Question

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

Answer 1

Hint: This question can be solved by forming a linear equation and then solving it to find the total number of winners. First we will list out all the given data values. Since we have to find the number of winners, we will denote it by a variable $ x $ . Using the given data and variable $ x $ we will form a linear equation in $ x $ .

Complete step by step solution:
According to the data given,
The total number of participants $ = 63 $
Total prize money distributed $ = 3000Rs. $
The cash prize given to a winner $ = 100Rs. $
The cash prize given to the participants who did not win $ = 25Rs. $
We have to find the total number of winners of the competition. Let $ x $ be the total number of winners, then the number of participants who do not win is $ 63 - x $ .
The total amount of cash prize given to winners $ = $ number of winners $ \times $ the money given to each winner $ = $ $ 100x $ .
The money given to the rest of $ 63 - x $ participants who did not win is $ (63 - x) \times 25 $ .
The total money distributed =Total amount given to winners + Total amount given to participants who did not win, i.e.
 $ $ 3000 = 100x + (63 - x)25 $ $ Solving for $ x $ by rearrangement and simplification we have,
 $
  100x + (63 \times 25) - 25x = 3000 \\
   \Rightarrow 75x + 1575 = 3000 \\
   \Rightarrow 75x = 3000 - 1575 = 1425 \\
   \Rightarrow x = \dfrac{{1425}}{{75}} = 19 \;
 $
Therefore the number of winners is 19
So, the correct answer is “ $ 19 $ ”.

Note: Equations of first order are called linear equations. They represent straight lines in the coordinate system. The standard form of linear equation in one variable is $ ax + b = 0 $ which can be solved by using basic arithmetic operations. The important step in solving this type of question is the formation of the equation , once a proper equation is formed it can easily be solved.