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I asked a question yesterday. It was

"If I have a spirit which is 38% and one which is 15%, what proportions do I need to mix it to get 20%?"

Reason: I had some blackcurrant gin which was 20% alcohol and I ran out. I had a bottle of gin and a bottle of cassis, 38% and 15% respectively. And I had man flu so I couldn't work it out.

So.

Clearly, you need to put more of the 15% in to get to 20% from 38%.

Therefore:

(38 +15x)/(x+1) =20.

38 + 15x = 20x + 20

38 - 20 = 20x - 15x

18 = 5x

x = 18/5

18 cassis to 5 gins, or 3 and 3/5, or pretty much 7 to 2.

But it is clear from the above that if you substitute H for 38 (high content) and L for 15 (low content), and T for 20 (target content), you get

(H + Lx)/(x+1)=T

H + Lx = Tx + T

H - T = Tx - Lx

H - T = x(T - L)

The proportion is then given by

(H - T)/(T - L)

QED.

There. That's 2 minutes of your life that you won't get back.