To find two resistors in parallel, add the reciprocals of their values and flip the result. With 12 Ω and 4 Ω, 1/12 + 1/4 = 1/3, so the total is 3 Ω. That is below both resistors, which is the sign the calculation went right.
The general formula is 1/R = 1/R₁ + 1/R₂ + … + 1/Rₙ. It applies to resistors connected between the same two points, so each one has the same voltage across it. It is a sum of fractions, so you can work it out with the adding fractions calculator and check each step.
A full example: 6 Ω and 3 Ω
Worked example: 6 Ω in parallel with 3 Ω
Two resistors of 6 Ω and 3 Ω are connected between the same two points. What is the total resistance?
- Write the reciprocal of each. They are 1/6 and 1/3.
- Find a common denominator. The LCM of 6 and 3 is 6, so 1/3 = 2/6.
- Add. 1/6 + 2/6 = 3/6 = 1/2.
- Flip the result. If 1/R = 1/2, then R = 2 Ω.
Result: 2 Ω.
Check: 2 × (1/6 + 1/3) = 2 × 1/2 = 1, and 2 Ω is less than 3 Ω.
Why the total goes down instead of up
Each new resistor opens another path for the current. That makes it easier to get through, not harder. The reciprocal 1/R measures that ease, and the ease of several branches adds up. That is why you add reciprocals and not the values themselves.
One result surprises people at first: with positive resistors, the parallel total never reaches the smallest value in the group. The reciprocal of that value is the largest of all, and the sum is bigger still, so its own reciprocal ends up below it. With 6 Ω and 3 Ω the total is 2 Ω, below 3 Ω. If your answer comes out above the smallest branch, something went wrong in a step.
Two shortcuts that save fractions
With exactly two resistors, the sum of reciprocals reduces to product over sum: R = (R₁ × R₂)/(R₁ + R₂). It comes from adding 1/R₁ + 1/R₂ over the common denominator R₁ × R₂ and flipping. With 10 Ω and 15 Ω: 150/25 = 6 Ω, the same as 1/10 + 1/15 = 1/6.
With n equal resistors of value R, the total is R/n. Five 10 Ω resistors give 2 Ω, and six 3 Ω resistors give 1/2 Ω. The product-over-sum shortcut does not work on three or more values at once. Apply it to pairs, or go back to the sum of reciprocals.
Three branches: 10 Ω, 15 Ω and 30 Ω
With three resistors you add three reciprocals. The LCM of 10, 15 and 30 is 30:
1/10 + 1/15 + 1/30 = 3/30 + 2/30 + 1/30 = 6/30 = 1/5
The total is 5 Ω, less than 10 Ω. With the pair shortcut, 10 and 15 give 6 Ω, and 6 Ω with 30 Ω gives 180/36 = 5 Ω. Both routes agree.
Series and parallel with the same pair
| Connection | Rule | Total |
|---|---|---|
| Series | R = R₁ + R₂ | 9 Ω, larger than both |
| Parallel | 1/R = 1/R₁ + 1/R₂ | 2 Ω, smaller than both |
The same pair gives opposite results, so look at how the resistors are connected before you calculate. If two 2 Ω resistors seemed to give 4 Ω in parallel, you added them as if they were in series; in parallel they give 1 Ω. The sum of reciprocals and the plain sum are two different calculations, and the circuit diagram tells you which one applies.
The same structure in other problems
Adding reciprocals and flipping also shows up in work-rate and faucet problems. A faucet that fills a tank in 3 h adds 1/3 of the tank per hour, and another that fills it in 6 h adds 1/6. Together they add 1/2 per hour, and the tank fills in 2 h. It is the same arithmetic with times in place of ohms.
How to check the result
Multiply the total by the sum of the reciprocals. You should get 1. For the last step, the reciprocal of 1/2 is 2. To review why flipping is the same as dividing, see the reciprocal of a fraction and the dividing fractions calculator. To find the common denominator when the values are not so round, use the least common denominator calculator. Work with exact fractions and round only at the end, because rounding the reciprocals early piles up error.
The formula assumes resistors connected between the same two points, which is the condition stated in the source below. The examples on this page also use ideal resistors with positive values. The guide to verifying a fraction answer explains how to confirm a calculation by a second route.
Sources
- Wikipedia, Series and parallel circuits, section on resistors in parallel: gives the reciprocal-sum formula, the product-over-sum form for two resistors, and R/N for N equal resistors (accessed October 17, 2026).