How Do You Calculate Total Resistance in Series and Parallel Circuits?

• French B
• Geography
• German A
• German A Lit
• German ab initio
• German B
• Global Politics
• History
• History (First Exam 2028)
• Mathematics Analysis and Approaches (AA)
• Mathematics Applications & Interpretation (AI)
• Philosophy
• Physics
• Psychology
• Psychology (First Exam 2027)
• Spanish A
• Spanish A Lit
• Spanish ab initio
• Spanish B
• Sports, exercise and health science (SEHS)
• Theory of Knowledge (TOK)
• World Religions

In series, the same current passes through every resistor, whereas the individual potential differences add to the supply potential difference. From $V=IR$, adding the potential differences produces the series rule.

In parallel, every branch has the same potential difference, but the branch currents add to the total current. Substituting $I=\frac{V}{R}$ into the current relationship produces the reciprocal rule.

Worked Examples

For three series resistors of $2.0\,\Omega$, $3.0\,\Omega$, and $5.0\,\Omega$:

For $6.0\,\Omega$ and $3.0\,\Omega$ in parallel:

Therefore, $R_{\text{total}}=2.0\,\Omega$. For exactly two parallel resistors, the shortcut is $R_{\text{total}}=\frac{R_1R_2}{R_1+R_2}$.

Now place a $4.0\,\Omega$ resistor in series with that parallel pair. The pair has an equivalent resistance of $2.0\,\Omega$, so the complete circuit has $R_{\text{total}}=4.0+2.0=6.0\,\Omega$. Simplify only resistors that genuinely share the same series path or the same two connection points.

For $n$ identical resistors of resistance $R$, the results provide useful shortcuts: series gives (nR), while parallel gives $\frac{R}{n}$. These also confirm that additional parallel branches reduce equivalent resistance.

A common misconception is to add parallel resistances directly. Parallel branches provide additional paths for charge carriers, so adding a branch decreases total resistance rather than increasing it.

Exam Technique

For a mixed circuit, simplify one identifiable section at a time. In a Calculate question, show the equation, substitution, and final answer in ohms. Apply the size checks in the table to detect an inverted reciprocal or an incorrectly classified arrangement.