How many cups of white vinegar do you need to add to the mixture?

Question

You have 10 cups of a cleaning solution that is 10% white vinegar and 90% water. You want the mixture to be 20% white vinegar and 80% water

Answer 1

Kris, the easiest way I find to think about these problems is to figure out how much of the original solution is "pure" white vinegar, and how much is "pure" water.

 

If I know the percentage, I can multiply that by the total to get the part that is just white vinegar:

 

10 cups is 10% white vinegar. 10% is 0.1, so I multiply 0.1 * 10 cups = 1 cup white vinegar.

10 cups is 90% water. 90% is 0.9, so I multiply 0.9 * 10 cups = 9 cups water.

 

That makes sense, because 1 cup vinegar + 9 cups water is 10 cups total solution.

 

We don't know how much white vinegar to add. Let's call that x.

 

1+x will be the total vinegar in the new solution (1 cup original white vinegar, plus another x cups white vinegar)

10 + x will be total volume of the new solution (10 cups original solution, plus another x cups white vinegar)

20% is the proportion that should be white vinegar (20/100). Let's divide the amount of white vinegar (1+x) by the total volume of the new solution (10+x), and that should be equal to the proportion (20/100).

(1+x) / (10 + x) = 20/100

 

I like to solve equations like this by simplifying 20/100 to 1/5, then we can multiply both sides by the least common denominator (5)*(10+x) to solve.

 

Are there any steps of this that I can help to clarify for you?