Question

For resistors of 10 Ω, 15 Ω and 20 Ω, two resistors are connected in parallel and one in series. Calculate all the possible total resistance values.

Answer 1

Given the resistors:

10Ω, 15 Ω and 20 Ω

Two resistors are connected in parallel and one in series.

Let's calculate all the possible total resistance values.

When resistors are connected in parallel, the equivalent resistance is:

`(1)/(R_4)=(1)/(R_1)+(1)/(R_2)`

After finding the equivalent resistance value in parellel, we now add the resistance value with the resistance in series:

`R_(eq)=R_4+R_3`

• When 10Ω and 15Ω are connected in parallel and 20Ω in series, we have:

R1 = 10 Ω

R2 = 15 Ω

R3 = 20 Ω

`\begin{gathered} (1)/(R_4)=(1)/(10)+(1)/(15)=(3+2)/(30)=(5)/(30)=(1)/(6) \\ \\ R_4=6\Omega \\ \\ \text{Now to find the total resistance, we have:} \\ R_(eq)=R_4+R_3=\text{ 6 }\Omega\text{ + 20}\Omega \\ \\ R_(eq)=26\text{ }\Omega \end{gathered}`

When 10Ω and 15Ω are connected in parallel and 20Ω in series the total resistance is 26 Ω.

• When 10Ω and 20Ω are connected in parallel and 15Ω in series, we have:

R1 = 10 Ω

R2 = 20 Ω

R3 = 15 Ω

Hence, we have:

`\begin{gathered} (1)/(R_4)=(1)/(R_1)+(1)/(R_2)=(1)/(10\Omega)+(1)/(20\Omega)=(2+1)/(20)=(3)/(20)\Omega \\ \\ R_4=(20)/(3)\Omega\approx6.7\Omega \\ \\ \text{Thus, we have:} \\ R_(eq)=R_4+R_3=6.7\Omega+15\Omega=21.7\Omega \end{gathered}`

When 10Ω and 20Ω are connected in parallel and 15Ω in series the total resistance is 21.7 Ω.

• When 20Ω and 15Ω are connected in parallel and 10Ω in series, we have:

R1 = 20 Ω

R2 = 15 Ω

R3 = 10 Ω

Thus, we have:

`\begin{gathered} (1)/(R_4)=(1)/(R_1)+(1)/(R_2)=(1)/(20)+(1)/(15)=(3+4)/(60)=(7)/(60) \\ \\ R_4=(60)/(7)\approx8.6\Omega \\ \\ R_(eq)=R_4+R_3=8.6+10=18.6\Omega \end{gathered}`

When 20Ω and 15Ω are connected in parallel and 10Ω in series the total resistance is 18.6Ω.

ANSWER:

• When 10Ω and 15Ω are connected in parallel and 20Ω in series the total resistance is 26 Ω.

,

• When 10Ω and 20Ω are connected in parallel and 15Ω in series the total resistance is 21.7 Ω.

,

• When 20Ω and 15Ω are connected in parallel and 10Ω in series the total resistance is 18.6Ω.