Assume I fill up a 1-liter bottle of water to the brim.

The next day, I empty $\frac 34$th of the bottle, and fill with fresh water until the bottle is full. I shake the bottle thoroughly to mix the water.

The next day, I again empty $\frac 34$th of the bottle and refill with fresh water. Mix again.


My question is: what percent of 1 liter is made up of the original water on the first day after exactly $n$ days?

I believed that the percentage is $\left(\frac 14\right)^n \cdot 100.$

Am I missing something? Or is this correct?

  • $\begingroup$ Assuming perfect mixing, you are correct $\endgroup$ – siegehalver Jun 10 '17 at 15:47
  • $\begingroup$ Should we separately take into account the amount of old water that is retained? Or is that included when we multiply by 1/4? $\endgroup$ – coldspeed Jun 10 '17 at 15:49
  • $\begingroup$ Yeah, you are correct. You can just imagine having two bottles that together always add up to 1L. The first bottle starts out with 1L of pure "old water", while the other just has 0L of fresh, new water. Removing 3/4 of your water bottle, could then be imagined as removing 3/4 of the old water battle and 3/4 the new water bottle, as they always add up to 1. You can now easily see that your reasoning is correct. $\endgroup$ – Imago Jun 10 '17 at 15:53
  • $\begingroup$ Whatever proportion of the the day before was, after emptying it and refilling the proportion will be 1/4 of the that. Before you start the proportion was all of it. After the first day it wias 1/4 of all. After the second day it was 14 of 14 of all. After n days is was 1/4 of 1/4 of 1/4 of ...... of 1/4 of all (n times). That's is $(\frac 14)^n$. To convert to precentages (which have no innate mathematical meaning and are just a bookkeeping convention) we multiply by 100 (i.e. a percentage is a "so many parts out of 100" ) So your reasoning is perfectly correct. $\endgroup$ – fleablood Jun 10 '17 at 15:57

Define a sequence of the number of water $$x_0=1$$ Now you take out 3/4 $$x_1=1/4$$ Now you take out another 3/4 but only out of the 1/4 so you leave only 1/4 of the water $$x_2=1/4*1/4$$ You continue that way, so in the n-th day you have $$x_n=(1/4)^n$$ If you want percentages so multiply by 100.

  • $\begingroup$ Would I not be taking 3/4 of the 1/4 meaning I take out 3/16 leaving 13/16 of the original amount? $\endgroup$ – coldspeed Jun 10 '17 at 15:51
  • $\begingroup$ Sorry, my bad. Please see my edit on my comment. $\endgroup$ – coldspeed Jun 10 '17 at 15:53
  • $\begingroup$ Here is another explanation: you take 3/4 of the water, only 1/4 out of the 3/4 is old water, so you take out 3/16 old water. Therefore the amount of old water is $1/4-3/16=1/16$ $\endgroup$ – Yanko Jun 10 '17 at 15:57
  • $\begingroup$ Thanks for the help! I've got it now. $\endgroup$ – coldspeed Jun 10 '17 at 15:59
  • 1
    $\begingroup$ "Would I not be taking 3/4 of the 1/4 meaning I take out 3/16 leaving 13/16 of the original amount? " No, because there were only 4/16th of the original left on that day. So if you substract 3/16 from 4/16 you have 1/16. $\endgroup$ – fleablood Jun 10 '17 at 16:02

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