V.I. Arnold's Problem:

From a barrel of wine, a spoon was poured into a cup of tea, and then the same spoon of the obtained (non homogeneous!) mixture was poured from the cup back into the barrel. Where did the amount of the foreign beverage become greater ?

Question

V.I. Arnold's Problem:

From a barrel of wine, a spoon was poured into a cup of tea, and then the same spoon of the obtained (non homogeneous!) mixture was poured from the cup back into the barrel. Where did the amount of the foreign beverage become greater ?

vishweshshrimali5 · Answer

@ganeshie8 @Hero @hartnn @Kainui @iambatman @mathslover

inkyvoyd · Answer

how does one pour a spoon? do you mean a spoonful?

vishweshshrimali5 · Answer

Well I think he meant he just first inserted the spoon in one container and then the other one and finally again in the first one.

vishweshshrimali5 · Answer

I think that this must depend on the properties of the liquid. The larger the amount of the liquid particles stuck to the spoon, the higher must be the amount of foreign beverage.

vishweshshrimali5 · Answer

But, this must make the problem a chemistry or a physics problem and its a maths question and he says that 
"children 4-5 years old would have solved them in half an hour"

vishweshshrimali5 · Answer

@Miracrown

ganeshie8 · Answer

One full spoon of `pure wine` was poured into the glass
One full spoon of `tea+wine` mixture was poured back into the barrel

ganeshie8 · Answer

hmm some wine that was poured earlier into the glass was taken back into the barrel as well

vishweshshrimali5 · Answer

Yeah if he meant spoonful then its easy to solve. But, what did he mean !!!

ganeshie8 · Answer

spoonful makes more sense...

vishweshshrimali5 · Answer

Yeah. I agree with you @ganeshie8 . If that's case, then, cup of tea must be the answer.

ganeshie8 · Answer

maybe, we need to prove it

anonymous · Answer

He's asking in which (cup or barrel) is the foreign liquid greater in.

vishweshshrimali5 · Answer

May be. But, I think I can do that.

vishweshshrimali5 · Answer

Let, the volume of the liquid in one spoonful = V mL

Then, the wine transferred in cup of tea due to first step = V mL.

Let, a cup of tea had tea = x mL 

and wine present initially = y mL

Then, cup has x mL tea + V mL wine after first step

Now, second step will transfer V mL mixture into barrel of wine.

Now, since V < x, thus, amount of wine would be less than amount of tea in the spoon. But, even then the amount of tea would be less than V.

So, after second step the barrel of wine would contain:

(y - V + wine from spoon) mL wine + (tea from spoon and this amount < V)

Clearly, foreign amount in tea cup = V mL

foreign amount in barrel of wine < V

Thus, tea of cup would have more foreign amount

vishweshshrimali5 · Answer

What do you think @ganeshie8 ?

vishweshshrimali5 · Answer

Here I would love to share this link for young students: http://jnsilva.ludicum.org/HMR13_14/Arnold_en.pdf (I recently found it after a lot of research)

anonymous · Answer

This is probably just a tricky question, where the answer is they both have equal amounts of the foreign substance.

vishweshshrimali5 · Answer

@iambatman But, I don't think they can have the same amount. Even if you think in terms of probability, then also the amount of wine in the spoon would be very less after second step but not zero.

ganeshie8 · Answer

key thing is to notice that the volumes of barrel and glass don't change after the experiment

anonymous · Answer

That's what I'm thinking, if you think of its initial state, there's no change.

vishweshshrimali5 · Answer

Yeah that's correct.

ganeshie8 · Answer

volume of water entered into the barrel =  volume of wine left in the glass

ganeshie8 · Answer

so i think iambatman has it !

vishweshshrimali5 · Answer

But, volume of the liquid is constant not the type of liquid. Isn't it ?

nincompoop · Answer

ASK me

vishweshshrimali5 · Answer

Ok @nincompoop ?

vishweshshrimali5 · Answer

Well I think I got it. Thanks @iambatman  and @ganeshie8 .

nincompoop · Answer

first, post this in Chemistry section

nincompoop · Answer

then I will break it down

vishweshshrimali5 · Answer

Why ?

anonymous · Answer

Np, glad I could help, we often tend to over complicate the problems, such as nins chess problem, yeah I said it ;).

nincompoop · Answer

because you used the word non-homogenous mixture

ganeshie8 · Answer

chess is complicated

kainui · Answer

pretend the mixture is a differential equation, and solve it here.

anonymous · Answer

Ik ganeshie, I was joking :p

mathslover · Answer

Oh I missed this one.. :( Sorry, I couldn't help you this time @vishweshshrimali5 :P lol

vishweshshrimali5 · Answer

That's it @Kainui. I was thinking of finding something to infuriate my cousins by asking them this question and saying that its very difficult; you have to use DE. :D

nincompoop · Answer

you use chemistry tricks

vishweshshrimali5 · Answer

Number of moles transferred by assuming molarity ?

nincompoop · Answer

concentration and dilution

vishweshshrimali5 · Answer

Ahh M*V = constant !!

Well, this could also help. Thanks @nincompoop .

nincompoop · Answer

the units you will use are barrel, cup and spoon

nincompoop · Answer

use this hypothetical conversion units 
$$1Barrel_w \times \frac{ 1000 cupful_w}{ 1Barrel_w } \times \frac{ 50 spoonful_w }{ 1 cupful_w }=50000spoonful_w$$

and since you're familiar with chem already, then compare the amounts just like how mole fractions are done 
$$MolefractionA= \chi_A = \frac{ n_A }{ n_A+n_B }$$

vishweshshrimali5 · Answer

OKay I will try solving the question using these units. Thanks again @nincompoop