Question

# The organizers of an essay competition decide that a winner in the competition gets a prize of Rs. 100 and a participant who does not win gets a prize of Rs. 25. The total prize money distributed is Rs. 3,000. Find the number of winners, if the total number of participants is 63.

Hint: In this question first assume any variable for the number of winners and assume another variable for the rest of the candidates, then the sum of these variables are the total number of participants, use this concept to reach the solution of the question.

Let the number of winners be x.
And the rest of the candidates be y.
Now it is given that the total participants is 63.
$\Rightarrow x + y = 63.................\left( 1 \right)$
Now according to the question winners get a prize of Rs. 100.
And the rest of the candidates get a prize of Rs. 25.
Total prize money is Rs. 3000
Now, convert this information into linear equation we have,
$\Rightarrow 100x + 25y = 3000$
Now, divide by 25 in above equation we have,
$\Rightarrow 4x + y = 120...............\left( 2 \right)$
From equation (1)
$y = 63 - x$
Substitute this value in equation (2) we have,
$\Rightarrow 4x + 63 - x = 120 \\ \Rightarrow 3x = 120 - 63 = 57 \\ \Rightarrow x = \dfrac{{57}}{3} = 19 \\$
So, the total number of winners in an essay competition is 19.

Note: Whenever we face such types of questions first assume the variables for winners and rest of the participants as above then convert the given information into linear equations as above then solve these two equation using substitution method as above or we can use elimination method by directly subtracting equation (1) from equation (2), we will get the required number of winners in an essay competition.