Understanding the Student Ranking Problem

This problem involves calculating the rank of a student from the top, given their position relative to another student and that student's rank from the bottom (end) in a class of a specific size.

We are given:

• Total number of students in the class: 64
• Shikha's rank from the end (bottom): 59th
• Komal's rank is 6 positions lower than Shikha (towards the bottom).

We need to find Komal's rank from the top.

Step-by-Step Solution for Ranking Calculation

1. Find Shikha's Rank from the Top

To find Shikha's rank from the top, we can use the formula relating rank from the top, rank from the bottom, and the total number of students:

\(\text{Total students} = \text{Rank from top} + \text{Rank from bottom} - 1\)

We know the total students (64) and Shikha's rank from the bottom (59). Let Shikha's rank from the top be \(\text{R}_{\text{Shikha\_top}}\).

\(64 = \text{R}_{\text{Shikha\_top}} + 59 - 1\)

\(64 = \text{R}_{\text{Shikha\_top}} + 58\)

Now, we solve for \(\text{R}_{\text{Shikha\_top}}\):

\(\text{R}_{\text{Shikha\_top}} = 64 - 58\)

\(\text{R}_{\text{Shikha\_top}} = 6\)

So, Shikha's rank from the top is 6th.

2. Find Komal's Rank from the Top

The problem states that Komal's rank is 6 positions lower than Shikha. Being 'lower' in rank means having a higher rank number when counting from the top. For example, rank 2 is lower than rank 1, rank 3 is lower than rank 2, and so on.

Since Shikha is at the 6th position from the top, Komal, being 6 positions lower, will have a rank number that is 6 more than Shikha's rank number from the top.

Let Komal's rank from the top be \(\text{R}_{\text{Komal\_top}}\).

\(\text{R}_{\text{Komal\_top}} = \text{Shikha's rank from top} + 6\)

\(\text{R}_{\text{Komal\_top}} = 6 + 6\)

\(\text{R}_{\text{Komal\_top}} = 12\)

Therefore, Komal's rank from the top is 12th.

Let's verify this. If Komal is 12th from the top in a class of 64, her rank from the bottom would be:

\(\text{Total students} = \text{Rank from top} + \text{Rank from bottom} - 1\)

\(64 = 12 + \text{Rank}_{\text{Komal\_bottom}} - 1\)

\(64 = 11 + \text{Rank}_{\text{Komal\_bottom}}\)

\(\text{Rank}_{\text{Komal\_bottom}} = 64 - 11 = 53\)

Komal is 53rd from the bottom. Shikha is 59th from the bottom. Is Komal 6 positions lower (towards the bottom) than Shikha? Yes, 53 is 6 positions higher (closer to the top) than 59 when counting from the bottom. This aligns with Komal being higher ranked (closer to the top) than Shikha. Let's re-read the question carefully: "Komal’s rank is 6 positions lower (i.e. towards bottom) than her friend Shikha". This means Komal is further down the list, i.e., has a higher rank number from the top, compared to Shikha.

Our calculation that Komal's rank from the top is 12th matches the requirement that she is lower than Shikha's 6th position from the top.

Result

Komal's rank from the top in the class is 12th.

Revision Table: Key Concepts in Ranking Problems

Additional Information on Ranking and Ordering

Ranking and ordering problems are common in quantitative aptitude. They test your ability to understand relative positions in a sequence.

• Understanding 'Lower' and 'Higher' Rank: When ranks are given from the top, a numerically smaller rank (e.g., 1st) is higher, and a numerically larger rank (e.g., 10th) is lower. When ranks are given from the bottom, a numerically smaller rank (e.g., 1st from bottom) is actually higher up the list, and a numerically larger rank (e.g., 10th from bottom) is lower down the list. It is crucial to read carefully whether 'lower' or 'higher' refers to the rank number itself or the position in the list (towards the top or bottom). The question specified "lower (i.e. towards bottom)", which clarifies the direction.
• Positions vs. Ranks: The 1st position means rank 1, the 2nd position means rank 2, and so on. The terms are often used interchangeably, but it's good to be clear.
• Overlapping: Sometimes, ranking problems involve two people and ask about the number of people between them. These can become complex if their positions from opposite ends overlap. However, this problem was simpler, focusing on relative position and conversion between ranks.