In a parallel resistor network, the total resistance is which of the following?

• A. It is greater than the largest resistance.
• B. It is equal to the largest resistance.
• C. It is equal to the sum of all the resistances.
• D. It is less than the smallest resistance.

Answer: (D)

When resistors share the same voltage in parallel, more paths for current open up, so the total current increases for a given voltage. Since the total current is the sum of the branch currents and the voltage across every branch is the same, the equivalent resistance must shrink. Mathematically, 1/R_eq = sum(1/R_i), which means R_eq = 1 divided by that sum. This value is always smaller than the smallest individual resistor, and it gets even smaller as you add more parallel paths. For two resistors in parallel, R_eq = (R1*R2)/(R1+R2), which is deliberately less than min(R1, R2). So the total resistance in a parallel network is less than the smallest resistance.