So yesterday, I posted a question but it seems I wrote it too confusingly. Now, I will simplified the question so that probably you have some opinion or suggestion.
Suppose, I have an function like below:
$\frac{a_{1_m}*a_{2_m}*A_m+b_{1_m}*b_{2_m}*B_m+c_{1_m}*c_{2_m}*C_m}{a_{1_n}*a_{2_n}*A_n+b_{1_n}*b_{2_n}*B_n+c_{1_n}*c_{2_n}*C_n}$
Given that, only the ratio of each corresponding variable is known, for example:
$\frac{a_{1_m}}{a_{1_n}}$ is known
$\frac{a_{2_m}}{a_{2_n}}$ is known
...
$\frac{C_m}{C_n}$ is known
Is it possible to estimate the value of the original function with that known information?
Thank you.
Let $\frac{a_{1m}}{a_{1n}}=r_1$
$\frac{a_{2m}}{a_{2n}}=r_2$
$\frac{A_{m}}{A_{n}}=r_3$
$\frac{b_{1m}}{b_{1n}}=r_4$....
$\frac{C_{m}}{C_{n}}=r_9$
Then, it would reduce to $$\frac{r_1r_2r_3A+r_4r_5r_6B+r_7r_8r_9C}{A+B+C}$$
Where $A=a_{1n}a_{2n}A_{n}$
So, if $r_1r_2r_3=r_4r_5r_6=r_7r_8r_9=r $ then it would reduce to $r$
else , it can't be reduced.