The formula to calculate the total resistance of a parallel circuit is:
![\frac{1}{R_T}=\frac{1}{R_1}+\frac{1}{R_2}+...+\frac{1}{R_n}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7BR_T%7D%3D%5Cfrac%7B1%7D%7BR_1%7D%2B%5Cfrac%7B1%7D%7BR_2%7D%2B...%2B%5Cfrac%7B1%7D%7BR_n%7D)
Where
are the individual resistances
Let's now apply the formula to each circuit:
A. ![6.67 \Omega](https://tex.z-dn.net/?f=6.67%20%5COmega)
A parallel circuit with a 20-ohm resistor and a 10-ohm resistor.
The equivalent resistance is:
![\frac{1}{R_T}=\frac{1}{20 \Omega}+\frac{1}{10 \Omega}=\frac{3}{20 \Omega}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7BR_T%7D%3D%5Cfrac%7B1%7D%7B20%20%5COmega%7D%2B%5Cfrac%7B1%7D%7B10%20%5COmega%7D%3D%5Cfrac%7B3%7D%7B20%20%5COmega%7D)
![R_T = \frac{20}{3}\Omega =6.67 \Omega](https://tex.z-dn.net/?f=R_T%20%3D%20%5Cfrac%7B20%7D%7B3%7D%5COmega%20%3D6.67%20%5COmega)
B. ![5.0 \Omega](https://tex.z-dn.net/?f=5.0%20%5COmega)
A parallel circuit with two 20-ohm resistors and a 10-ohm resistor
The equivalent resistance is:
![\frac{1}{R_T}=\frac{1}{20\Omega}+\frac{1}{20 \Omega}+\frac{1}{10 \Omega}=\frac{4}{20 \Omega}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7BR_T%7D%3D%5Cfrac%7B1%7D%7B20%5COmega%7D%2B%5Cfrac%7B1%7D%7B20%20%5COmega%7D%2B%5Cfrac%7B1%7D%7B10%20%5COmega%7D%3D%5Cfrac%7B4%7D%7B20%20%5COmega%7D)
![R_T = \frac{20}{4}\Omega =5.0 \Omega](https://tex.z-dn.net/?f=R_T%20%3D%20%5Cfrac%7B20%7D%7B4%7D%5COmega%20%3D5.0%20%5COmega)
C. ![8.57 \Omega](https://tex.z-dn.net/?f=8.57%20%5COmega)
A parallel circuit with a 15-ohm light bulb and a 20-ohm resistor.
The equivalent resistance is:
![\frac{1}{R_T}=\frac{1}{20 \Omega}+\frac{1}{15 \Omega}=0.1167 \Omega ^{-1}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7BR_T%7D%3D%5Cfrac%7B1%7D%7B20%20%5COmega%7D%2B%5Cfrac%7B1%7D%7B15%20%5COmega%7D%3D0.1167%20%5COmega%20%5E%7B-1%7D)
![R_T = \frac{1}{0.1167}\Omega =8.57 \Omega](https://tex.z-dn.net/?f=R_T%20%3D%20%5Cfrac%7B1%7D%7B0.1167%7D%5COmega%20%3D8.57%20%5COmega)
D. ![14.3 \Omega](https://tex.z-dn.net/?f=14.3%20%5COmega)
A parallel circuit with two 100-ohm resistors and a 20-ohm resistor.
The equivalent resistance is:
![\frac{1}{R_T}=\frac{1}{20\Omega}+\frac{1}{100 \Omega}+\frac{1}{100 \Omega}=\frac{7}{100 \Omega}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7BR_T%7D%3D%5Cfrac%7B1%7D%7B20%5COmega%7D%2B%5Cfrac%7B1%7D%7B100%20%5COmega%7D%2B%5Cfrac%7B1%7D%7B100%20%5COmega%7D%3D%5Cfrac%7B7%7D%7B100%20%5COmega%7D)
![R_T = \frac{100}{7}\Omega =14.3 \Omega](https://tex.z-dn.net/?f=R_T%20%3D%20%5Cfrac%7B100%7D%7B7%7D%5COmega%20%3D14.3%20%5COmega)
E. ![6.1 \Omega](https://tex.z-dn.net/?f=6.1%20%5COmega)
A parallel circuit with a 10-ohm, 20-ohm, 100-ohm and 200-ohm resistor.
The equivalent resistance is:
![\frac{1}{R_T}=\frac{1}{10\Omega}+\frac{1}{20 \Omega}+\frac{1}{100 \Omega}+\frac{1}{200 \Omega}=\frac{33}{200 \Omega}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7BR_T%7D%3D%5Cfrac%7B1%7D%7B10%5COmega%7D%2B%5Cfrac%7B1%7D%7B20%20%5COmega%7D%2B%5Cfrac%7B1%7D%7B100%20%5COmega%7D%2B%5Cfrac%7B1%7D%7B200%20%5COmega%7D%3D%5Cfrac%7B33%7D%7B200%20%5COmega%7D)
![R_T = \frac{200}{33}\Omega =6.1 \Omega](https://tex.z-dn.net/?f=R_T%20%3D%20%5Cfrac%7B200%7D%7B33%7D%5COmega%20%3D6.1%20%5COmega)