John W. answered • 08/04/20
High School STEM Tutor
This honestly feels like more of a chemistry question than an algebra question, and it also simplifies some important concepts from chemistry that probably shouldn't be simplified, so I don't blame you for having trouble with it!
Let's call the volume of pure alcohol you need to add "x". That would make the volume of the final mixture 5 L + x. Let's assume that the 60% refers to percentage BY VOLUME. If the mixture is 60% alcohol, then the amount of alcohol in the final mixture will be 0.6 * (5 L + x), and the amount of alcohol in the initial mixture would be 5 L * 0.1 = 0.5 L. So, we are adding x alcohol to 0.5 L of alcohol and getting 0.6 * (5 L + x) alcohol. We can set this up as an equation:
0.5 L + x = 0.6 * (5 L + x)
Patrick's answer shows the steps to solve it from here, but I've shown it below again so you don't have to scroll between answers:
0.5 L + x = 0.6 * (5 L + x)
0.5 L + x = 3 L + 0.6 x
0.4 x = 2.5 L
x = 6.25 L
So, you would need to add 6.25 L of pure alcohol.
Let's check the answer to make sure it makes sense. Assuming we add 6.25 L of alcohol, the total amount of the alcohol in the final mixture would be 0.5 L + 6.25L = 6.75 L. The total volume of the final mixture would be 5 L + 6.25 L = 11.25 L. 60% of 11.25 L is 6.75 L, and so the solution WOULD be 60% alcohol by volume.
Side Note: The issue I have with this question is that it never states that the alcohol is 60% BY VOLUME. 60% can refer to 60% by mass, 60% mole fraction, or 60% by volume (and maybe even other things too). If the 60% were not by volume, you would need to know a lot more information of the mixture's properties, the density of the pure alcohol, etc. to answer the question. Alcohol percentages in mixtures are usually taken to mean by volume implicitly, but it would still be nice to state it in the problem statement for clarity.
