How to solve this parameter finding problem?

Question

I have the following equalities:

$A_1=R_1\frac{D_1}{D_1+D_2+D_3+\cdots+D_N}B_T$

$A_2=R_2\frac{D_2}{D_1+D_2+D_3+\cdots+D_N}B_T$

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$A_N=R_N\frac{D_N}{D_1+D_2+D_3+\cdots+D_N}B_T$

Let the values of $D_1$, $D_2$, $D_3$ ... $D_N$ and $B_T$ are given.

I need to find the values of $R_1$, $R_2$, $R_3\cdots R_N$ such that

$A_1:A_2:A_3:\cdots:A_N=\rho_1:\rho_2:\rho_3:\cdots:\rho_N$ where $\rho_1$, $\rho_2$, $\rho_3\cdots\rho_N$ are known

Answer 1

The equation of $A_i$ can be written as, $$ A_i=R_i\frac{D_i}{D_1+D_2+D_3+\cdots+D_N}B_T$$ $$=>R_i=\frac{D_1+D_2+D_3+\cdots+D_N}{D_i*B_T}A_i$$

for $i=1,2,3.....N$.

And as all $A_i$s are present in ratio so they can be written as, $$A_1=\rho_1*x ,A_2=\rho_2*x,.............,A_N=\rho_N*x$$ Adding all these equations of $A_i$s we get, $$A_1+A_2+A_3+....+A_N=(\rho_1+\rho_2+\rho_3+....+\rho_N)*x $$ As $A_1+A_2+A_3+...+A_N=B_T$ (you said this in the comment) so, $$x=\frac{B_T}{\rho_1+\rho_2+\rho_3+....+\rho_N}$$ and hence for all $A_i$s , $$A_i=\rho_i*x$$ $$=> A_i=\frac{\rho_i*B_T}{\rho_1+\rho_2+\rho_3+....+\rho_N}$$ Putting these value of $A_i$ in equation of $R_i$ (see at the top)so, $$R_i=\frac{D_1+D_2+D_3+\cdots+D_N}{D_i*B_T}*\frac{\rho_i*B_T}{\rho_1+\rho_2+\rho_3+....+\rho_N}$$ $$=>R_i=\frac{\rho_i}{D_i}*\frac{D_1+D_2+D_3+\cdots+D_N}{\rho_1+\rho_2+\rho_3+....+\rho_N}$$ In this all values are known so this solves the problem.....

Answer 2

First note that $$A_i=R_i D_i\frac{A_1+A_2+\cdots +A_n}{D_1+D_2+\cdots +D_n}$$ After applying some algebra, we have $$R_i=\frac{A_i\left(D_1+D_2+\cdots +D_n\right)}{D_i\left(A_1+A_2+\cdots +A_n\right)}$$ Since $$A_1:A_2:\dots:A_n=\rho_1:\rho_2:\dots:\rho_n$$ Then $$A_i\propto \rho_i\Rightarrow A_i=k\rho_i$$ $$\text{such that}\ k\in\mathbb R, k\neq 0$$ Therefore $$R_i=\frac{k \rho_i \left(D_1+D_2+\cdots +D_n\right)}{D_i k\left(\rho_1+\rho_2+\cdots +\rho_n\right)}$$ $$=\frac{\rho_i \left(D_1+D_2+\cdots +D_n\right)}{D_i\left(\rho_1+\rho_2+\cdots +\rho_n\right)}$$ This solution assumes that $$D_i\sum\limits_{j=1}^n \rho_j\neq 0$$