In parallel circuits, the same voltage is applied to every branch, and the current from the source splits among those branches. The total current leaving the source is the sum of the currents through each branch, because at the junction where branches meet, currents recombine and Kirchhoff’s current law applies. Each branch carries a current I_i = V / R_i, since the branch voltage is V. Adding up all branch currents gives It = I1 + I2 + …. For a quick check, if you have 12 V across two branches with resistances of 6 Ω and 3 Ω, the branch currents are 2 A and 4 A, respectively, so the total current is 6 A. The equivalent resistance is 12 V / 6 A = 2 Ω, which also satisfies 1/R_eq = 1/6 + 1/3. The voltage across each branch is equal to the source voltage, not greater. The idea of dividing the source voltage by the sum of resistors applies to a single path, not to a parallel network, where the current distribution depends on each branch’s resistance.