Resistors combine differently depending on how they are wired. In series they add. In parallel they combine to less than the smallest one. Mixed circuits are solved by collapsing one group at a time until a single equivalent resistance remains.
Series: simply add
For resistors in series, the same current flows through each one, and total resistance R = R1 + R2 + R3 and so on. Example: 10 + 20 + 30 = 60 ohms. Series always increases resistance, and the total is larger than the largest single resistor.
A series chain is also a voltage divider: the voltage splits across resistors in proportion to their values. The 30-ohm resistor in our example takes half the total voltage, because 30 is half of 60.
Parallel: reciprocals
For resistors in parallel, the voltage is the same across each, and 1/R = 1/R1 + 1/R2 + 1/R3 and so on. Example with two resistors, 10 and 20 ohms: 1/R = 1/10 + 1/20 = 3/20, so R = 20/3 = 6.67 ohms. Note the total is smaller than the smallest resistor, 10 ohms. That surprises beginners but it is correct: parallel paths give current more routes, so resistance falls.
For exactly two resistors there is a shortcut: R = (R1 x R2) / (R1 + R2). For 10 and 20 ohms: 200 / 30 = 6.67 ohms. For three or more, use the reciprocal method.
Worked example: three in parallel
Resistors of 10, 20, and 30 ohms in parallel: 1/R = 1/10 + 1/20 + 1/30. As fractions over 60: 6/60 + 3/60 + 2/60 = 11/60. So R = 60/11 = 5.45 ohms. Check: the result is below the smallest resistor (10), which is the expected pattern for parallel combinations.
Mixed example: a 10-ohm and a 20-ohm resistor in parallel (6.67 ohms) in series with a 30-ohm resistor gives 36.67 ohms. Solve the parallel group first, replace it with its equivalent, then add the series part.
Mistakes to avoid
The classic error is adding parallel resistors directly, giving 30 ohms instead of 6.67 for the 10 and 20 pair. The second is forgetting that the reciprocal method gives 1/R, not R, so you must invert at the end. The third is mixing units: convert kilohms to ohms before combining, or keep everything in kilohms consistently. And always sanity-check: a parallel total below the smallest resistor, a series total above the largest.
Voltage dividers and current dividers
Series resistors divide voltage. In a divider with R1 = 10 ohms and R2 = 20 ohms across 12 volts, the voltage across R2 is 12 x 20 / (10 + 20) = 8 volts, and across R1 it is 4 volts. The voltages split in proportion to resistance, and they always add up to the supply voltage.
Parallel resistors divide current the other way: the smaller resistor takes the larger share. With 10 and 20 ohms in parallel fed by 3 amps total, the 10-ohm branch takes 3 x 20 / (10 + 20) = 2 amps, and the 20-ohm branch takes 1 amp. Dividers are the building blocks of sensors, volume controls, and bias circuits.
One more combination worth knowing: equal resistors in parallel divide neatly. Two equal resistors in parallel give half of one resistor’s value; three give a third; four give a quarter. Four 100-ohm resistors in parallel give 25 ohms. This shortcut is handy for checking a reciprocal calculation at a glance before trusting it.
Try it faster with the tool
You can follow every step above by hand, but the free Online Series and Parallel Circuit Resistance Calculator does the arithmetic for you. Enter your numbers and it gives the answer right away, so you can check your hand calculation or skip straight to the result.
Quick tips
- Series: add the values.
- Parallel: add the reciprocals, then invert.
- Parallel total is always below the smallest resistor.