If you’re looking for a technical, precise explanation on handling concentrated liquid problems, look elsewhere. For the rest of you, this may be helpful.

When I say concentrated liquid problems, I mean problems that take the following form:

5 gallons of a mixture of cranberry juice and water is 15 percent juice. How much water will have to be added in order to create a mixture that is 10 percent juice?

You see these problems in various forms, sometimes asking you to evaporate out some water, or sometimes to add some water. Sometimes you’re given the total volume of the desired mixture and asked to determine the concentration. Other times you’re given the desired concentration and asked to figure out the total volume in some form. However, despite the specific form that you’re given, I have some very eloquent advice that’s going to make tackling these problems much simpler: Find out how much stuff you’ve got.

Yup, that’s right “stuff”. Almost all of the problems of this type that I’ve encountered involve a mixture of water and something else. That something else is the “stuff” we’re interested in. Water will simply be a filler that will take up all of the space not occupied by “stuff”. Let’s take a look at the problem I mentioned earlier to illustrate this concept.

Here we have 5 gallons of mixture and we know that it’s 15% cranberry juice. That’s our stuff. We need to find out how much cranberry juice we have. The simple calculation is:

Once we know how much stuff we have, we just need to set that 0.75 gallons equal to 10 percent of x to find out the volume of the new mixture. This one is relatively straightforward, so we know that the total volume will equal 7.5 gallons. Since we started with 5 gallons, and all we’re adding is water to get to 7.5 gallons, we need to add 2.5 gallons of water and there’s your correct answer.

Regardless of the question type or form, when you see one of these concentrated liquids problems make sure you find out how much “stuff” you have! The rest will be water and you’ll be well on your way to a correct answer!