Question

Students of a class are made to stand in rows. If one student is extra in a row, there would be $2$ rows less. If one student is less in a row there would be $3$ rows more. Find the number of students in the class.

Answer 1

Let the number of students be $x$ and the number of rows be $y$. Then,
Number of students in each row $=\frac{x}{y}$

When one student is extra in each row, there are $2$ rows less i.e., when each row has $(\frac{x}{y}+1)$ students, the number of rows will be $(y-2)$.

$\therefore$ Total number of students $=$ No. of rows $\times$No. of students in each row

$x=(\frac{x}{y}+1)(y-2)$

$x=x-\frac{2x}{y}+y-2$

$-\frac{2x}{y}+y-2=0\dots \left(1\right)$

If one student is less in each row, then there are $3$ rows more i.e., when each row has $(\frac{x}{y}-1)$ students, the number of rows will be $(y+3)$.

$\therefore$ Total number of students $=$ No. of rows $\times$ No. of students in each row
$\Rightarrow x=(\frac{x}{y}-1)(y+3)$

$x=x+\frac{3x}{y}-y-3$

$\frac{3x}{y}-y-3=0\dots \left(2\right)$

Putting $\frac{x}{y}=u$ in $\left(1\right)$ and $\left(2\right)$, we get:
$-2u+y-2=0\dots \left(3\right)$
$3u-y-3=0\dots \left(4\right)$

Adding $\left(3\right)$ and $\left(4\right)$, we get:
$u-5=0$

$\Rightarrow u=5$

Putting $u=5$ in $\left(3\right)$, we get $y=12$

Now, $u=5$

$\frac{x}{y}=5$

$\frac{x}{12}=5$

$x=60$

Hence, the number of students in the class is $60$.