Question

The organizers of an essay competition decide that a winner in the competition gets a prize of '$₹100$' and a participant who does not win gets a prize of '$₹25$‘. The total prize money distributed is ’$₹3000$‘. Find the number of winners, if the total number of participants is ’$63$'.

Answer 1

Let the number of winners be ‘$\mathrm{W}$’

$\therefore$ The number of participants who didn’t win$=\left(63-\mathrm{W}\right)$

$\therefore$Total money was given to the winners$=\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{winners}\times \mathrm{Prize}\mathrm{given}\mathrm{to}\mathrm{each}\mathrm{winner}$

$=\mathrm{W}\times 100$

$=₹100\mathrm{W}$

$\therefore$Total money was given to participants who didn’t win$=\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{participants}\mathrm{who}\mathrm{didn}\text{'}\mathrm{t}\mathrm{win}\times \mathrm{Prize}\mathrm{given}\mathrm{to}\mathrm{each}\mathrm{participant}\mathrm{who}\mathrm{didn}\text{'}\mathrm{t}\mathrm{win}$

$=₹\left(63-\mathrm{W}\right)\times 25$

According to the given details, the equation becomes:

$\begin{array}{rcl}\therefore 100\mathrm{W}+\left(63-\mathrm{W}\right)\times 25& =& 3000\\ 100\mathrm{W}+1575-25\mathrm{W}& =& 3000\\ 75\mathrm{W}& =& 3000-1575\\ 75\mathrm{W}& =& 1425\\ \mathrm{W}& =& \frac{1425}{75}\\ \mathrm{W}& =& 19\end{array}$

Therefore, the total number of winners is'$19$'