A recent question on http://homebrew.stackexchange.com/ asked for a simple formula for calculating the percentage alcohol by volume (ABV) of a beer given the original gravity (OG) and final gravity (FG). Here was the complicated formula which was to be simplified:
OK, so it's not that complicated, but a linearization would be good. The original post suggested a range of 1.035-1.095 for OG and 1.002-1.028 for FG. I chose the midpoints of these intervals to get the point (1.065,1.015) and linearized the given formula at this point. Here's what I got:
It's simpler, but not easy to remember. If we round the coefficients, we get an easier to handle expression.
Analysis using Maple shows that this simplified linear approximation has an error of at most 0.78 from the original equation. This is not too hard to remember, since 147-130=17. Also, if both OG and FG are equal to one (the specific gravity of water), then the formula predicts 0% ABV.