Let's say I have a 100 liter container of water. I extract 10 liters of water, and replace it with 10 liters of juice. Now, the container is now 10% juice / 90% water.
I can keep repeating this to increase the percentage of total juice (Extract 10 liters of mixture, and replace with 10 liters of pure juice)
So, for round two, extracting 10 liters means there are 90 liters left of 90% solution (1 part juice, 9 parts water). (90 liters)(.9) = 81 liters of water. 9 liters of juice. I have removed 10 liters (1 liter juice, 9 liters water) I now replace this with 10 liters of juice. Total of 9+10 = 19 liters of total juice. So, after doing this a 2nd time, I have 19% juice.
If I do this a 3rd time, I remove 10 liters, and the remaining 90 liters have (90)(.81)=73 liters of water and 17 liters of juice (total 90 liters). Replace with 10 liters of juice = 27 liters of juice. 27% juice.
1) How many times do I need to do this to have 90% juice?
2) Is there a general formula that can determine percentage of fluid #2 as a function of number of extract/refill cycles completed?
3) Is there a general formula that can determine how many extract/refill cycles are needed to attain x% of fluid #2?
Also, what would be the correct tag for this question? What category of math is this called? Mixture problem?