Where did 1$ go?

We all have heard about this problem once in our life, the problem goes like this, there were three friends Saransh, Ameya and Mayank who went for eating outside in a restaurant because the food in their mess was a mess, so after they had eaten up the waiter comes with the bill of 30$, so they decided to split the bill and pay 10$ each, when waiter was on his way back he realizes that he had swapped their bill with someone's else, their original bill was only 25$ so he rushed back to them to return 5$, now the three friends divided the 3$ with themselves and return back 2$ to the waiter as a tip. Now each friend had paid 10$ and they got 1$ back so in nutshell each friend has paid 9$, now 9 times 3 is 27$ and the 2 dollars are in the waiters pocket that makes 29$, but the total money they had paid initially was 30$ then where did the 1$ go?

Let us solve this problem by considering the real money, since I live in India I only have Indian rupee as currency so we will imagine one rupee coin as one dollar, here is the distribution :

Now, they have got 1 rupee back hence the situation will look like :

Now the left over scene will be like this :

So the original bill was 25$ and here we have 5$ which add upto 30$ hence there is no confusion here, we are clear no 1$ is lost, then why 1$ is less according to problem, the answer is simple after giving 9 + 9 + 9 = 27$ they had paid 27$ instead of 25$ and that 2$ goes in the pocket of waiter, so if we are considering the second(original) bill whose value was 25 which is equivalent to paying 9 + 9 + 9 = 27$ and 27 - 2 = 25$

Now if we consider the first bill which was misplaced whose value was 30$ then as each friend had paid 10 + 10 + 10 = 30 $, the waiter returns with 5$ gives 5 - 2 = 3$ back and keep 2$ with him so we have the original bill = 25$ the paid bill=27$ and difference 2$ with waiter. So in nutshell we can say

25$(original bill) + 3$(money given back) + 2$(money with the waiter) = 30$(wrong bill),

Now I want you to concentrate on the above equation, what the question is doing is we subtract 3$ from the original bill to get 27$ paid amount as hence the about equation can be re-written as:

25$(original bill) + 2$(money with the waiter) = 30$(wrong bill) - 3$(money given back)

Now what we are doing we are adding 2$ of the waiter to get 29$ which is horribly wrong:

25$(original bill) + 2$(money with the waiter) + 2$(money with the waiter)

= 30$(wrong bill) - 3$(money given back) + 2$(money with the waiter)

Hence we are getting 29$ on R.H.S which does not make any sense, as it will be equivalent to saying Original bill added to twice the money with waiter which will obviously give us wrong answer and can never be equal to 30$.

So that's it guys! Hope you understood the problem and as always have a nice day!