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:

$\frac{1}{R_4}=\frac{1}{R_1}+\frac{1}{R_2}$

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

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

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:

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

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:

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

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.