Proving the formula for parallel resistors
It might be easier to understand how it works if you use the inverse of resistance - conductance.
In a parallel circuit each component has the same voltage across it, so the current flowing through it is independent of the others. Total current is the sum of the individual currents, so the total conductance is simply the sum of the individual conductances.
Take the example below:-



simulate this circuit &ndash; Schematic created using CircuitLab
R1 has a conductance of 1 / 1 Ω = 1 A/V, R2 is 1 / 2 Ω = 0.5 A/V, and R3 is 1 / 4Ω = 0.25 A/V. Add them all together to get a total conductance of 1.75 A/V, then invert the result to get the total resistance of 1 / 1.75 = 0.571Ω.
This gives the general formula for parallel resistances:-
\$\frac{1}{Rt} = \frac{1}{R1} + \frac{1}{R2} + ...\frac{1}{Rn}\$
In the special case of only two resistors in parallel you can rearrange the formula from
\$\frac{1}{Rt} = \frac{1}{R1} + \frac{1}{R2}\$
to
\$Rt = \frac{R_1 R_2}{R_1 + R_2}\$