Resistors in parallel
If you have many resistors in parallel, the arithmetic is easier if we work with conductance instead of resistance. The two concepts are really just different ways of looking at the same thing -- conductance is how easily the electrons can flow, and resistance is how difficult it is for the electrons to flow.
Resistance is measured in ohms, and conductance is measured in siemens. Normally you won't be using siemens, but I introduce it here because it makes calculating parallel resistances easier. To convert resistance to conductance, you simply take the reciprocal -- that is, you divide 1 by the number of ohms.
To get the value for the conductance of parallel resistors, we simply add them. This makes parallel resistors as simple as series resistors, as long as we are using conductance instead of resistance.
Let's look at our earlier example of two 100 ohm resistors in parallel. Each resistor has a conductance of 0.01 siemens. Adding them together in parallel makes the combination more conductive, giving us 0.02 siemens. Taking the reciprocal of that to convert back to ohms gives us 50 ohms.
If you like, you can still just work in ohms. To find out the resistance of a 100 ohm resistor in parallel with a 200 ohm resistor, we add 1/(100 ohms) and 1/(200 ohms), and then divide 1 by that. To put it in a simple formula, we just say:
resistance = 1 / (1/resistor1 + 1/resistor2)
It's only a bit more complicated than working with conductances, but it is easy to enter into a calculator, or into Google.
In mathematical form, it looks like this:
resistance = 1 1 resistor 1 + 1 resistor 2
Which can be rearranged as resistance = resistor 1 resistor 2 resistor 1 + resistor 2
This latter form is called the product over sum method, and many electronics students memorize it and have no idea how it comes about, or that understanding parallel resistances is really as easy as adding the conductances.
We can mentally check the result by realizing that the resistors in parallel will always have less resistance than the smallest of the parallel resistors. In our example, we got 66 and 2/3rds ohms, which is less than the smaller of the two resistors (100 ohms).
If you have more than two resistors in parallel, you just add up all of their conductances, and divide that into 1 to convert back to ohms.