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 ?

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- Jul 28th 2011, 12:25 AM #1

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- Jul 28th 2011, 12:49 AM #2
## Re: Purity

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 #3
## 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 #4

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## Re: Purity

Hello, PASCALfan!

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.

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__Spoiler__:

- Jul 29th 2011, 05:56 AM #5

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## 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 #6

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