What is the intuition behind the sum of the inverse values of the resistors in a parallel circuit? I wish to understand in the area of parallel circuits why this formula works:
$$\frac{1}{R_T} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \cdot\cdot\cdot+$$
in particular what is the meaning of $\frac{1}{R_x}$. I understand that it derives from
\begin{align}
I_T &= I_1 + I_2 + I_3 \\
\frac{V}{R_{t}} &= \frac{V}{R_1} + \frac{V}{R_2} + \frac{V}{R_3} \\
\frac{V}{R_{t}} &= V \left(\frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}\right)
\end{align}
and so I know how it works. But I'm trying to intuitively understand what is the meaning of the sum of the inverse values of the individual resistors in a parallel circuit. I'm not looking for a mathematical explanation.
 A: The resistance $R = \frac{U}{I}$.
The inversed resistance $G = \frac{1}{R} = \frac{I}{U}$ is the  electrical conductance .
Resistances are additive in serial scenarios, as voltages  at the same current are additive.
$$R_\mathrm{T} = \frac{U_\mathrm{T}}{I} = \frac {U_1 + U_2 + U_3}{I} = \frac{U_1}{I} + \frac{U_2}{I} + \frac{U_3}{I} = R_1 + R_2 + R_3$$
Conductances are additive in parallel scenarios, as currents at the same voltage are additive.
$$G_\mathrm{T} = \frac{I_\mathrm{T}}{U} =  \frac{I_1 + I_2 + I_3}{U} =  \frac{I_1}{U} + \frac{I_2}{U} + \frac{I_3}{U} = G_1 + G_2 + G_3$$
If there are parallel resistors 1 Ohm, 2  Ohm, 5  Ohm,  they have the respective conductances 1 S, 0.5 S, 0.2 S, with the summary conductance 1.7 S, which is equivalent to the resistance 1/1.7 Ohm.
For $n$ general resistors:
$$R_\mathrm{T} = \frac 1{G_\mathrm{T}} = \frac {1}{ \sum_{i=1}^{n} G_i } = \frac {1}{ \sum_{i=1}^{n} \frac{1}{R_i} }$$
It is valid even for other passive components like capacitors and inductors, if the respective complex arithmetic and generalized quantities impedance (complex resistance) and admittance (complex conductance) are involved.

Few analogies:

*

*Imagine your car fuel tank has a leaking hole with some flow resistance and gasoline flows out of the tank. Does it helps to slow down leaking if you punch out many other holes ?

*If the above tank is leaking at one place, does it help to totally stop leaking if it does not leak on other places at all ?

*An insulated wire is a parallel connection of the wire and its insulation. If their resistance were additive, you could not use insulated wires to conduct electricity.

