Question

how many gallons of distilled water must be mixed with 50 gallons of 30% alcohol solution to obtain a 25% solution. distilled water is 0% alcohol.

Answer 1

You need the situation where\[(alcohol)/(alcohol+water)=0.25\] Write A for alcohol and W for water. Now, you're not changing the amount of alcohol in the solution, but we need to find out how much you have. From what you're given, currently, \[\frac{A}{A+W^*}=\frac{A}{50 gal}=0.3 \rightarrow A=0.3 \times 50gal = 15gal\]where we've identified A+W* as the total volume of solution.

You need\[\frac{A}{A+W^*}=\frac{15}{15+W^*}=0.25\]where W* is the amount of total water now needed. Solving for W* gives, \[15=.25 \times 15+.25W^*\rightarrow W^*=45gal\]Now, this 45 is the final amount of water needed to be present in the solution. You already have (50-15)gal = 35 gal sitting there, so to make water volume equal to 45, you need to add 10 gallons.

Answer 2

To check, you have 15 gallons of alcohol, and now 45 gallons of water, so your percent volume of alcohol will be,\[\frac{15}{15+45}=\frac{15}{60}=\frac{1}{4}=0.25\]