Some fun on Happy hour prices.

Note:

Before reading this note I suggest you to solve happy hour prices, it's second part and the third part . . . .
This note is posted just for some fun and relief from all the daily activites, the characters in this story(there are only two!) are: Calvin Lin and an anonymous bartender, please don't take this story seriously!

You are eligible to read this now,

One fine morning, some illiterate thought of starting a bar(he had long since cherished a secret fantasy about this).

As he was and illiterate(he never heard of math in his life), he put up the price board haphazardly and that too, not caring about the prices(see the question) he put on them .Whatsoever, his business continued. . . .
Consequently, most of the people bought the $\$\dfrac{1}{2}$ drink, but this guy didn't understand why (You may ask Calvin sir why?). He just cared whether the drinks are sold or not. . .

1 Year later: He started noticing, that almost all of the people(with few exceptions) bought the $\$3$ drink. So that night, he decided that he would ask this question, "why everyone preferred the $\$3$ drink?" to the next customer. . .

Subsequently, the next morning had come. And the first person he saw was certain Calvin Lin , who was a generous person in his thirties and liked solving math problems on Brilliant.org.
He came up to the shop and glanced at the price board, being a "challenge master" he did the math quite quickly and ordered $2$ , $\$3$ drinks instead of $1$ , $\$10$ drink. And as decided, the barkeeper asked this guy, "why so?".
At first, Calvin thought he would better go away, but as he was a generous person and moreover he had time to spare, he started explaining why. Here goes the conversation:

See Mr., I hope you're familiar with "Mathematics."

What-matics"?

"Mathe"-matics.

Sorry, but what's that?

It's the abstract science of number, quantity, and space, either as abstract concepts, or as applied to other disciplines such as physics and engineering.

So, what have we got to do with that?

Let's consider your example, you have priced the $\dfrac{1}{2}$ liter drink at $\$3$ and the $1$ liter drink at $\$10$ . Can you explain what's wrong with that?

Sorry, but I don't see anything wrong, can you explain that?

Alright, you see that $10$ liters is double the $\dfrac{1}{2}$ liter. So, you should have priced the $\dfrac{1}{2}$ liter drink, at half the cost of the $10$ liter one, i.e, at $\dfrac{10}{2}=\$5$ , but you'd placed it at less that one-third the cost. That means you've incurred a $\$2$ loss!

Oh my god! I understand it. This is so horrible, what should I do now!

After, seeing the barkeep's reaction, Calvin left. But, the Barkeeper was awestruck.
And then he broke down and cried for an hour, finally the long silence was broken by a customer, who bought a pen and changed the price board, he was Calvin . . .
The barkeeper, moved forward to get a closer look, and found that the $\dfrac{1}{2}$ liter drinks' price was changed to $\$6$ , the $1$ liters' to $\$12$ and the $2$ liters' to $\$24$
He then explained that the business would get better now, and also asked him,

"how many, drinks were sold every day? and how many years the shop has been running?"

The barkeeper replied, "(still sobbing)around 20 drinks. . . and it's been on for an year. . . "
As a finance person Calvin calculated the loss,

"that means, a loss of $20 \times 365 \times 2 = \boxed{14600}$ .

"Oh my god! When will I recover from all this. . . "

Again Calvin did the math and said,

"maybe within 8 months or so . . . "

"Thanks my son, I don't know what would have been my fate, if you weren't present today, you are an "angel" for me!"

"No, no sir, no need to mention that, I hope your business gets better soon . . . And as a debt from my side, I'll pay you $\$12$ for that drink."

"No my son! Your help was the biggest reward I could ever get . . ."

Two years later: The barkeep came out of all his difficulties and losses and learnt a whole lot of math, he learnt tricks to draw more customers as he learnt about discounts , here goes his new price board:

$CALVIN'S \quad BAR :\quad MENU$
$\dfrac{1}{2}$ Liter - $10 - 30% Discount!
$1$ Liter - $20 - 30% Discount!
$2$ Liter - $40 - 30% Discount!

Three years later: The barkeep went mad with math, he started learning about limits and never ending sequences.

And with Calvin sir's blessings, he created a new and attractive offer:

Here it goes,

SPECIAL OFFER! For only $\$45$ , you have the option of getting drinks infinite number of times which goes as $1$ litre on the first drink, $\dfrac 12$ litre on the second drink, $\dfrac 14$ litre on the third drink... and so on. LIMITED TIME OFFER! HURRY!

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`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Some fun on Happy hour prices.

Hope you enjoyed it! Also, the last part(three years later) was suggested by Prasun Biswas

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