Resistors in Series and Parallel

In one sentence: Resistors in a line add up, and resistors side by side always give a total smaller than the smallest one.

$R_{\text{series}} = R_1 + R_2 + \cdots + R_n \qquad \frac{1}{R_{\text{parallel}}} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots + \frac{1}{R_n}$

Read it out loud: “R series equals R 1 plus R 2 plus all the way to R n. R series is the total resistance in series. One over R parallel equals one over R 1 plus one over R 2 plus all the way to one over R n, where R parallel is the total resistance in parallel.”

Before you start

The ideas and signs this page uses, explained from zero. For powers of ten, Greek letters and other notation in general, see Math basics.

Subscripts and the three dots $\cdots$: A subscript is a small label written under a letter. $R_1$ is read R one, and it means the first resistor. $R_2$ is the second resistor, and $R_n$ is the last of $n$ resistors, where $n$ is the number of resistors in the group. The three dots, $\cdots$ , mean ‘and so on’ for the items in between, so $R_1 + R_2 + \cdots + R_n$ adds every resistor in the list.
Reciprocal, or one over a number: The reciprocal of a number is one divided by that number, written $\frac{1}{x}$ and read ‘one over x’. The reciprocal of 4 is $\frac{1}{4} = 0.25$ . The reciprocal of 0.25 is 4 again, so flipping twice returns the starting number. The parallel formula uses reciprocals of each resistance, flips the sum at the end, and so gets its answer back in ohms.
Adding fractions with a common bottom: To add fractions, you first write them over the same bottom number. For $\frac{1}{100} + \frac{1}{300}$ , multiply the top and bottom of $\frac{1}{100}$ by 3 to get $\frac{3}{300}$ . Then $\frac{3}{300} + \frac{1}{300} = \frac{4}{300}$ . The bottom number stays the same, and only the top numbers are added together.
Resistance, the ohm ( $\Omega$ ) and the resistor: Resistance is how strongly a part opposes electric current, and it is measured in ohms, written $\Omega$ . One ohm means one volt drives one ampere through the part. A resistor is a small part made to have a set resistance, such as $100\ \Omega$ or $300\ \Omega$ , and it is used to limit current in a circuit.
Conductance and the siemens (S): The reciprocal of a resistance, $\frac{1}{R}$ , is called conductance. It says how easily current gets through a part, and its unit is the siemens (S). One siemens equals one ampere per volt, which is the same as one per ohm. A resistance of 100 $\Omega$ has a conductance of $\frac{1}{100} = 0.01$ S, and parallel resistors add their conductances.
Series and parallel paths: A series connection puts parts one after another on a single path, so the same current passes through each part in turn. A parallel connection joins parts to the same two points, so each part sees the same voltage across it, and the current splits between the branches. Tracing the wires from one end of a group to the other shows which connection you have.

What each symbol means

Symbol	Meaning
$R_1$	The resistance of the first resistor in the group, measured in ohms ( $\Omega$ ). The small 1 is a subscript, a label that tells this resistor apart from the others. The example uses $R_1 = 100\ \Omega$ .
$R_2$	The resistance of the second resistor in the group, measured in ohms ( $\Omega$ ). The subscript 2 tells it apart from $R_1$ . The example uses $R_2 = 300\ \Omega$ , which is three times the first resistor.
$R_n$	The resistance of the last resistor in the group, measured in ohms ( $\Omega$ ). The letter $n$ counts how many resistors there are, so with four resistors, $R_n$ is $R_4$ .
$n$	A whole number that counts how many resistors are in the group. It is not a resistance and has no unit. If there are three resistors, $n = 3$ , and the formulas add or add reciprocals of three values.
$R_{\text{series}}$	The total resistance when the resistors are connected in series, measured in ohms ( $\Omega$ ). It is one equivalent resistor that would let the same current pass as the whole series group.
$R_{\text{parallel}}$	The total resistance when the resistors are connected in parallel, measured in ohms ( $\Omega$ ). It is one equivalent resistor that would let the same total current pass at the same voltage as the whole parallel group.
$\cdots$	Three dots that mean ‘and so on’. They stand for the resistors in the middle of a long list, so $R_1 + R_2 + \cdots + R_n$ means add every resistor from the first to the last, without writing each one out.

The formula piece by piece

$R_1$: Start with one resistor. $R_1$ is its resistance in ohms, a fixed number for that part. Every formula on this page adds or reciprocates this number as one of its inputs.
$R_1 + R_2$: Two resistors placed in series. Their resistances add, because the same current has to push through both, so the opposition builds up one resistor after another.
$\frac{1}{R_1}$: The conductance of one resistor, in siemens. It is one divided by the resistance, and it measures how easily current gets through that resistor.
$\frac{1}{R_1} + \frac{1}{R_2}$: Two resistors placed in parallel. Their conductances add, because each branch gives the current one more route to flow through. The sum runs on to $\cdots + \frac{1}{R_n}$ for more resistors.
$R_{\text{series}} = R_1 + R_2 + \cdots + R_n$: The series rule in one sentence: the total resistance of resistors in a line is the sum of their resistances. Adding one more resistor in series always increases the total.
$\frac{1}{R_{\text{parallel}}} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots + \frac{1}{R_n}$: The parallel rule in one sentence: the reciprocal of the total resistance equals the sum of the reciprocals of the resistors, so the total is found by adding conductances and then flipping the answer. The total is always smaller than the smallest resistor.
$R_{\text{series}} = R_1 + R_2 + \cdots + R_n \qquad \frac{1}{R_{\text{parallel}}} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots + \frac{1}{R_n}$: The whole formula in one sentence: resistances in series add together, and for resistors in parallel you add their reciprocals and then take the reciprocal of that sum to get the total resistance. Both rules are needed, and each one applies to a different way of wiring.
$R_{\text{parallel}} = \frac{R_1 R_2}{R_1 + R_2}$: A standard shortcut for exactly two resistors in parallel. It is the same parallel rule, worked out in one step: multiply the two resistances and divide by their sum. It does not work for three or more resistors.

What the formula is doing, and why

A resistor limits current. When several resistors are wired together, you can replace the whole group with one equivalent resistor that behaves the same way from outside. The total is what you would measure between the two ends of the group. There are two basic ways to connect resistors: in series and in parallel.

In series, the resistors sit one after another on a single path, so the same current flows through each one. Why add? Each resistor makes the current work harder to get through, and the extra opposition from each one stacks up along the path. Two 100 ohm resistors in series act like one 200 ohm resistor. Adding any resistor in series always increases the total, so $R_{\text{series}} = R_1 + R_2 + \cdots + R_n$ .

In parallel, the resistors share the same two connection points, so each one has the same voltage across it, and the current splits between them. Here it is easier to add the conductances, $\frac{1}{R}$ , than the resistances. Why reciprocals? Each branch adds a route for current, and the routes add their ability to carry current. The total conductance is the sum of the branch conductances, and $\frac{1}{R_{\text{parallel}}} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots + \frac{1}{R_n}$ turns that sum back into a resistance.

The total resistance in parallel is always smaller than the smallest resistor in the group. Each extra branch gives the current another path to flow through, so the total opposition drops. Two equal resistors in parallel give half the value of one of them. For exactly two resistors there is a shortcut: $R_{\text{parallel}} = \frac{R_1 R_2}{R_1 + R_2}$ . For three or more, use the reciprocal rule.

An everyday picture

Series is like people lining up for one turnstile. Everyone passes through the same gate, one after another, so the waiting times add up. Parallel is like several checkout lanes in a supermarket. Shoppers spread across the lanes, so the store as a whole gets through more people per minute than one lane alone. The more lanes there are, the less the whole store is held up.

Worked example: A 100 ohm and a 300 ohm resistor

Two resistors, $R_1 = 100\ \Omega$ and $R_2 = 300\ \Omega$ , are used first in series and then in parallel. Find the total resistance in each case. The symbol $\Omega$ is the ohm, the unit of resistance, so 100 $\Omega$ means one hundred ohms.

Series: add the resistances. The two resistors sit one after another on a single path, so the same current passes through both. Their resistances add: $R_{\text{series}} = 100\ \Omega + 300\ \Omega = 400\ \Omega$ . Each resistor adds its opposition to the path, so the total opposition is the sum.
Parallel: add the reciprocals. The reciprocal of each resistance is one over it. $\frac{1}{100} = 0.01$ and $\frac{1}{300} = 0.00333$ . Adding them gives $0.01333$ , and as one fraction it is $\frac{1}{100} + \frac{1}{300} = \frac{3}{300} + \frac{1}{300} = \frac{4}{300}$ , because $\frac{1}{100}$ equals $\frac{3}{300}$ .
Flip the sum back to a resistance. The sum is a reciprocal, so the total resistance is its reciprocal: $R_{\text{parallel}} = \frac{300}{4} = 75\ \Omega$ . Flipping a fraction turns it upside down, and the reciprocal of $\frac{4}{300}$ is $\frac{300}{4}$ , which is 75.
Check the parallel total against the smallest resistor. The smallest resistor is 100 $\Omega$ , and the parallel total is 75 $\Omega$ , which is smaller. This is expected, because the current has two routes, so the total opposition falls below the smallest branch.
Check the parallel answer with the two-resistor shortcut. The shortcut is $\frac{R_1 R_2}{R_1 + R_2} = \frac{100 \times 300}{100 + 300} = \frac{30\,000}{400} = 75\ \Omega$ . It gives the same answer as the reciprocal method, so the result is consistent.
Check the currents with 10 V across the parallel pair. The 100 $\Omega$ branch takes $10 \div 100 = 0.1$ A, and the 300 $\Omega$ branch takes $10 \div 300 = 0.0333$ A. Together they take $0.1333$ A. The 75 $\Omega$ equivalent takes $10 \div 75 = 0.1333$ A, the same total, so 75 $\Omega$ is the right equivalent.

Answer: In series the total is 400 ohm. In parallel the total is 75 ohm, which is smaller than the smallest resistor, 100 ohm. The parallel branches give the current a second route, so the total drops.

The numbers in this example are recomputed by script every time the site is built.

Common mistakes

Adding resistors in parallel as if they were in series. 100 ohm and 300 ohm in parallel give 75 ohm, not 400 ohm.
Stopping after adding the reciprocals. The sum is $\frac{1}{R}$ , not $R$ . For 100 ohm and 300 ohm the sum is 0.01333 per ohm, and the answer is its reciprocal, 75 ohm.
Expecting a parallel total bigger than the smallest resistor. It can never be bigger, because each new branch only adds a path for current.
Using the two-resistor shortcut $\frac{R_1 R_2}{R_1 + R_2}$ for three or more resistors. The shortcut works only for two; for more, add the reciprocals and flip the sum.

Where you will meet it

Voltage dividers, where two resistors in series share a supply voltage to give a smaller voltage for a sensor or an input pin.
Household wiring, where outlets and appliances are connected in parallel so that each one gets the full mains voltage.
Circuit analysis, where a group of resistors is replaced by one equivalent resistor to make the rest of the calculation simpler.
Meter design, where extra resistors in series or a low-resistance shunt in parallel set the range of a measuring instrument.

Questions people ask

How do I know whether two resistors are in series or in parallel?

Trace the current path. If there is only one route through both resistors, they are in series. If both share the same two connection points, they are in parallel.

Why is the parallel total always smaller?

The current has more than one way to go. More paths mean less opposition overall, so the total resistance drops below the smallest branch.

Can I use the shortcut $\frac{R_1 R_2}{R_1 + R_2}$ for three resistors?

No. For three or more, add the reciprocals: $\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}$ , then flip the total to get the resistance.

Do capacitors follow the same rules?

No, the rules are reversed. A capacitor stores charge, and its size is its capacitance, measured in farads. Capacitances add when capacitors are in parallel, while in series you add the reciprocals, which is the opposite of what resistors do.

Ohm's Law: Ohm’s law says the push on a wire, called voltage, equals the flow through it, called current, times how hard the wire resists that flow, called resistance.
Kirchhoff's Current Law: Kirchhoff’s current law says that the electricity flowing into a junction has to flow out again, because nothing piles up or disappears at a junction.
RC Time Constant: The RC time constant tells you how long a capacitor takes to fill up through a resistor. You get it by multiplying the resistance in ohms by the capacitance in farads, and the answer comes out in seconds.