Effective resistance calculation worksheet featuring six circuit diagrams with resistors in series and parallel.

To solve the problem of calculating the effective resistance for each circuit, we need to analyze the circuits step by step. Let's go through each one:

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Circuit (a)

The circuit consists of three resistors in series:
- Two resistors of \(3 \, \Omega\) are in series.
- Another resistor of \(3 \, \Omega\) is in series with the first two.

The formula for resistors in series is:
\[
R_{\text{series}} = R_1 + R_2 + R_3
\]
Here, \(R_1 = 3 \, \Omega\), \(R_2 = 3 \, \Omega\), and \(R_3 = 3 \, \Omega\).

\[
R_{\text{effective}} = 3 + 3 + 3 = 9 \, \Omega
\]

Effective resistance for (a): \(\boxed{9 \, \Omega}\)

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Circuit (b)

The circuit consists of three resistors in parallel:
- All three resistors have a resistance of \(3 \, \Omega\).

The formula for resistors in parallel is:
\[
\frac{1}{R_{\text{parallel}}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}
\]
Here, \(R_1 = 3 \, \Omega\), \(R_2 = 3 \, \Omega\), and \(R_3 = 3 \, \Omega\).

\[
\frac{1}{R_{\text{effective}}} = \frac{1}{3} + \frac{1}{3} + \frac{1}{3} = \frac{3}{3} = 1
\]
\[
R_{\text{effective}} = \frac{1}{1} = 1 \, \Omega
\]

Effective resistance for (b): \(\boxed{1 \, \Omega}\)

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Circuit (c)

The circuit consists of two sets of resistors in parallel, which are then connected in series:
- The top branch has two resistors of \(3 \, \Omega\) in parallel.
- The bottom branch has a single resistor of \(3 \, \Omega\).
- These two branches are in series.

First, calculate the effective resistance of the top branch (two \(3 \, \Omega\) resistors in parallel):
\[
\frac{1}{R_{\text{top}}} = \frac{1}{3} + \frac{1}{3} = \frac{2}{3}
\]
\[
R_{\text{top}} = \frac{3}{2} = 1.5 \, \Omega
\]

Now, the effective resistance of the entire circuit is the sum of the top branch (\(1.5 \, \Omega\)) and the bottom branch (\(3 \, \Omega\)):
\[
R_{\text{effective}} = 1.5 + 3 = 4.5 \, \Omega
\]

Effective resistance for (c): \(\boxed{4.5 \, \Omega}\)

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Circuit (d)

The circuit consists of two sets of resistors in series, which are then connected in parallel:
- The left branch has two resistors of \(3 \, \Omega\) in series.
- The right branch has two resistors of \(3 \, \Omega\) in series.
- These two branches are in parallel.

First, calculate the effective resistance of each branch (two \(3 \, \Omega\) resistors in series):
\[
R_{\text{left}} = 3 + 3 = 6 \, \Omega
\]
\[
R_{\text{right}} = 3 + 3 = 6 \, \Omega
\]

Now, calculate the effective resistance of the entire circuit (two \(6 \, \Omega\) resistors in parallel):
\[
\frac{1}{R_{\text{effective}}} = \frac{1}{6} + \frac{1}{6} = \frac{2}{6} = \frac{1}{3}
\]
\[
R_{\text{effective}} = 3 \, \Omega
\]

Effective resistance for (d): \(\boxed{3 \, \Omega}\)

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Circuit (e)

The circuit consists of two sets of resistors in parallel, which are then connected in series:
- The top branch has two resistors of \(2 \, \Omega\) in parallel.
- The bottom branch has two resistors of \(2 \, \Omega\) in parallel.
- These two branches are in series.

First, calculate the effective resistance of each branch (two \(2 \, \Omega\) resistors in parallel):
\[
\frac{1}{R_{\text{top}}} = \frac{1}{2} + \frac{1}{2} = 1
\]
\[
R_{\text{top}} = 1 \, \Omega
\]
\[
\frac{1}{R_{\text{bottom}}} = \frac{1}{2} + \frac{1}{2} = 1
\]
\[
R_{\text{bottom}} = 1 \, \Omega
\]

Now, calculate the effective resistance of the entire circuit (two \(1 \, \Omega\) resistors in series):
\[
R_{\text{effective}} = 1 + 1 = 2 \, \Omega
\]

Effective resistance for (e): \(\boxed{2 \, \Omega}\)

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Circuit (f)

The circuit consists of two sets of resistors in series, which are then connected in parallel:
- The left branch has two resistors of \(2 \, \Omega\) in series.
- The right branch has two resistors of \(2 \, \Omega\) in series.
- These two branches are in parallel.

First, calculate the effective resistance of each branch (two \(2 \, \Omega\) resistors in series):
\[
R_{\text{left}} = 2 + 2 = 4 \, \Omega
\]
\[
R_{\text{right}} = 2 + 2 = 4 \, \Omega
\]

Now, calculate the effective resistance of the entire circuit (two \(4 \, \Omega\) resistors in parallel):
\[
\frac{1}{R_{\text{effective}}} = \frac{1}{4} + \frac{1}{4} = \frac{2}{4} = \frac{1}{2}
\]
\[
R_{\text{effective}} = 2 \, \Omega
\]

Effective resistance for (f): \(\boxed{2 \, \Omega}\)

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Final Answers:

\[
\boxed{9 \, \Omega, 1 \, \Omega, 4.5 \, \Omega, 3 \, \Omega, 2 \, \Omega, 2 \, \Omega}
\]