# Purity

• Jul 28th 2011, 12:25 AM
PASCALfan
Purity
Two vessals contains equal amount of water and beer. 1spoon beer is added to water and mixed well. From this mixture 1 spoon is added to beer. Which of the liquids is more pure ?
• Jul 28th 2011, 12:49 AM
alexmahone
Re: Purity
Quote:

Originally Posted by PASCALfan
Two vessals contains equal amount of water and beer. 1spoon beer is added to water and mixed well. From this mixture 1 spoon is added to beer. Which of the liquids is more pure ?

Let the volume of the vessel be V and the volume of the spoon be v.

After 1 spoon beer is added to water, vessel 1 contains V parts of water and v parts of beer.

After 1 spoon mixture is added to beer, vessel 2 contains V-v parts of beer and v parts of (beer+water in the ratio v:V).
Parts of beer contained in vessel 2 = $\displaystyle V-v+v\times\left(\frac{v}{v+V}\right)=\frac{V^2-v^2+v^2}{v+V}=\frac{V^2}{v+V}$
Parts of water contained in vessel 2=$\displaystyle V-\frac{V^2}{v+V}=\frac{Vv+V^2-V^2}{v+V}=\frac{Vv}{v+V}$
The ratio is $\displaystyle \frac{V^2}{v+V}:\frac{Vv}{v+V}=V:v$

So, both the liquids are equally pure.
• Jul 28th 2011, 02:49 AM
kalyanram
Re: Purity
Let the volume of the vessel be $\displaystyle V$ and the volume of the spoon be $\displaystyle v$.
Purity of water by the end of adding a spoon of beer and mixing = $\displaystyle \frac{V}{V+v}$
Purity of beer by the end of adding a spoon of water containing beer and mixing = $\displaystyle \frac{V-v +v.(\frac{v}{V+v})}{V}$ = $\displaystyle \frac{V}{V+v}$

So by my equation I get that they are of equal purity.

Kalyan.
• Jul 28th 2011, 07:33 AM
Soroban
Re: Purity
Hello, PASCALfan!

Quote:

Two vessels contains equal amount of water and beer.
1 spoon beer is added to the water and mixed well.
From this mixture 1 spoon is added to the beer.
Which of the liquids is more pure?

This is the classic "Wine and Water" problem.

As long as the amount transferred each time remains the same,
. . the final concentrations are equal.

This is the basis of a stunning card trick.

You are seated at a small table across from your volunteer.

You show him a deck of cards, fanning them
. . so he can see they are all facing one way.

Both of you place your hands under table.
You hand him the deck.

Instruct him to select a secret number from 1 to 20,
. . and keep it a secret. .Call it n.

Have him count off the top n cards,
. . turn them over and replace them on the deck.

Have him shuffle the deck thoroughly.
Then count off the top n cards
. . and hand them to you under the table.

Now you remind him:
. . you don't know his secret number,
. . he doesn't know how many face-up cards he has,
. . you don't know how many face-up cards you have.

Despite this, you will try to make your number of face-up cards
. . equal his number of face-up cards.

He hears the sound of cards being counted and moved.

After a few moment, you bring your cards to the top of the table
. . and count the face-up cards.

Instruct him to do the same . . . and the numbers match!

[Acknowledge the thunderous applause ... modestly, of course.]

Spoiler:

Secret: When he hands you the packet, turn it over.
• Jul 29th 2011, 05:56 AM
emakarov
Re: Purity
In fact, the answer does not change regardless of how well the liquids were mixed in the process. Indeed, the volume of the liquid in each vessel after the two operations is the same as it was initially. Therefore, in the end the volume of beer in water has to equal the volume of water in beer.

This is one of my favorite math problems because it shows that there is more to math than handling large formulas.
• Jul 29th 2011, 07:10 AM
PASCALfan
Re: Purity
(Clapping)
Thanks for the card trick..

Regards (Bow)