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janos comments on Inverse Speed - Less Wrong
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The way I'd try to do this problem mentally would be:
Relative to the desired concentration of 55%, each unit of 40% is missing .15 units of alcohol, and each unit of 85% has .3 extra units of alcohol. .15:.3=1:2, so to balance these out we need (amount of 40%):(amount of 85%)=2:1, i.e. we need twice as much 40% as 85%. Since we're using 1kg of 40%, this means 0.5kg of 85%.
That's clever! Changing your frame of reference is a useful tool - there are a lot of problems which become simpler if you use measurements from a 'zero' that you pick.