# Ch. 4 :- Comparison of Ranks : Introduction & Formulas

### 1. Fixed Ranking

For understanding the fixed ranking, study the example given below carefully.

**Ex. 1. In an examination Amit scores more than Arnan and less than Rohit. Sudha scores more than Rohit and less than Geeta. Who among them scored the highest marks ?**

sol.,: The information given in the question may be interpreted systematically as follows:

Amit > Arnan ; Arnit < Rohit ; Sudha > Rohit ; Sudha < Geeta

On ranking the marks in ascending order, we further get Arnan, Arnit, Rohit, Sudha, Geeta.

This final ranking fulfils all the conditional relationship given in the question. Hence, we find that Geeta scored the maximum marks in the examination.

In the fixed ranking of objects, we get the position of ranks as fixed. As in the above example, the ranking of marks of each of the students IS fixed. In other words, by fixed ranking we mean that answer about the position of one’s rank is definite and certain. Therefore, on the basis of ranking of marks of students in the above question, it can be interpreted definitely that Geeta gets the highest marks and Arnan gets the lowest marks.

### 2. Variable Ranking

For easy understanding of this type, go through the example given below carefully and try to differentiate this type with the previous one

**Ex.2. In an examination, Arnit scores more than Arnan and less than Rohit. Sudha scores more than Rohit and Geeta scores more than Rohit. Who among them gets the maximum marks ?**

sol.: If we arrange the marks of all the students in ascending order, we get two orders of ranking:

(a) Arnan, Arnit, Rohit, Sudha, Geeta

(b) Arnan, Arnit, Rohit, Geeta, Sudha.

Order of ranks of marks of the students in this case is different in one way from the order of ranks in the previous case. And this difference lies in the varying ranks of Geeta and sudha .. And hence it cannot be said definitely who scores the highest marks. However, it may be either Geeta or Sudha.

Therefore, in varying ranking, the position of anyone of the ranks keeps varying. The variable ranking is the consequence of inadequate information given in the question. Students are therefore, required to be cautious while answering the questions of this type as these questions are basically designed to confuse the students.

### Methodology:

Ranking of objects sometimes seems to be tedious for the students when information given in the question is complicated and distorted. In such a situation, establishing the relationship among ranks with the use of symbols > (greater than) and < (less than) proves to be insufficient. We therefore, introduce an easy approach to deal with the questions of comparison of ranks in the following steps. Students are advised to go through the following steps carefully and practice all questions given in the exercise with this method only.

**1.** First we solve the question given in the example 1 with the help of this method. Read the question step by step carefully and place the ranks in a column so as to show the highest rank at the top and the lowest at the bottom. The information given in the first line of the question. “Arnit scores more than Aman and less than Rohit” can be placed in the column, as discussed above, in the following way:

Rohit Arnit Arnan

Here, the relation among one another is automatically established and hence ranks of marks of each student get fixed. Second information “Sudha scores more than Rohit and less than Geeta” can easily be inserted in the present ranking and thus we get final ranking as under:

Geeta Highest marks

Sudha Rohit Amit

Arnan Lowest marks

Here we see that the position of ranks of each student’s marks is fixed.

**2.** If we solve example 2 with the method as discussed above, we get the order of rank of ma rks of the students as shown in the column. Since we are given in the question that Geeta scores more than Rohit but her relationship with Sudha has not been given, therefore, Geeta’s position of marks may occupy three positions which are equal to, greater than and less than Sudha’s position. Therefore, it cannot be said definitely that who scores the maximum marks.