Question

₹$7500$ 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.

Answer 1

Step-1: Make equation from given condition:

Given total money = ₹$7500$

Let original number of children = $x$

Money received by each children = ₹$\frac{7500}{x}$

Since total number of children is $20$ less,

Now, total number of children = ($x-20$).

Money received by each children = ₹$\frac{7500}{x-20}$

Step-2: Solve for $\mathit{x}$:

According to question,

$\begin{array}{rcl}\frac{7500}{x}& =& \frac{7500}{x-20}-100\\ \Rightarrow \frac{7500}{x}-\frac{7500}{x-20}& =& -100\\ & \Rightarrow & \frac{7500(x-20)-7500x}{x(x-20)}=-100\\ \Rightarrow \frac{-7500\times 20}{x(x-20)}& =& -100\\ \Rightarrow x(x-20)& =& \frac{-7500x20}{-100}\\ \Rightarrow x(x-20)& =& 1500\\ \Rightarrow x^2-20x-1500& =& 0\\ \Rightarrow x^2-50x+30x-1500& =& 0\\ \Rightarrow (x-50)(x+30)& =& 0\end{array}$

$x-50=0$ or $x+30=0$

$x=50,-30$

But $x$ is number of children , It cannot be negative, so $x=50$

Hence, the number of children is equal to $50$.