The organizers of an essay competition decide that a winner in the competition gets a prize of Rs100 and a participant who does not win

Solution: Given, Total number participants = 63
Total prize money distributed = Rs 3000
Winner gets a prize of Rs 100
Loser gets a prize of Rs 25
Number of winners = ?

Let the number of winners = m
Since,
Number of winners + Number of losers = Total number of participants
Or, m + Number of losers = 63

By transposing ‘m’ to RHS, we get
Number of losers = 63 – m
Now, Total Prize money distributed to winners
= Number of winners X prize money distributed to each winner = m X 100 = 100m
Total prize money distributed to losers
= Number of losers X prize money distributed to each loser
= (63 – m) X 25 = (63 x 25) – 25 m = 1575 – 25 m

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 losers, we get
100 m + 1575 – 25 m = 3000
⇒ 100 m – 25 m + 1575 = 3000

By transposing 1575 to RHS, we get
100 m – 25 m = 3000 – 1575
⇒75 m = 1425

After dividing both sides by 75, we get

Thus, number of winners = 19

Let the no. of winners be x 
& the no. of runner ups be 63-x
ATQ,
100(x) + 25(63-x) = 3000
100x + 1575 - 25x = 3000
75x + 1575 = 3000
75x = 3000 - 1575
75x = 1425
x = 1425/75
x = 19
No. of winners = 19
No. of runner ups = 44.