How To Solve a Ratio Equation

Question

This question actually comes from backyard composting. When choosing materials to compost the ideal ratio of Carbon to Nitrogen of the raw materials to make compost is 30:1. There are basically no materials that are 30:1 naturally, so you have to work with what you have. For instance, grass clippings are 20:1 and wood chips are 400:1.

My question is how to you determine the quantity $x_1$ of a material with a $C:N$ ratio $r_1$ as a function of the quantity $x_2$ of a second material with $C:N$ ratio of $r_2$ where the sum of the two materials will be $r_{1+2}$, in this case $30:1$?

Answer 1

If you have $x_2$ with $\frac CN=r_2$, you get $\frac {x_2r_2}{1+r_2}$ of $C$ and $\frac {x_2}{1+r_2}$ of $N$ and similarly with subscripts $1$. The ratio of the mix is then $r_{1+2}=\frac {x_2r_2(1+r_2)+x_1r_1(1+r_1)}{x_2(1+r_2)+{x_1(1+r_1)}}$ Now solve for $x_1$ and you have what you need.

Answer 2

Another approach could be used considering the composition of the two elements to be mixed. In the first component (we shall use it a s the reference) the fraction of $C$ is given by $\frac {a_1}{a_1+b_1}=\frac {r_1}{r_1+1}$ ($r_1=\frac {a_1}{b_1}$) and the fraction of $N$ is given by $\frac {b_1}{a_1+b_1}=\frac {1}{r_1+1}$. The same for the second component (change $1$ to $2$).

Now consider that you mix $1$ quantity of the first component (this is your $x_1$) and a quantity $x$ of the second component (this is your $x_2$). You then have $(1+x)$ of mixture which contains $$C=\frac {r_1}{r_1+1}+x\frac {r_2}{r_2+1}$$ and $$N=\frac {1}{r_1+1}+x\frac {1}{r_2+1}$$ and you want that the ratio of this last quantities to be equal to $r_3$. So, $$r_3=\frac{\frac {r_1}{r_1+1}+x\frac {r_2}{r_2+1}} {\frac {1}{r_1+1}+x\frac {1}{r_2+1} }$$ from which can be extracted $$x=\frac{({r_2}+1) ({r_3}-{r_1})}{({r_1}+1) ({r_2}-{r_3})}$$