Determine the total resistance of the circuit.

Circuit Analysis:

• Resistors R1 and R2 are in parallel.
• Resistors R4 and R5 are in parallel.
• The parallel combination of R1 and R2 is in series with R3.
• The series combination of (R1 || R2) + R3 is in parallel with R4.
• The parallel combination of [(R1 || R2) + R3] and R4 is in series with R5.

Step-by-step calculation:

1. Step 1: Calculate the equivalent resistance of R1 and R2 in parallel (R12).
   \( R_{12} = \frac{R_1 \times R_2}{R_1 + R_2} = \frac{3 \Omega \times 6 \Omega}{3 \Omega + 6 \Omega} = \frac{18 \Omega^2}{9 \Omega} = 2 \Omega \)
2. Step 2: Calculate the equivalent resistance of R12 and R3 in series (R123).
   \( R_{123} = R_{12} + R_3 = 2 \Omega + 2 \Omega = 4 \Omega \)
3. Step 3: Calculate the equivalent resistance of R4 and R5 in parallel (R45).
   \( R_{45} = \frac{R_4 \times R_5}{R_4 + R_5} = \frac{12 \Omega \times 6 \Omega}{12 \Omega + 6 \Omega} = \frac{72 \Omega^2}{18 \Omega} = 4 \Omega \)
4. Step 4: Calculate the equivalent resistance of R123 and R45 in parallel (R_AB).
   \( R_{AB} = \frac{R_{123} \times R_{45}}{R_{123} + R_{45}} = \frac{4 \Omega \times 4 \Omega}{4 \Omega + 4 \Omega} = \frac{16 \Omega^2}{8 \Omega} = 2 \Omega \)
5. Step 5: Calculate the total resistance of the circuit by adding R5 (which is in series with the parallel combination of R123 and R45). Wait, looking at the diagram again, R5 is in parallel with R4 and the combination of R1, R2, R3. The point B is after R3 and R4 and R5. So R4 and R5 are in parallel and the whole thing is in series with R3 which is in series with R1||R2. Let's re-evaluate.
6. Step 1 (Revised): Calculate the equivalent resistance of R1 and R2 in parallel (R12).
   \( R_{12} = \frac{R_1 \times R_2}{R_1 + R_2} = \frac{3 \Omega \times 6 \Omega}{3 \Omega + 6 \Omega} = \frac{18 \Omega^2}{9 \Omega} = 2 \Omega \)
7. Step 2 (Revised): Calculate the equivalent resistance of R12 and R3 in series (R123).
   \( R_{123} = R_{12} + R_3 = 2 \Omega + 2 \Omega = 4 \Omega \)
8. Step 3 (Revised): Calculate the equivalent resistance of R4 and R5 in parallel (R45).
   \( R_{45} = \frac{R_4 \times R_5}{R_4 + R_5} = \frac{12 \Omega \times 6 \Omega}{12 \Omega + 6 \Omega} = \frac{72 \Omega^2}{18 \Omega} = 4 \Omega \)
9. Step 4 (Revised): Calculate the total resistance of the circuit (R_total) by adding R123 and R45 in parallel.
   \( R_{total} = \frac{R_{123} \times R_{45}}{R_{123} + R_{45}} = \frac{4 \Omega \times 4 \Omega}{4 \Omega + 4 \Omega} = \frac{16 \Omega^2}{8 \Omega} = 2 \Omega \)

Answer: The total resistance of the circuit is 2 Ω.