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.
$
  (a){\text{ 14}} \\
  (b){\text{ 16}} \\
  (c){\text{ 19}} \\
  (d){\text{ 12}} \\
$

Answer 1

Hint – In this question let the total number of winners be a variable and the rest candidates who don't win as other variables. A linear equation can easily be formed using the fact that the total number of participants is 63, and a participant can be categorized only in two categories that is either a winner or a loser. The total price money concept will help in formation of a second linear equation in two variables.

Complete step-by-step answer:

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.
Hence option (C) is correct.

Note – Whenever we have to solve a linear equation in two variables then we can have two different approaches to solve this problem. The first one can be a method of substitution, in which one variable is taken out in terms of another and is then substituted into another equation to obtain the value of variables. The second method is of elimination, in this initially the coefficient of one variable in both the equations is made the same and then both the equations are added or subtracted in such a manner that one variable gets eliminated.