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Mustafa Bilgic · Physics & Engineering Calculator Specialist · Updated 20 Feb 2026

Resistors in Series and Parallel: Complete GCSE Guide

Understanding how resistors combine is fundamental to circuit analysis at GCSE and A-Level. The behaviour of current and voltage differs completely depending on whether components are connected in series (one after the other) or parallel (side by side).

Resistors in Series

When resistors are connected end-to-end in a single loop, they are in series. The same current flows through every resistor, and the supply voltage is shared between them.

Rtotal = R1 + R2 + R3 + …

The voltage drop across each resistor: Vn = I × Rn, where I = Vsupply / Rtotal.

V+ ──── R1 ──── R2 ──── R3 ──── V– (3Ω) (5Ω) (2Ω) R_total = 3 + 5 + 2 = 10 Ω At 10V: I = 10/10 = 1A throughout

Resistors in Parallel

When resistors are connected across the same two nodes, they are in parallel. The same voltage appears across every resistor, and current splits between branches.

1/Rtotal = 1/R1 + 1/R2 + 1/R3 + …

Shortcut for exactly two resistors:

Rtotal = (R1 × R2) / (R1 + R2)

Example: Two 10 Ω resistors in parallel: R = (10×10)/(10+10) = 100/20 = 5 Ω. The total is always less than the smallest individual resistor.

Mixed (Series-Parallel) Circuits

Complex circuits combine series and parallel elements. Solve them by working from the inside out: (1) Identify parallel groups and calculate their equivalent resistance. (2) Treat each group as a single series resistor. (3) Apply Ohm's Law to find total current, then work backwards for each branch.

Kirchhoff's Laws

Kirchhoff's Current Law (KCL): The sum of currents entering a node equals the sum leaving. Itotal = I1 + I2 + I3.

Kirchhoff's Voltage Law (KVL): The sum of all voltage changes around any closed loop equals zero. Vsupply = VR1 + VR2 + …

E12 and E24 Preferred Value Series

Resistors are manufactured in standard preferred values. The E12 series (±10% tolerance) includes 12 values per decade: 10, 12, 15, 18, 22, 27, 33, 39, 47, 56, 68, 82 — and their multiples (100, 120, 150 … 820, 1k, 1.2k …). The E24 series (±5%) doubles this to 24 values. When a circuit calculation gives a non-standard value, choose the nearest E12 or E24 value.

Resistor Colour Code Quick Reference

Resistor values are often marked with colour bands. Use the Resistor Colour Code Calculator for full decoding. Memory aid: "Bad Boys Race Our Young Girls But Violet Generally Wins" = Black(0), Brown(1), Red(2), Orange(3), Yellow(4), Green(5), Blue(6), Violet(7), Grey(8), White(9).

Worked GCSE Example — Series

Q: Resistors of 3 Ω, 5 Ω and 2 Ω are connected in series to a 10 V supply. Find total resistance, current and voltage across each resistor.
  1. Rtotal = 3 + 5 + 2 = 10 Ω
  2. Total current: I = V/R = 10/10 = 1 A
  3. V across 3 Ω: 1 × 3 = 3 V
  4. V across 5 Ω: 1 × 5 = 5 V
  5. V across 2 Ω: 1 × 2 = 2 V
  6. Check: 3 + 5 + 2 = 10 V ✓ (KVL satisfied)

Worked GCSE Example — Parallel

Q: Resistors of 6 Ω and 12 Ω are connected in parallel to a 12 V supply. Find total resistance and current in each branch.
  1. Rtotal = (6 × 12)/(6 + 12) = 72/18 = 4 Ω
  2. Total current: I = 12/4 = 3 A
  3. Branch 1 (6 Ω): I1 = 12/6 = 2 A
  4. Branch 2 (12 Ω): I2 = 12/12 = 1 A
  5. Check: 2 + 1 = 3 A total ✓ (KCL satisfied)

Voltage Divider Rule

In a series circuit, the voltage across resistor Rn is: Vn = Vsupply × Rn / Rtotal. This is the voltage divider rule — used in sensors, audio circuits, and biasing transistors.

Frequently Asked Questions

What is the formula for resistors in series?
Rtotal = R1 + R2 + R3 + … The total resistance is the sum of all individual resistances. The same current flows through every component. Adding more resistors in series always increases total resistance. Example: 10 Ω + 22 Ω + 47 Ω = 79 Ω total.
What is the formula for resistors in parallel?
1/Rtotal = 1/R1 + 1/R2 + 1/R3. Invert the sum of reciprocals to get Rtotal. For two resistors: R = (R1 × R2)/(R1 + R2). The total resistance is always less than the smallest individual resistor. Adding more parallel branches always decreases total resistance — each extra path makes it easier for current to flow.
Why is parallel resistance always less than the smallest resistor?
Every additional parallel branch creates a new path for current. More paths mean less total opposition to current flow. Mathematically, because 1/Rtotal = 1/R1 + 1/R2, and all terms are positive, Rtotal must be smaller than either R1 or R2 individually. Physically: connecting a second wire in parallel alongside a thin wire provides more conductance (= 1/R), so resistance falls.
How do I find current through each branch in a parallel circuit?
Using Kirchhoff's current law and Ohm's Law: each branch sees the same voltage V. For branch n: In = V / Rn. The branch with the smallest resistance carries the most current. Total current = sum of all branch currents = V / Rtotal. Example with 12 V across 4 Ω and 6 Ω: I1 = 12/4 = 3 A, I2 = 12/6 = 2 A, total = 5 A, Rtotal = 12/5 = 2.4 Ω.
What are E12 and E24 preferred resistor value series?
Standard resistors are manufactured to preferred values within tolerance bands. E12 (±10%): 12 values per decade — 10, 12, 15, 18, 22, 27, 33, 39, 47, 56, 68, 82. E24 (±5%): 24 values per decade, filling gaps between E12 values. E96 (±1%): 96 values per decade for precision work. When designing circuits, always choose the nearest standard value to your calculated requirement.
What is Kirchhoff's Voltage Law (KVL)?
KVL states: the algebraic sum of all voltages around any closed loop equals zero. In practice for a series circuit: Vsupply = VR1 + VR2 + VR3. You can use KVL to verify calculations — if your voltage drops don't sum to the supply voltage, something is wrong. KVL is one of the two fundamental laws (along with KCL) for solving circuit networks.
How do I analyse a mixed series-parallel circuit?
Follow these steps: (1) Identify groups of purely parallel resistors and replace each group with its equivalent resistance. (2) The circuit is now fully series — add all equivalent resistances. (3) Use Ohm's Law to find total current. (4) Work backwards: at each parallel group, the voltage across the group is Iseries × Rgroup_equiv. Then apply Ibranch = Vgroup / Rbranch to find individual branch currents.

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Official Sources

Data verified against official UK government sources. Last checked April 2026.