### The students of a school are made to stand in rows. If the number of students in each row is increased by $6$, there would be $4$ row less. If the number of students in each rows is reduced by $6$, there would be $5$ rows more. Find the number of students in the school.

$2160$

Step by Step Explanation:
1. Let us assume that there are $x$ students in each row and $y$ rows in total.

Thus, the total number of students in the school is $xy$.
2. If the number of students in each row is increased by $6$, then the number of rows becomes $(y - 4)$.

As the total number of students remain same, we have \begin{aligned} & xy = (x + 6)(y - 4) \\ {\implies} &xy = xy - 4 x + 6 y - 24 &&[\text{ Cancelling } xy {\space} ]\\ {\implies} &4 x - 6 y = -24 &&\ldots{\text{(i)}} \\ \end{aligned}
3. If the number of students in each row is reduced by $6$, then the number of rows becomes $(y + 5)$. \begin{aligned} &\therefore &&xy = (x - 6)(y + 5) \\ &{\implies} &&xy = xy + 5 x - 6 y - 30 &&[\text{ Cancelling } xy {\space} ] \\ &{\implies} &&-5 x + 6 y = -30 &&\ldots{\text{(ii)}} \\ \end{aligned}
4. Adding $\text{eq(i)}$ and $\text{eq(ii)}$, we get \begin{aligned} {\implies}& (4 - 5) x = (-24 -30) \\ {\implies}& -x = -54 \\ {\implies}& x = 54 \end{aligned}
5. Substituting the value of $x$ in $\text{eq(i)}$, we get \begin{aligned} &4 \times 54 - 6 y = -24 \\ {\implies} & - 6 y = -24 - 216 \\ {\implies} & - 6 y = -240 \\ {\implies} & y = 40 \end{aligned}
6. Therefore, the total number of students in the school = $xy$ = 54 ${\times}$ 40 = $2160$. 