Question

The students of a class are made to stand in rows. If $4$ students are extra in each row, there would be $2$ rows less. If four students are less in each row, there would be $4$ more rows. Find the number of students in the class.

Answer 1

Step 1: Find the linear equation to obtain the number of students:

Let the number of students in each row be $x$ and the number of rows be $y$.

$\Rightarrow$The number of students will be $xy$.

It is given that if $4$ students are extra in each row, then there would be $2$ rows less.

$\begin{array}{rcl}& \Rightarrow & \left(x+4\right)\left(y-2\right)=xy\\ & \Rightarrow & xy+4y-2x-8=xy\\ & \Rightarrow & -x+2y-4=0...\left(1\right)\end{array}$

Also, given that if four students are less in each row, then there would be $4$ more rows.

$\begin{array}{rcl}& \Rightarrow & \left(x-4\right)\left(y+4\right)=xy\\ & \Rightarrow & xy-4y+4x-16=xy\\ & \Rightarrow & x-y-4=0...\left(2\right)\end{array}$

Hence, equations (1) and (2) formed the required system of linear equations.

Step 2: Solve the obtained system of linear equation by elimination method.

$\begin{array}{rcl}-x+2y-4& =& 0\\ x-y-4& =& 0\end{array}$

Now, by adding both equations, we will get:

$\begin{array}{rcl}\left(-x+2y\right)+\left(x-y\right)& =& 4+4\\ & \Rightarrow & y=8\end{array}$

Now, substitute the value of $y$ in equation (2) to get:

$\begin{array}{rcl}x-y-4& =& 0\\ x-8-4& =& 0\\ & \Rightarrow & x=12\end{array}$

$\therefore$ The total number of students, $xy=12\times 8=96$.

Hence, the total number of students in the class is $96$.