The Memory Trick
'Series Stacks, Parallel Splits'
Resistors in SERIES stack up like cars in a single-lane road: every car (charge) must pass through each one, so resistances add. Resistors in PARALLEL split the traffic into extra lanes: more paths means LESS total resistance, so reciprocals add. Series shares the same CURRENT; parallel shares the same VOLTAGE.
1
Series
R_total = R₁ + R₂ + R₃ + … The same current flows through each resistor, and the total voltage divides in proportion to resistance (V = IR for each).
2
Parallel
1/R_total = 1/R₁ + 1/R₂ + 1/R₃ + … Every branch has the same voltage, and current divides inversely to resistance (the smallest resistor carries the most current). The total is always LESS than the smallest resistor.
Shortcuts: two resistors in parallel = (R₁R₂)/(R₁ + R₂); n identical resistors R in parallel = R/n.
3
Reduce step by step
For combination circuits, find groups that are purely in series or purely in parallel, replace each with its equivalent, redraw, and repeat until one resistor remains.
4
Expand back out
Use Ohm's law on the single equivalent resistor to get the total current, then work backward through your redrawn circuits to find each resistor's current and voltage. Check that the voltages around the loop add up to the source voltage.
🏥 Worked Example
A 12 V battery drives a 6 Ω resistor in series with a parallel pair of 3 Ω and 6 Ω resistors. Find the total resistance, the battery current, and the current through each resistor in the parallel pair.
STEP 1
Collapse the parallel pair — (3 × 6)/(3 + 6) = 18/9 = 2 Ω.
STEP 2
Add the series resistor — R_total = 6 Ω + 2 Ω = 8 Ω, so the battery current is I = 12 V / 8 Ω = 1.5 A.
STEP 3
Find the voltage across the parallel pair — V = (1.5 A)(2 Ω) = 3 V, and the 6 Ω series resistor drops (1.5 A)(6 Ω) = 9 V — together 12 V, which checks out.
CONCLUDE
Each branch of the parallel pair has 3 V across it: the 3 Ω branch carries 1.0 A and the 6 Ω branch carries 0.5 A. They add back to 1.5 A, confirming the answer.
⚠️ Most Common Resistor Combinations Mistakes
Adding parallel resistances directly (instead of reciprocals) — and forgetting to flip the final 1/R_total back to R_total — are the two most common errors. Sanity check: a parallel total must be smaller than the smallest branch.
✓ Quick Self-Test
1) What is the formula for resistors in series? In parallel? 2) What stays the same across all resistors in series? In parallel? 3) What is the shortcut for two resistors in parallel? 4) What happens to total resistance when you add another resistor in parallel? 5) Find the equivalent of three 9 Ω resistors in parallel.