrangements
| Concept | Formula | Notes |
|---|---|---|
| Total people from one person's position from both ends | \(\text{Total} = \text{Left Position} + \text{Right Position} - 1\) | Subtract 1 because the person is counted twice (once from each end). |
| Number of people between two persons (positions from same end) | \(\text{Between} = |\text{Position 1} - \text{Position 2}| - 1\) | Absolute difference minus 1. Valid when positions are from the same end. |
| Finding one person's position from the other end | \(\text{Position from other end} = (\text{Total people} - \text{Position from one end}) + 1\) | Useful if total is known. |
Additional Information: Understanding "Exactly Between"
When it's stated that person 'B' is "exactly between" persons 'A' and 'C', it means that the number of individuals between A and B is equal to the number of individuals between B and C. This implies that B is the midpoint in terms of count of people between A and C. In our problem, Amit is exactly between Prince and Aditya. This allowed us to deduce that the 9 people between Amit and Prince must also be the number of people between Amit and Aditya.
It's important to note the difference between the "number of people between" and the "distance" in terms of positions. If Amit is 15th and Prince is 5th from the right, the difference in positions is \(15 - 5 = 10\). This difference includes Amit, Prince, and the people between them. The number of people *between* them is one less than this difference: \(10 - 1 = 9\).
Always clarify if positions are given from the left end or the right end, as this affects calculations involving the relative placement of individuals in the row arrangement.
Important Questions from Ordering and Ranking
In a row, Mansi is 29th from the left and 33rd from the right. How many students are there in the row?
A class has a total of 60 students. Student 'B' is at 41st rank in merit order from the bottom. What is the rank of student 'B' from the top?
A medical representative plans to visit each of six hospitals — A, B, C, D, E and F — exactly once during the course of one day. He is setting up his schedule for the day according to the following conditions:
i. He must visit A before B and E.
ii. He must visit B before D.
iii. The third hospital he visits must be C.
Which of the following could be the possible order in which the medical representative visits the six hospitals?Sonam is older than Renu. Komal is younger than Renu. Priya is older than Sonam. Who is the eldest of them?
Five men P, Q, R, S and T are reading a newspaper. The one who reads it first, gives it to T. The one who reads it last, had taken it from R. Q is not the first or last one to read it. There were two readers between P and R. Who is the third one to have read the newspaper?
