Question

The organiser of an essay competition decide that a winner in the competition gets a prize of 100 rupees and a participant who does not win gets a prize of 25 rupees. the total price money distributed is 3000. find the number of winners, if the total number of participant is 63.

Answer 1

Let's denote the number of winners as \( w \) and the number of non-winners as \( p \). According to the problem, we have the following equations:

1. The total number of participants:
\[
w + p = 63
\]

2. The total prize money distributed:
\[
100w + 25p = 3000
\]

We can solve these equations step by step.

First, from equation (1), we can express \( p \) in terms of \( w \):
\[
p = 63 - w
\]

Next, we can substitute \( p \) in equation (2):
\[
100w + 25(63 - w) = 3000
\]

Expanding this gives:
\[
100w + 1575 - 25w = 3000
\]

Combining like terms:
\[
75w + 1575 = 3000
\]

Now, subtract 1575 from both sides:
\[
75w = 3000 - 1575
\]
\[
75w = 1425
\]

Next, divide both sides by 75 to find \( w \):
\[
w = \frac{1425}{75} = 19
\]

So, the number of winners is \( \boxed{19} \).

Now, we can find the number of non-winners \( p \):
\[
p = 63 - w = 63 - 19 = 44
\]

To verify, we can check the total prize money:
- Prizes for winners: \( 19 \times 100 = 1900 \) rupees.
- Prizes for non-winners: \( 44 \times 25 = 1100 \) rupees.
- Total prize money: \( 1900 + 1100 = 3000 \) rupees.

The calculations confirm the solution is correct: the number of winners is \( \boxed{19} \).