₹ 7500 were divided equally among a certain number of children. Had there been 20 less children, each would have received ₹ 100 more. Find the original number of children.
Let the original number of children $=x$
Total amount $=$ ₹ 7500
$\therefore$ Amount received by each child $=$ ₹ $\dfrac{7500}{x}$
New number of children $=x-20$
$\therefore$ New amount received by each child $=$ ₹ $\dfrac{7500}{x-20}$
$\therefore$ As per question,
$\dfrac{7500}{x-20} = \dfrac{7500}{x}+100$
i.e. $\dfrac{75}{x-20} = \dfrac{75}{x}+1$
i.e. $75x=75\left(x-20\right)+x\left(x-20\right)$
i.e. $75x=75x-1500+x^2-20x$
i.e. $x^2-20x-1500=0$
i.e. $x^2 -50x+30x-1500=0$
i.e. $x\left(x-50\right)+30\left(x-50\right)=0$
i.e. $\left(x-50\right) \left(x+30\right)=0$
i.e. $x=50$ or $x=-30$
Since the number of children cannot be negative, therefore, $x=-30$ is not an acceptable solution.
$\therefore$ Original number of children $=50$