Question

$₹7500$ were divided equally among a certain numeber of children. Had there been $20$ less children, each would have received $₹100$more. Find the original amount of children.

Answer 1

Step 1- Given data in the question

The amount of money to be distributed is$₹7500$ and there are $20$ less children than each would have received$₹100$ more.

Let the number of children be $x$, and the amount of money received by each child is $y$.

$y=\frac{7500}{x}$

Step 2- Find an equation when the number of children is reduced by $20$ :

According to the equation

The amount received by each child will be $100$ more when the number of children is reduced by $20$.

$\begin{array}{rcl}(y+100)& =& \frac{7500}{x-20}\\ \frac{7500}{x}+100& =& \frac{7500}{x-20}\\ \frac{7500}{x-20}-\frac{7500}{x}& =& 100\\ \frac{75x-75(x-20)}{x(x-20)}& =& 1\\ 75x-75x+1500& =& x(x-20)\\ x^2-20x& =& 1500\\ x^2-20x-1500& =& 0\end{array}$

Step 3-Solve the equation to find the original number of children :

$$\begin{gathered} \Rightarrow x^2-50x+30x-1500=0 \\ \Rightarrow x(x-50)+30(x-50)=0 \\ \Rightarrow (x+30)(x-50)=0 \\ \Rightarrow x=-30orx=50 \end{gathered}$$

Neglect $x=-30$as the number of children cannot be negative

Hence the original number of children is $50$.