Question

You have 1 liter of 80 proof vodka (40% by weight ethanol in water) and someone removes 3 shots (100ml). What is the resulting concentration (proof)? Next someone fills the bottle back up with water, what is the resulting proof?

Answer 1

I think the proof  is defined as twice the percentage of alcohol by volume. Therefore, it should be 40% by volume and not weight.

Now when someone removes 3 shots there will not be any change in concentration because we assume that ethanol and water are mixed completely and when we take out shots than each shot contains same % of ethanol.

Now, in 1 liter or 1000 ml Amount of water = 0.6*1000 =600 mL

Amount of ethanol = 0.4*1000 = 400 mL

Now each shot of 100 ml again contains 40% by volume ethanol. i.e. 40 mL and 60 mL water. Therefore, 3 shots contain 120 ml ethanol and 180 mL water.

Therefore, after removing the 3 shots ethanol left in the bottle = 400 - 120 mL

= 280 mL

And water left in the bottle = 600-180 mL

= 420 mL

However, we added 300 mL of water, Therefore new volume of water = 420 +300 mL

= 720 ml

Therefore Concentration of ethanol = [(Volume of ethanol) / (Volume of ethanol + volume of water)] * 100 %

= [280 ml/ (280 + 720 mL)] *100%

= 28 % by Volume ethanol

Proof  is defined as twice the percentage of alcohol by volume. Therefore,

Proof = 2 * 28 proof

= 56 proof vodka