Question

A teacher attempting to arrange the students for a mass drill in the form of a solid square found that 24 students were left. When he increased the size of the square by one student, he found that he was short of 25 students. Find the number of students. Find the number of students.

Answer 1

Hint: Assume the total number of students as ‘N’ and the number of students standing initially as ‘x’. Take the area of this square and add 24 to it to form equation (1). Now, increase the number of students standing in each row and column by 1 to make it (x + 1). Take the area of the new square and subtract 25 from it to form equation (2). Equate the two equations to find the value of x and N.

Complete step-by-step solution
Let us assume the total number of students required for mass drill as ‘N’.
Now, in the first case, it is given that students are arranged to form a solid square. So, let us assume the number of students standing in each row and column is ‘x’. The arrangement can be shown as: -

So, the total number of students in the above arrangement will be the area of the square with side ‘x’.
\[\Rightarrow \] Area = \[{{x}^{2}}\]
It is given that in this arrangement 24 students were left.
Therefore, the total number of students will be,
\[\Rightarrow N={{x}^{2}}+24\] ------- (1)
Now, in the second case, it is given that the teacher increased the size of the square by one student. So, the number of students in each row and column will become (x + 1).
\[\Rightarrow \] Area = \[{{\left( x+1 \right)}^{2}}\]
It is given that this time he was short of 25 students.
Therefore, the total number of students will be,
\[\Rightarrow N={{\left( x+1 \right)}^{2}}-25\] ------ (2)
Equating equations (1) and (2), we get,
\[\Rightarrow {{x}^{2}}+24={{\left( x+1 \right)}^{2}}-25\]
Applying the identity, \[{{\left( a+b \right)}^{2}}={{a}^{2}}+{{b}^{2}}-2ab\], we get,
\[\Rightarrow {{x}^{2}}+24={{x}^{2}}+1+2x-25\]
Cancelling the like terms, we get,
\[\begin{align}
  & \Rightarrow 2x=48 \\
 & \Rightarrow x=24 \\
\end{align}\]
Substituting, x = 24 in equation (1), we get,
\[\begin{align}
  & \Rightarrow N={{\left( 24 \right)}^{2}}+24 \\
 & \Rightarrow N=576+24 \\
 & \Rightarrow N=600 \\
\end{align}\]
Hence, the total number of students is 600.

Note: One may note that we have added 24 to the area of the square in equation (1) because to form the square they needed x students in each row and column and 24 students were in extra. In equation (2) he needed 25 more students to complete the square with (x + 1) students in each row and column and that is why 25 is subtracted from the area. So, we have to read the question carefully so that we do not get confused.