Question

There are 500 children in a school. For a P.T. drill they have to stand in such a manner that the number of rows is equal to the number of columns. How many children would be left out in this arrangement?

Answer 1

Hint: If there are an equal number of rows and columns then the arrangement will look like a square in shape. Use the property of squares. And they use the information that the total number of students in the school are 500. So the number of students present in the P.T. cannot be more than 500.

Complete step by step solution:
It is given in the question that there are a total 500 children in a school.
They have to stand for a P.T. in such a manner that the number of rows is equal to the number of columns.
Let, the number of rows be $ x $
Then the number of column will also be $ x $
We can observe that the number of students in one row is equal to the number of columns.
Therefore, we can say that
Total number of students is equal to the product of the number of rows and total number of students in one row.
i.e. the total number of students is equal to the product of the number of rows and the total number of columns.
 $ = x \times x $
 $ = {x^2} $
Since the total number of students in the school are 500. Total number of students present in the P.T. cannot be more than 500.
 $ \therefore {x^2} \leqslant 500 $
Now, $ {22^2} = 484 < 500 < 529 = {23^2} $
therefore, we can write
 $ {x^2} = {22^2} $ (since, number of students must be a natural number. It cannot be in decimals.)
 $ \Rightarrow x = 22 $
Therefore, there will be 22 rows and 22 columns.
Hence, the total number of students present in the P.T. will be $ {x^2} = 484 $
Hence, the number of students that would be left out in this arrangement will be
 $ 500 - 484 = 16 $
Therefore, 16 students will be left out in this arrangement.
So, the correct answer is “16”.

Note: In this question. The key point was to understand that the arrangement will look like a square and hence the total number of students present in the P.T. must be equal to a perfect square closest to 500 but less than 500. And the total number of students in the P.T. will be equal to the product of the number of rows to the number of columns.