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]
