Series prob: You have two 3-L vessels. 1st contains oil while 2nd contains vinegar.

[hhjr]

[FONT=&quot]You have two 3 litre vessels. The first contains oil while the second contains vinegar. [/FONT]
[FONT=&quot]You have two spoons, each can scoop out k litres of liquid, and you use the spoons to [/FONT]
[FONT=&quot]transfer k litres between each vessel. That is, you take a spoon from one vessel, a spoon [/FONT]
[FONT=&quot]from the other and then you transfer their contents over to the other vessel. You mix [/FONT]
[FONT=&quot]the vessels thoroughly so that each vessel contains a mixture of oil and vinegar. This [/FONT]
[FONT=&quot]process is then repeated any number of times. [/FONT]

[FONT=&quot]The question asks: [/FONT]

[FONT=&quot]a) let x = vinegar in first vessel and u = vinegar in second vessel [/FONT]
[FONT=&quot]show that for n rounds of the game [/FONT]
[FONT=&quot]x(n+1) = x(n) - (k/3)x(n) +(k/3)u(n). [/FONT]
[FONT=&quot]u(n+1) = u(n) - (k/3)u(n) +(k/3)x(n)


[/FONT]

 

[Deleted member 4993]

You have two 3 litre vessels. The first contains oil while the second contains vinegar.
You have two spoons, each can scoop out k litres of liquid, and you use the spoons to
transfer k litres between each vessel. That is, you take a spoon from one vessel, a spoon
from the other and then you transfer their contents over to the other vessel. You mix
the vessels thoroughly so that each vessel contains a mixture of oil and vinegar. This
process is then repeated any number of times.

The question asks:

a) let x = vinegar in first vessel and u = vinegar in second vessel
show that for n rounds of the game
x(n+1) = x(n) - (k/3)x(n) +(k/3)u(n).
u(n+1) = u(n) - (k/3)u(n) +(k/3)x(n)

What are your thoughts?

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[hhjr]

Thanks for your response.

I understand that x(0) = 0 and y(0) = 3 to begin with as one barrel is filled with oil only and the other is filled with vinegar.
Moreover, after the first round of the game, I know that x(1) = k and y(1) =3-k it is relatively simple to determine the amount of vinegar that will be transferred from one vessel to the other at this point.

However, I am having a bit of trouble in being able to determine the amount of vinegar that will be transferred in between the vessels after that.

The answer suggests that it would be k/3*x(n), is it because the proportion of vinegar in the vessel is given by x(n)/3 where 3 is the total volume of the vessel, hence if multiplied by k, we will be able to determine the amount of vinegar transferred and received ?

Sorry for not explaining that earlier

 

[hhjr]

so basically i know that x(0) = 0 and y(0) = 3
x(1) = k and y(1) = 3 - k
but not so sure how to continue from there

 

[Deleted member 4993]

You have two 3 litre vessels. The first contains oil while the second contains vinegar.
You have two spoons, each can scoop out k litres of liquid, and you use the spoons to
transfer k litres between each vessel. That is, you take a spoon from one vessel, a spoon
from the other and then you transfer their contents over to the other vessel. You mix
the vessels thoroughly so that each vessel contains a mixture of oil and vinegar. This
process is then repeated any number of times.

The question asks:

a) let x = vinegar in first vessel and u = vinegar in second vessel
show that for n rounds of the game
x(n+1) = x(n) - (k/3)x(n) +(k/3)u(n).
u(n+1) = u(n) - (k/3)u(n) +(k/3)x(n)

Let's do some transfer and check:

Let's assume k = 1 litre and vessels hold 3 L fluids but are big enough to hold 4 L.

Then 1 L vinegar is moved from Vv1 to Vo2. I am left with 2 L of vinegar in Vv1.

After mixing, in Vo2, total fluid 4 L and concentration of vinegar is 1/4.

Now move 1 L of this mixture (from Vo2) to Vv1. This mixture has 0.75 L of O and 0.25 L of V. In Vo2 (the mixture left - 3 L) - the amount of vinegar is 0.75 L and and amount of oil is 2.25 L.

So in Vv1 - the amount of oil is 0.25 L and and amount of vinegar is 2.75 L. (Oil concentration 1/12)

Now pick up one litre from Vv1 (11/12 L of vinegar and 1/12 L of oil) and mix it in Vo2.

After mixing - total fluid in Vo2 = 4 liter of which Oil (= 2.25+1/12 = 7/3 L) and amount of Vinegar in Vo2 is (3/4+11/12 = 5/3 L) - so the concentration of vinegar is 5/12 .... and so on....

Look at the progression of the changes in concentrations and

 

[hhjr]

That was very helpful, cheers

 

[Ishuda]

Thanks for your response.

I understand that x(0) = 0 and y(0) = 3 to begin with as one barrel is filled with oil only and the other is filled with vinegar.
Moreover, after the first round of the game, I know that x(1) = k and y(1) =3-k it is relatively simple to determine the amount of vinegar that will be transferred from one vessel to the other at this point.

However, I am having a bit of trouble in being able to determine the amount of vinegar that will be transferred in between the vessels after that.

The answer suggests that it would be k/3*x(n), is it because the proportion of vinegar in the vessel is given by x(n)/3 where 3 is the total volume of the vessel, hence if multiplied by k, we will be able to determine the amount of vinegar transferred and received ?

Sorry for not explaining that earlier

Approaching it from a slightly different direction: Have you had proof by induction yet?

Also, If something (your vessels) holds a 'well mixed' mixture of vinegar (v_n) and oil (o_n) so that v_n+o_n=3, and you take a spoon full holding k from that mixture, how much vinegar will you have and how much oil will you have in the mixture in the spoon [think of proportions]?