Series vs parallel resistance: formulas and intuition
How resistors combine in series and in parallel, why the totals move in opposite directions, and how to reduce a mixed network step by step.
Series: resistances simply add
When resistors are wired end to end in a single path, the same current flows through every one of them. Their resistances add directly, so the total is R1 + R2 + ... + Rn. Because you are only ever adding positive numbers, a series total is always larger than any individual resistor. This is the intuition behind the series mode here: three resistors of 100, 200 and 300 ohm in series give a straightforward 600 ohm.
Parallel: reciprocals add
When resistors sit side by side across the same two nodes, the current splits among them and each offers an extra route for it to flow. The conductances (the reciprocals of resistance) add, which is why the total obeys 1 / R_total = 1/R1 + 1/R2 + ... Inverting that sum gives a value smaller than the smallest resistor in the group. For two resistors there is a handy shortcut, R_total = (R1 x R2) / (R1 + R2), which for 60 and 40 ohm gives 2400 / 100 = 24 ohm.
Why the totals move in opposite directions
It can feel backwards that adding resistors in parallel lowers the resistance, but it follows from what each configuration does to the current. In series you force the current through more obstacles in a row, so resistance climbs. In parallel you open more lanes for the current to travel at once, so overall resistance drops. Keeping this mental picture makes it easy to sanity check a result: a parallel answer larger than your smallest resistor, or a series answer smaller than your largest, means something went wrong.
Reducing a mixed network
Most real circuits are neither purely series nor purely parallel but a mix of both. The way to handle them is to work from the inside out. Find a group that is clearly all in series or all in parallel, replace it with its single equivalent value using this calculator, then redraw the circuit with that simpler value in place. Repeat, collapsing one group at a time, until a single resistance remains. This step-by-step reduction turns an intimidating network into a short sequence of easy calculations.