Ask Uncle Colin: These alcohol-related figures look a bit fuzzy

Ask Uncle Colin is a chance to ask your burning, possibly embarrassing, maths questions -- and to show off your skills at coming up with clever acronyms. Send your questions to colin@flyingcoloursmaths.co.uk and Uncle Colin will do what he can.

Dear Uncle Colin,

I was recently on a tour of a whisky distillery where the guide told us about the ‘angel’s share’: every year, 2% of the alcohol in a bottle evaporates. He went on to explain that a 25-year-old single malt would have half the alcohol it did originally.

That doesn’t seem right, though — it would mean a 51-year-old whisky would make you slightly more sober for drinking it! What’s going on?

— Got A Lot Of Inverse Scotch

You’re quite right, GALOIS: if there was such a thing as inverse scotch, a shot of it would probably go through your stomach lining and give you peritonitis.

The tour guide is making a very common mistake of mixing up simple evaporation (which isn’t a thing) with compound evaporation (which is what actually happens.)

As you say, his/her assumption that losing 2% of the alcohol each year corresponds to the same amount doesn’t make sense — in the extreme case, alcohol that isn’t there evaporates, and you get a ridiculous answer, which is a sure sign that there’s something wrong.

Instead of $A = 100 – 2n$, as your guide thought, the correct formula for the percentage of alcohol left is $A = 100(0.98)^n$. The two graphs look very similar for small $n$, but the correct curve gradually slopes less steeply.

For instance, after 25 years, a little less than 40% of the alcohol has evaporated1; after 50 years, it’s about 63% — quite a lot less alcoholic, but certainly not non-alcoholic!

I see students make similar mistakes when dealing with financial interest — it’s the same as the difference between simple interest (repeated adding on the same percentage of the original investment) and compound interest (adding on a percentage of whatever’s in the bank at that point in time — so you get interest on the interest!)

— Uncle Colin

Colin

Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.

  1. The Mathematical Ninja says $A \approx e^{-\frac{n}{50}}$ []

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