Question

In a school a P.T. the teacher wants to arrange 2000 students in the form of rows and columns for P.T. display. If the number of rows is equal to the number of columns and 64 students could not be accommodated in this arrangement. Find the number of rows.

Answer 1

Hint: Here, we have to find the number of rows where the students are arranged for P.T. Display. We will first find the total number of students in the arrangement. Then we will multiply the number of rows to that of the column and equate it to the number of students in the arrangement. We will simplify this to get the answer.

Complete step-by-step answer:
We have a total of 2000 students. These students should be arranged in the form of rows and columns for P.T. Display.
We are given that the number of rows equals the number of columns.
64 students could not be accommodated in this arrangement.
So, Number of students in the arrangement \[ = 2000 - 64\]
Number of students in the arrangement \[ = 1936\]
Let \[x\] denote the number of rows.
Since, Number of rows is equal to Number of Columns, we have
 Number of rows \[ = \] Number of Columns
So, Number of Rows \[ \times \] Number of Columns \[ = \]Number of Students in the arrangement
Substituting the values in above equation, we get
\[ \Rightarrow x \cdot x = 1936\]
\[ \Rightarrow {x^2} = 1936\]
Taking Square root on both the sides, we have
\[ \Rightarrow x = \sqrt {1936} \]
By factorization method, we have
\[ \Rightarrow x = \sqrt {2 \times 2 \times 2 \times 2 \times 11 \times 11} \]
\[ \Rightarrow x = 2 \times 2 \times 11\]
By multiplying the terms, we get
\[ \Rightarrow x = 44\]
Therefore, the number of rows where students are arranged for a P.T. Display is 44.

Note: Here we have found out the square root of 1936 using factorization method. We have two methods to find the square root of a number. If the number is small, it is quite easy to find the square root of a number. Any number can be expressed as a product of prime numbers. This method of representation of a number in terms of the product of prime numbers is termed as the prime factorization method.