What is the total resistance of a 6-ohm resistor and a 12-ohm resistor in a parallel series?

In series like so ---6 ohms ---- 12 ohms --- , the total resistance is just 6 ohms + 12 ohms.

assuming you mean in parallel like this:

_|---6 ohms-----|

-|~|-

_|---12 ohms---|

then the resistance of this can be calculated like so:

1/6 ohms + 1/12 ohms = 1/R (where R is the resistance of the circuit as a whole)

2/12 ohms + 1/12 ohms = 1/R

3/12 ohms = 1/R

1/4 ohms = 1/R

so R = 4 ohms

A few notes, if the resistors are in parallel the total resistance will always be less than or equal to the lowest resistance in parallel (i.e 6 ohms in parallel with 12 ohms will have resistance less than 6 ohms).

Also if two resistances in parallel are the same, then the resistance is half of the resistance of both resistors (i.e. 1/2 ohms + 1/2 ohms = 1/R; 1 = 1/R, R=1 ohm which is half of 2 ohms).

This process can be extended to 2 or more resistors in parallel.

i.e if we had a 6 ohm, 6 ohm and 12 ohm resistor in parallel we could go

1/6 ohms + 1/6 ohms + 1/12 ohms = 1/R

(1/6 ohms + 1/6 ohms) + 1/12 ohms = 1/R

1/3 ohms + 1/12 ohms = 1/R

4/12ohms + 1/12 ohms = 1/R

5/12 ohms = 1/R

so R = 12/5 ohms or 2.4 ohms