To create an ink refresher for dye-based ink, some crafters will mix pure vegetable glycerine with water to get a mixture that is \(12\%\) vegetable glycerine, by volume. Kathy does not have pure vegetable glycerine, but she does have
a \(90\) mL mixture that is \(10.5\%\) vegetable glycerine,
a \(120\) mL mixture that is \(30\%\) vegetable glycerine, and
a \(1\) L mixture that is \(7.5\%\) vegetable glycerine.
Since Kathy is a math teacher, she knows she can use the contents of these three mixtures to create a mixture that is \(12\%\) vegetable glycerine, by volume. She combines the contents of the entire \(90\) mL mixture with the contents of the entire \(120\) mL mixture, and then adds some of the \(1\) L mixture. How many millilitres of the \(1\) L mixture should she add to create a new mixture that is \(12\%\) vegetable glycerine, by volume?
Let \(x\) be the number of millilitres needed from the \(1\) L mixture.
The \(90\) mL mixture that is \(10.5\%\) vegetable glycerine has \(0.105\times 90 = 9.45\) mL of vegetable glycerine.
The \(120\) mL mixture that is \(30\%\) vegetable glycerine has \(0.30\times 120 = 36\) mL of vegetable glycerine.
In the \(x\) mL from the \(1\) L mixture, there is \(0.075\times x =~0.075x\) mL of vegetable glycerine.
Therefore, the total amount of vegetable glycerine in the new mixture is \(9.45 + 36 + 0.075x = (45.45 + 0.075x)\) mL.
The new mixture contains \(90 + 120 + x =
(210+x)\) mL of liquid, of which \(12\%\) is vegetable glycerine.
Therefore, \(0.12\times (210+x) = (25.2 +
0.12x)\) mL of the new mixture is vegetable glycerine.
Since we have shown that the amount of vegetable glycerine in the new mixture is \((45.45 + 0.075x)\) mL and \((25.2 + 0.12x)\) mL, we must have $$\begin{align} 45.45 + 0.075x &= 25.2 + 0.12x \\ 0.075x-0.12x &= 25.2-45.45 \\ -0.045x &= -20.25 \\ x &= 450 \end{align}$$ Therefore, she should add \(450\) mL of the \(1\) L mixture that is \(7.5\%\) vegetable glycerine.