Two Buckets

[MattcAnderson]

Here is one I learned as a child.

One bucket contains a gallon of water, another a gallon of alcohol. A cup of alcohol from the second bucket is poured into the bucket of water. A cup of the resulting mixture is then poured back into the bucket of alcohol. Is there:
a) More water in the alcohol than alcohol in the water;
b) More alcohol in the water than water in the alcohol;
c) The same amount of water in the alcohol as alcohol in the water?

First one two answer correctly gets money from the second one to answer :)

Regards,
Matt

[retina]

The percentage of each liquid in the buckets is the same. Since they both contain one gallon at the end of the mixing the amount taken from one must be equal to the amount taken from the other. I only answered because I am greedy and want money from the next poster. Thanks in advance for your generosity.

[LaurV]

That is wrong. If you do this, then you have two buckets of water.
I see it like this: if you put a drop cup of whiskey (or wine) in a bucket of pee, then you still have a bucket of pee. If you put a drop cup of pee in a bucket of whiskey (or wine), then you have a bucket of pee. If you don't agree, you are welcome to drink it.

[kladner]

You start with 16 cups of each substance, or 16/16 gallon.

1/16 alcohol + 16/16 water = total fluid 17/16
Mixture has a ratio of 1 to 16, alcohol/water, 1/17 alcohol/mix

1/16 (cup) of the mix added to 15 cups of water makes the total volume of each the same. However, pure alcohol was added to the water, whereas 'dilute' water was added to the alcohol.

There is more alcohol in the water, than there is water in the alcohol.

This leaves aside that alcohol tends to be hygroscopic. Except in an ideal setting with no evaporation to, or absorption from the air, the alcohol will always be somewhat diluted. In the short term, this probably does not affect the outcome.

[LaurV]

Originally Posted by kladner

However, pure alcohol was added to the water, whereas 'dilute' water was added to the alcohol.

There is more alcohol in the water, than there is water in the alcohol.

That is wrong. Retina's solution was right and this is a 2500 years old problem (at least). The problem is quite famous and appeared in IQ tests of both Mensa and High IQ Society, many times.

Letting apart my joke in the former post, the concentration of alcohol in water in the first bucket is 1 part alcohol to 16 parts water, as you said, and using your example. So, if you take a cup of it, you have 1 part alcohol and 16 parts water in your cup (or 1/17 concentration, as you said). So, in the bucket remained 16/17 cups of alcohol in water. You just took 1/17 cup of alcohol with your cup. Adding this cup to the second bucket, you get 16 cups in the second bucket (as you said), the same total quantity. How much water you put in the alcohol? Well, you added 16/17 from a cup (the rest was alcohol). But in the first bucket you also have 16/17 of a cup of alcohol (because you took 1/17 with your second cup). The concentrations are always equal.

If it is not clear, try with half bucket. Or, say, to make the calculus easier, have two buckets of 60 liters filled with water and pure alcohol respectively. You put 40 liters of water in 60 liters of pure alcohol. Mix it well. You have 40% water there. Now take 40 liters of it, you will have left 60 liters, from which 36 are alcohol and 24 are water, because the mixture is 40% water in alcohol in the bucket, but also in the quantity you took. So, in the 40 liters you took, you have 16 liters of water and 24 of alcohol. Put it over the 20 liters of water remained in the other bucket, and you have 36 water and 24 alcohol. Qed.

[TheMawn]

Retina is correct.

There is only a gallon of water and a gallon of alcohol by the end, no matter how they end up being mixed.

Bucket A has 1-x gallons of water (some is missing) and x gallons of alcohol. It MUST be x gallons of alcohol because we know the bucket contains one gallon total.

Bucket B contains the missing x gallons of water (because there is 1-x + x = 1 total gallons of water) and 1-x gallons of alcohol (this last fact can be shown in two different ways).

[S485122]

Forget it

[wblipp]

Originally Posted by retina

Since they both contain one gallon at the end of the mixing

I know that we have at least one chemist here, so I will defer to him. But I believe that this statement is false. To see why, suppose you started with a bucket of sand and a bucket of water.

[LaurV]

Hehe, we said in the beginning that we ignore the fact that mixing equal volumes of alcohol and water results in lower than 2.0 of volume (dependent of alcohol, for pure ethanol, mixing one liter of it with one liter of water will only result in a volume of 1.92 liters of mixture).

Homework is how this influences our puzzle? (assuming we end up with equal volume in both buckets).

(I go to bed, 2:30 AM here... half sleeping)

[Uncwilly]

The problem is, each bucket is exactly 1 gallon in size.

[Batalov]

Water and ethanol have distinctly different densities (>20% different).
What is more relevant is that the density of a mixture at any concentration is higher than expected by linear (first-degree) effect (see attached chart).

In other words, taken any volume x of water and volume y of ethanol, the volume after putting them together will be less than x+y. These substances "like" each other and form a denser packing than each alone.