1. **State the problem:** We have a 30V battery connected in series with a 3Ω resistor, followed by four resistors in parallel with resistances 3Ω, 4Ω, 5Ω, and 6Ω. We need to find the currents $I_1$, $I_2$, $I_3$, $I_4$ through each resistor and the total current supplied by the battery. 2. **Identify the circuit and formula:** The total current $I$ flows through the series resistor $R_s = 3\Omega$ first, then splits into four parallel branches with resistors $R_1=3\Omega$, $R_2=4\Omega$, $R_3=5\Omega$, and $R_4=6\Omega$. 3. **Calculate the equivalent resistance of the parallel resistors:** $$\frac{1}{R_p} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \frac{1}{R_4} = \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6}$$ Calculate each term: $$\frac{1}{3} = 0.3333, \quad \frac{1}{4} = 0.25, \quad \frac{1}{5} = 0.2, \quad \frac{1}{6} = 0.1667$$ Sum: $$0.3333 + 0.25 + 0.2 + 0.1667 = 0.95$$ So, $$R_p = \frac{1}{0.95} \approx 1.0526\Omega$$ 4. **Calculate total resistance in the circuit:** $$R_{total} = R_s + R_p = 3 + 1.0526 = 4.0526\Omega$$ 5. **Calculate total current from the battery using Ohm's law:** $$I_{total} = \frac{V}{R_{total}} = \frac{30}{4.0526} \approx 7.405\text{ A}$$ 6. **Calculate voltage drop across the series resistor:** $$V_s = I_{total} \times R_s = 7.405 \times 3 = 22.215\text{ V}$$ 7. **Calculate voltage across the parallel resistors:** $$V_p = V - V_s = 30 - 22.215 = 7.785\text{ V}$$ 8. **Calculate currents through each parallel resistor using Ohm's law:** $$I_1 = \frac{V_p}{R_1} = \frac{7.785}{3} = 2.595\text{ A}$$ $$I_2 = \frac{7.785}{4} = 1.946\text{ A}$$ $$I_3 = \frac{7.785}{5} = 1.557\text{ A}$$ $$I_4 = \frac{7.785}{6} = 1.298\text{ A}$$ 9. **Verify total current:** Sum of parallel currents: $$I_1 + I_2 + I_3 + I_4 = 2.595 + 1.946 + 1.557 + 1.298 = 7.396\text{ A}$$ This is approximately equal to $I_{total} = 7.405$ A (small difference due to rounding). **Final answers:** $$I_1 \approx 2.60\text{ A}, \quad I_2 \approx 1.95\text{ A}, \quad I_3 \approx 1.56\text{ A}, \quad I_4 \approx 1.30\text{ A}, \quad I_{total} \approx 7.41\text{ A}$$