I'm doing some calculations for a chemistry experiment and have decided that I need solutions of 1%, 5%, 10%, 20% and 40% consisting of x mL of ethanol and y mL of solution y. Are there any possible methods for finding a combination of x and y to give me nice, whole numbers or half numbers? As of now, I've only been doing trial and error and it's doing my head in...
eg. 180 mL solution y:
$$180 mL + 120 mL \text{ ethanol } = 40\text{%}$$ $$180 mL + 45 mL \text{ ethanol } = 20\text{%}$$ $$180 mL + 20 mL \text{ ethanol } = 10\text{%}$$ $$180 mL + 9.47 mL \text{ ethanol } = 5\text{%}$$ $\rightarrow \text{this is where it starts to get messy... is there any faster method to find a solution?}$
Say you want the concentration to be $p\%$ ethanol, then $$\frac{x}{x+y}=p\%$$ You can solve for $x$ to get $$x=\frac{p\%}{1-p\%}y$$
Example
With $p=60$ and $y=180$, you have $$x=\frac{0.60}{1-0.60}\cdot 180=270$$
$$\begin{array} {|r|r|}\hline y & p & x=\frac{p\%}{1-p\%}y \\ \hline 180 & 99 & 17820 \\ \hline 180 & 90 & 1620 \\ \hline 180 & 80 & 720 \\ \hline 180 & 70 & 420 \\ \hline 180 & 60 & 270 \\ \hline 180 & 50 & 180 \\ \hline 180 & 40 & 120 \\ \hline 180 & 30 & 77.\overline{142857} \\ \hline 180 & 20 & 45 \\ \hline 180 & 10 & 20 \\ \hline \end{array}$$