Total resistance of a parallel circuit:

Find out the total resistance of a parallel circuit which has one 12 ? and one 4 ? resistor.

R_T =    R₁ R₂ / R₁ + R₂ = (12) (4)/12+4 = 48/16 = 3?     

In some cases including two resistors in parallel, it is meaningful to search an unknown resistor, R_x, to obtain a certain R_T.  For find the suitable formula, we begin along with equation (2-10) and let the known resistor be R and the unknown resistor is R_x.

R_T =    R R_x / R + R_x

Cross multiply:        R_TR + R_TR_X = RR_X

Transpose:     RR_X - R_TR_X = R_TR

Factor:            R_X (R - R_T) = R_TR

Solve for R_x:  R_X = R_TR / R            - R_T

Example: What value of resistance must be added, in parallel, with an 8     resistor to provide a total resistance of 6 in figure?

Figure: Example Parallel Circuit

Solution:

R_X = RR_T / R - R_T = (8) (6) /8- 6 = 48 /2 =24 ?