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You're a fellow salesman who happens to have to sell a barrel full of wine. The barrel has a capacity of 5 Litres. Unfortunately, your apprentice is a wine lover and steals some wine each night, hoping you will not notice. He does it by taking out 1 Litre of the liquid in the barrel and replaces it with 1 Litre of Water. He repeats this procedure every night. When you notice his theft, the concentration of wine in the barrel has fallen down to 1%. How many days have you been fooled?

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    $\begingroup$ The apprentice is a wine lover but doesn't mind drinking diluted wine after the first day ? Yikes ! $\endgroup$ – Betcheg Jul 26 '18 at 16:04
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    $\begingroup$ Well, if you really like the Element Gold for instance, wouldn't you also like to have a mineral with ~5% Gold in it? ;) $\endgroup$ – Quotenbanane Jul 26 '18 at 19:05
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Let $x$ be the number of days that you have been fooled. After day 1, there is

1L of wine removed (20%, leaving 4L, 80%) and 1L of water in the barrel.

After day 2 (assuming continuous mixing), there is

80% wine and 20% water before any of the mixture is removed. Hence when a litre is removed, it will be composed of 0.8L of wine and 0.2L of water. This means that (adding 1L of water back), there will be 3.2L of wine and 1.8L of water after day 2.

We notice that we can model this using the formula

Amount of Wine = $5(0.8)^x$

Hence we need to find $x$, where

1% of 5 = 0.05, so $5(0.8)^x = 0.05$ means $0.8^x = 0.01$.

Then we have

$x = \frac{\ln 0.01}{\ln 0.8} = 20.63 \approx 21$ days. Hence you've been fooled for 3 weeks!

BONUS:

I think you can take advantage of the fact that the wine and water will have different freezing points...if you freeze it and pick out the slush/ice, what's left will be (stronger) 100% wine...? Either this or a fancier distillation technique should do the trick.

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    $\begingroup$ Well you should have told me that 3 weeks earlier.... Good job tho. $\endgroup$ – Quotenbanane Jul 26 '18 at 13:08
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    $\begingroup$ I think it's worth noting that by the 4th day, your assistant is going to be pulling mostly water (it'll be almost 60% water on the 4th day already!) out of the barrel, so at that point is it even worth it? Diminishing returns, indeed! (Also, welcome to Puzzling.SE - although I'm pretty new here too! Just for your reference, if you like my answer please feel free to upvote/accept accordingly!) $\endgroup$ – El-Guest Jul 26 '18 at 13:15
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    $\begingroup$ Re bonus question/answer: Would that be worth the trouble? You have only 1% of 5 litres left, or 50 ml. Normal glasses contain 150ml or more of wine, so you have less than a third of a glass left. $\endgroup$ – Jaap Scherphuis Jul 26 '18 at 13:17
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    $\begingroup$ @El-Guest does it help stating that the apprentice is not the brightest bulb? I cannot think of another excuse haha. Jaap you're right. It does not make sense to do so. I'll edit the Bonus-Question out $\endgroup$ – Quotenbanane Jul 26 '18 at 19:04
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    $\begingroup$ @Quotenbanane the removal of the water from the wine can be the punishment for your apprentice :D $\endgroup$ – LinuxBlanket Jul 27 '18 at 10:04

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