A parallel circuit contains an 18 ohm and a 9 ohm resistor. What is the total resistance?

In a parallel circuit, the total resistance is not determined by simply adding the resistances of the individual resistors. Instead, the correct formula for calculating the total resistance (R_total) in a parallel circuit involving multiple resistors is given by the reciprocal sum of their individual resistances:

[ \frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} ]

For the given resistors (18 ohms and 9 ohms), the equation can be set up as follows:

[ \frac{1}{R_{total}} = \frac{1}{18} + \frac{1}{9} ]

Converting the second term (1/9) to have a common denominator of 18 yields:

[ \frac{1}{R_{total}} = \frac{1}{18} + \frac{2}{18} = \frac{3}{18} ]

Simplifying this gives:

[ \frac{1}{R_{total}} = \frac{1}{6} ]

To find R_total, we take the reciprocal of both sides:

[ R_{total} = 6 \text{ ohms} \