Series and parallel
Two resistors, two arrangements, two completely different circuits. One adds up the way you would expect and the other gets smaller as you add parts — which is the rule people refuse to believe until they watch it happen.
Switch between the two and drag either resistor. Watch what happens to the total when you add resistance in parallel.
Series: one path
Everything is in a line, so there is one current and it is the same at every point in the loop. The resistors add: R = R₁ + R₂. The supply voltage is then spent across them in proportion — the bigger resistor takes the bigger share.
That sharing is the whole of the voltage divider, and it is also why an LED and its resistor work: the LED takes about 2 V and the resistor is sized to eat whatever is left.
Parallel: two paths
Both parts see the full supply voltage, and the current splits between them. The total resistance is always smaller than the smallest one, because you did not add an obstacle, you added another way through.
The shortcut worth memorising. Two equal resistors in parallel are half of one. 10 kΩ ∥ 10 kΩ = 5 kΩ. That covers most of what you will meet, and the full formula — R₁R₂ ÷ (R₁+R₂) — covers the rest.
Where this bites in real builds
Three I2C modules on one bus. Every breakout carries its own pair of pull-up resistors, and they are all in parallel. Three modules with 4.7 kΩ pull-ups is 1.6 kΩ, so the bus draws three times what it should, and the third module is the one that "broke everything". Remove the on-board resistors from all but one.
LEDs in parallel with one resistor. Do not. The two LEDs are never identical, the one with the slightly lower forward voltage hogs the current, and it gets brighter and hotter until it fails. One resistor per LED.
Batteries. In series the voltages add and the capacity does not — four 1.5 V cells make 6 V at the same mAh. In parallel the capacity adds and the voltage does not, which is how power banks get to 20,000 mAh at 3.7 V.
Reading a schematic quickly
Trace one loop with a finger. If your finger passes through both parts without choosing, they are in series. If it reaches a junction where it could go two ways and both routes rejoin, that is parallel.
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