Compound Direct Current Circuits

Compound circuits are solved by collapsing them. Two elements are in parallel when they share both nodes, so they have one voltage and split the current; they are in series when they share one node that connects nothing else, so they have one current and split the voltage. Series resistances add, parallel conductances add, and a parallel equivalent always comes out smaller than every branch. Reduce a ladder from the far end inward, relabeling after each collapse, then work back out to recover each branch's current and drop.

Four errors dominate. Classifying elements from how the diagram is drawn instead of from the node list. Adding parallel resistances directly, or stopping at $1/R_{\text{eq}}$ and reporting it as $R_{\text{eq}}$. Collapsing two resistors as series when a third branch taps the node between them, which poisons every later step of the reduction. And splitting a source voltage evenly among unequal series resistors, when the drops go as $IR_i$ and are therefore in the ratio of the resistances. Behind several of them sits the intuition that more components must mean more resistance, when a parallel branch is another road and lowers $R_{\text{eq}}$ while raising the total current.