When the resistances are connected in parallel, the (voltage) potential difference is same across each resistor.
Consider three resistors $R_1$, $R_2$ and $R_3$ connected in parallel. All resistors have the same potential difference, but the current through each resistor is different.
The total current, $I$ flowing in the circuit is the sum of the current in different resistors. i.e.
$I = I_1+I_2+I_3$ .......... (i)
Let $V$ be the potential difference across each resistors and $I_1$, $I_2$, $I_3$ be the current passing through $R_1$, $R_2$ and $R_3$ respectively.
From the Ohm's law:
$V = I_1\,R_1$
$I_1 = \frac{V}{R_1}$ .......... (ii)
$I_2 = \frac{V}{R_2}$ .......... (iii)
$I_3 = \frac{V}{R_3}$ .......... (iv)
From the equation (i), (ii), (iii) and (iv), we get
$I = \frac{V}{R_1} + \frac{V}{R_2} + \frac{V}{R_3}$
$I = V(\frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3})$
$\frac{I}{V} = \frac{1}{R_p} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}$
Where, $\frac{1}{R_p}$ $ = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}$ be the equivalence resistance of the parallel combination.
The equivalent resistance of the parallel combination is equal to the sum of reciprocal of the resistance of individuals resistors.
The effective resistance in the parallel combination is smaller than the smallest resistance in the combination (Principle of PHYSICS).
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