# Container with a 2 part mixture.

1. Oct 7, 2009

### Averagesupernova

My dad and I were having a discussion about filling a container with a 2 part mixture (50/50), draining off X amount of it and then refilling it with only 1 part of the mixture. For example:
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Suppose we have a container that holds 10 gallons. We fill it with 5 gallons of water and 5 gallons of ethyline-glycol anti-freeze. (Yes, this discussion was born out of a real world scenario.) Now suppose we drain off 2 gallons of this mixture, add 2 gallons of water and allow it to mix completely. Then, we drain off 2 gallons of this mixture and add 2 gallons of the pure anti-freeze. What is the current mixture in the tank? Is it extra water, extra anti-freeze, or back to 50/50? At first I thought it was one way, then a different way, and when I did a little quick math/logic I determined that it was in fact it was neither of those.
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It's not exactly a brain-teaser so I thought it would be appropriate to post it in GD. I'm curious to know how all of you think. So please post what you would intuitively think the end mix would be without throwing math at the problem, and finally post what you find using math or logic. And please post the computed answer with the appropriate text as to not spoil it for others.
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Incidentally, dad and I didn't agree clear up until the end.

The final mix will be extra anti-freeze. The logic being that the first 2 gallons drained will be 50/50. The second 2 gallons drained is is extra-water. We went through a total of 7 gallons of each with the whole experiment and we know that the 4 gallon mixture outside the tank that we drained is extra-water so the mixture inside the tank has to be extra anti-freeze.

Last edited: Oct 7, 2009
2. Oct 8, 2009

### lewdtenant

I guessed wrong, but here's what I discovered.

30 iterations of the procedure are shown below (in volume):

5 5
6 4
4.8 5.2
5.84 4.16
4.672 5.328
5.7376 4.2624
4.59008 5.40992
5.672064 4.327936
4.5376512 5.4623488
5.63012096 4.36987904
4.504096768 5.495903232
5.603277414 4.396722586
4.482621932 5.517378068
5.586097545 4.413902455
4.468878036 5.531121964
5.575102429 4.424897571
4.460081943 5.539918057
5.568065555 4.431934445
4.454452444 5.545547556
5.563561955 4.436438045
4.450849564 5.549150436
5.560679651 4.439320349
4.448543721 5.551456279
5.558834977 4.441165023
4.447067981 5.552932019
5.557654385 4.442345615
4.446123508 5.553876492
5.556898806 4.443101194
4.445519045 5.554480955
5.556415236 4.443584764

After an infinite amount of iterations, the system will oscillate between (5/9 water, 4/9 antifreeze) and (4/9 water, 5/9 antifreeze) with each successive iteration. For all practical purposes it will show this tendency after about 25 iterations.