Question

The organisers 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

Hint: The following steps are used to manipulate an equation and isolate the unknown variable, so that the last line reads x=_________, if x is the unknown. There is no set order, as the steps used depend on what is given:
* We may add, subtract, multiply, or divide an equation by a number or an expression as long as we do the same thing to both sides of the equal sign. Note that we cannot divide by zero.
* Apply the distributive property as needed
* Isolate the variable on one side of the equation.
* When the variable is multiplied by a coefficient in the final stage, multiply both sides of the equation by the reciprocal of the coefficient.

Complete step-by-step answer:
Given, Total number participants = 63
Total prize money distributed =Rs 3000
winner gets a prize of Rs 100
Who does not win gets a prize of Rs 25
Number of winners = ?
Let the number of winners be \[x\]
Since, Number of winners + Number of losers = Total number of participants
Or, x + Number of participants who did not win = 63
Therefore, the number of participants who did not win will be \[63{\text{ }} - {\text{ }}x\]
Now, Total Prize money distributed to winners= Number of winners X prize money distributed to each winner = Amount given to the winners = ₹ \[\left( {100{\text{ }} \times {\text{ }}x} \right)\]= ₹ \[100x\]
Total prize money distributed to losers = Number of participants who did not win X prize money distributed to each participants who did not win = Amount given to the participants who did not win = ₹ \[25\left( {63{\text{ }} - {\text{ }}x} \right)\] = ₹ \[\left( {1575{\text{ }} - {\text{ }}25x} \right)\]
According to the condition, given
Now, Total Prize money of winners + Total Prize money of losers = Total prize money
By substituting the total prize money distributed to winners and total prize money distributed to participants who did not win, we get
 \[100x{\text{ }} + {\text{ }}\left( {1575{\text{ }} - {\text{ }}25x} \right){\text{ }} = {\text{ }}3000\]
\[ \Rightarrow 75x{\text{ }} + {\text{ }}1575{\text{ }} = {\text{ }}3000\]
On transposing 1575 to RHS, we get
\[\begin{array}{*{20}{l}}
  { \Rightarrow 75x{\text{ }} = {\text{ }}3000{\text{ }} - {\text{ }}1575} \\
  { \Rightarrow 75x{\text{ }} = {\text{ }}1425}
\end{array}\]
On dividing both sides by 75, we get
$ \Rightarrow x = \dfrac{{1425}}{{75}} = 19$

Therefore, the number of winners = 19

Note: Solving linear equations is all about isolating the variable. Depending on the equation, this may take as little as one step or many more steps. Always check if you need to simplify one or both sides of the equation first, and always check your answer. We can substitute the answer which we got back in the equation, if the substituted value satisfies the equation then the answer is correct.