When a resistor is in series with a parallel network, the total resistance is equal to

• A. The product of the series resistor and the parallel network's resistance
• B. The sum of the series resistor and the parallel network's equivalent resistance
• C. The difference between them
• D. The quotient of them

Answer: (B)

Resistors in series add their resistances because the same current flows through every element and the total voltage is the sum of the drops across each part. When a resistor is in series with a parallel network, first treat the parallel portion as its equivalent resistance Rp. The current through the series resistor is the same as the current entering the parallel network, so the total voltage across the whole path is the sum of the voltage drops: Vr = Itotal × Rseries and Vp = Itotal × Rp. Therefore, the overall resistance seen by the source is Rt = Vt / Itotal = Rseries + Rp. For example, a 4-ohm resistor in series with a parallel network that has an equivalent resistance of 2 ohms yields a total of 6 ohms. The other options would imply multiplication, subtraction, or division, which don’t describe how a series resistor combines with a parallel network.