combining proportions

[Wideflange]

Hello, Math Gods -
My brain hurts. Mywife is making an herbal "tincture" (herbs soaked in vodka to drawout healing compounds). The vodka is 80 proof (40% alcohol). To this she willbe adding a separate solution that is boiled in water (to draw out otherspecific water-soluble stuff... 0 % alcohol). What is a formula should she goby as to how much of the boiled water solution to add to the final alcoholmixture to achieve a 25% alcohol product?

 

[Ishuda]

Hello, Math Gods -
My brain hurts. Mywife is making an herbal "tincture" (herbs soaked in vodka to drawout healing compounds). The vodka is 80 proof (40% alcohol). To this she willbe adding a separate solution that is boiled in water (to draw out otherspecific water-soluble stuff... 0 % alcohol). What is a formula should she goby as to how much of the boiled water solution to add to the final alcoholmixture to achieve a 25% alcohol product?

Call the amount of vodka you have x. The amount of alcohol you have in the mixture is 0.4x [40% of x]. Call the amount of vodka and boiled water you want y. When you mix the water and vodka together, you want to have 0.25y alcohol. Since you have 0.4x alcohol to mix, the two amounts are equal
0.25 y = 0.4 x
or
y = 1.6 x
That means you have to add 0.6 x (1.6 x - 1.0 x) water since you already have 1.0 x vodka.

For example, suppose you have a 500 ml [what some refer to as a (metric) pint] of 80 proof vodka. If you add 0.6 * 500 = 300ml of boiled water for a total of 800 ml, the mixture would be 50 proof (25% alcohol)

 

[Wideflange]

Thanks

Thanks all. In retrospect, I see that it was relatively straightforward (overly simple ?) question.