Understanding Rank and Position in a Class

This question asks us to find the rank of a student, X, from the bottom of a class, given their rank from the top and the total number of students. This is a common type of ranking or position problem often found in aptitude sections.

Calculating Rank from the Bottom

To find a person's rank from the bottom when you know their rank from the top and the total number of people, you can use a simple formula. The logic is that the number of students below X plus X himself equals the total number of students minus the number of students above X.

Let's break down the formula:

• Total number of students = Number of students above X + X + Number of students below X

From this, we can derive the rank from the bottom. The rank from the bottom is the number of students below X plus X's position (which is 1).

• Rank from bottom = Number of students below X + 1

Also, the number of students above X is simply X's rank from the top minus 1.

• Number of students above X = Rank from top - 1

Substituting this back into the first equation:

• Total number of students = (Rank from top - 1) + 1 + Number of students below X
• Total number of students = Rank from top + Number of students below X

Rearranging to find the number of students below X:

• Number of students below X = Total number of students - Rank from top

Now, using the formula for rank from bottom:

• Rank from bottom = (Total number of students - Rank from top) + 1

So, the standard formula to find the rank from the bottom is:

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

Applying the Formula to Find X's Rank

We are given the following information:

• X's rank from the top = 12
• Total number of students in the class = 16

Using the formula:

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

\(\text{Rank from bottom} = 16 - 12 + 1\)

\(\text{Rank from bottom} = 4 + 1\)

\(\text{Rank from bottom} = 5\)

So, X's rank from the bottom of the class is 5.

Summary of Calculation

Alternatively, using the direct formula:

\(\text{Rank from bottom} = 16 - 12 + 1 = 5\)

Both methods confirm that X's rank from the bottom is 5.

Revision Table: Key Ranking Concepts

Additional Information on Ranking Problems

Ranking and position-based problems are common in various competitive exams and aptitude tests. They test your understanding of relative positions within a sequence or group. Here are a few points to remember:

• Always clearly identify what position is being asked (from top, from bottom, from left, from right).
• Note down the total number of items (students, people, objects) in the sequence.
• Be careful with the "+1" or "-1" in formulas. Rank includes the person themselves, while counting the number of people before or after someone does not include them.
• Problems can sometimes involve more than one person's position, or positions from both ends (left/right, top/bottom) being given to find the total number of people.
• Visualizing the sequence with placeholders can sometimes help in solving complex problems.

Mastering these basic formulas and concepts will help you solve a variety of ranking and order-based questions efficiently.