You have 10 cups of a cleaning solution that is 10% white vinegar and 90% water. You want the mixture to be 50% white vinegar and 50% water. How many cups of white vinegar do you need to add to the mixture?____ cups

whatever% of anything is just (whatever/100) * anything.

let's say we'll add "x" cups of pure vinegar, pure vinegar meaning is 100% vinegar.

how much vinegar is there in "x" cups of 100% vinegar?  well (100/100) * x, or just "x".

how much vinegar is there in 10 cups of 10% vinegar?  well, (10/100) * 10, or just 1.

say after adding the "x" cups to the 10 cups, we end up with "y" cups, of 50% of vinegar.

how much is 50% of y?  well, (50/100) * y or 0.5y.

bearing in mind that x+10 = y, both cups amounts added up, and that x+1=0.5y, both vinegar amounts added up, thus 

[tex]\bf \begin{array}{lccclll} &\stackrel{cups}{amount}&\stackrel{vinegar~\%}{amount}&\stackrel{vinegar~cups}{quantity}\\ \textit{100\% vinegar}&x&1.00&x\\ \textit{10\% cups}&10&0.10&1\\ ------&------&------&------\\ mixture&y&0.5&0.5y \end{array} \\\\\\ \begin{cases} x+10=\boxed{y}\\ x+1=0.5y\\ ----------\\ x+1=0.5\left( \boxed{x+10} \right) \end{cases} \\\\\\ x+1=0.5x+5\implies 0.5x=4\implies x=\cfrac{4}{0.5}\implies x=8[/tex]