In a row, Mansi is 29th from the left and 33rd from the right. How many students are there in the row?
61
Finding Total Students in a Row Based on Ranking
This question asks us to find the total number of students in a row given the position of one student, Mansi, from both the left and the right ends of the row. This is a common type of problem in ranking and order.
Understanding Ranking in a Row
When a person's rank is given from both sides of a linear arrangement (like a row), we can find the total number of items (students, in this case) using a simple formula. Let's consider Mansi's position.
- Mansi is 29th from the left end. This means there are 28 students to her left.
- Mansi is 33rd from the right end. This means there are 32 students to her right.
If we just add her rank from the left (29) and her rank from the right (33), we are counting Mansi herself twice - once as part of the count from the left and once as part of the count from the right. To get the total number of students in the row, we need to count Mansi only once.
Formula for Total Number in a Row
The formula to calculate the total number of individuals in a row when the rank of one person from both ends is known is:
\( \text{Total Number} = (\text{Rank from Left}) + (\text{Rank from Right}) - 1 \)
We subtract 1 because the person whose rank is given from both ends is included in both counts. Subtracting 1 corrects this double counting.
Calculating the Total Number of Students
Let's apply the formula using Mansi's ranks:
- Mansi's rank from the left is 29.
- Mansi's rank from the right is 33.
Using the formula:
\( \text{Total Students} = 29 + 33 - 1 \)
First, add the ranks:
\( 29 + 33 = 62 \)
Now, subtract 1 to correct for double counting Mansi:
\( 62 - 1 = 61 \)
So, there are 61 students in the row.
Verification
Let's quickly verify this:
- Students to Mansi's left: 28 (since she is 29th)
- Students to Mansi's right: 32 (since she is 33rd)
- Mansi herself: 1
Total students = (Students to left) + (Mansi) + (Students to right)
Total students = \( 28 + 1 + 32 = 61 \)
The calculation using the formula matches this direct count, confirming the total number of students is 61.
Revision Table: Ranking Formula
| Concept | Formula | Explanation |
|---|---|---|
| Total Number in a Row | \( \text{Total} = L + R - 1 \) (where L = Rank from Left, R = Rank from Right) | Used when one person's rank is known from both ends. Subtract 1 because the person is counted in both L and R. |
Additional Information: Why Subtract 1?
Subtracting 1 in the formula \( \text{Total} = L + R - 1 \) is crucial because of how ranks are defined. When we say Mansi is 29th from the left, it means there are 28 people before her, plus Mansi herself makes it 29. Similarly, being 33rd from the right means there are 32 people after her, plus Mansi herself makes it 33.
If you simply add the ranks \( L + R \), you are counting everyone to the left of Mansi, Mansi herself, everyone to the right of Mansi, AND Mansi herself again. She is included in the 'L' count and also in the 'R' count. By subtracting 1, you remove the one extra count of Mansi, ensuring each person in the row is counted exactly once.
Think of a small row of 3 people: A, B, C. If B is 2nd from the left and 2nd from the right:
- L = 2, R = 2
- \( \text{Total} = L + R - 1 = 2 + 2 - 1 = 3 \)
This matches the actual total of 3 people (A, B, C). If we didn't subtract 1 (\(2 + 2 = 4\)), it would be incorrect.
Important Questions from Ordering and Ranking
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?
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?
