Calculating resistances in parallel

Question

I was playing around with a question involving resistances in parallel. I realized that $n$ equal resistors (resistance $R$) in parallel have combined resistance $R/n$. I proved this using (mathematical) induction. Then I considered two resistors with ratio $1:2$ in parallel, e.g. a $3\Omega$ and $6\Omega$ resistor in parallel. Since $3=6/2$, we can view the $3\Omega$ resistor as two 6 ohm resistors in parallel. So the problem boils down to finding the resistance of three $6\Omega$ resistors in parallel, which is $6/3=2$.

So this law can be used to make resistances easy to calculate. Above we just easily calculated the combined resistance of a $3\Omega$ and $6\Omega$ resistor in parallel. Yes, we could also have done $(3*6)/(3+6)$, but I think what I did is much easier.

Question: Is this algorithm known/is there a general version of it?

Answer 1

In general the effective resistance of resistors in parallel can be calculated with,

$$ R_{eff} = \frac{1}{\sum{\frac{1}{R_i}}}. $$

When the ratio between any of the resistors, $R_i$, is an irrational number it will not be possible to use the method you mentioned. For methods, which can be used to calculate something, the most robust method is usually preferred because it works in most, if not all cases. Even if every ratio between resistors can be written as factions of two integers, it might take more time finding those than it would to use the above equation. So the method you mentioned will only have limited application.

The method you mentioned basically factors out common factors. In mathematics with would be equivalent to prime factorization/greatest common divisor. But again in general this will often take more time than using the equation above.