A parallel combination of three resistors takes a current of `7.5 A` from a `30 V` supply. If the two resistors are `10 Omega and 12 Omega`, find third one.

Explanation

To find the value of the third resistor in a parallel combination of three resistors, given that the total current is 7.5 A from a 30 V supply, and the other two resistors are 10 Ω and 12 Ω, we can follow these steps:

### Step 1: Understand the Circuit
In a parallel circuit, the voltage across all components is the same. Thus, the voltage across the 10 Ω, 12 Ω, and the unknown resistor (let's call it R3) is 30 V.

### Step 2: Calculate the Current through the Known Resistors
Using Ohm's Law (V = IR), we can calculate the current through each known resistor.

1. **For the 10 Ω resistor:**
\[ I_1 = \frac{V}{R_1} = \frac{30 \text{ V}}{10 \text{ Ω}} = 3 \text{ A} \]

2. **For the 12 Ω resistor:**
\[ I_2 = \frac{V}{R_2} = \frac{30 \text{ V}}{12 \text{ Ω}} = 2.5 \text{ A} \]

### Step 3: Use the Total Current Equation
The total current (I_total) flowing from the supply is the sum of the currents through each resistor:
\[ I_{total} = I_1 + I_2 + I_3 \]
Substituting the known values:
\[ 7.5 \text{ A} = 3 \text{ A} + 2.5 \text{ A} + I_3 \]

### Step 4: Solve for I3
Rearranging the equation to find I3:
\[ I_3 = 7.5 \text{ A} - 3 \text{ A} - 2.5 \text{ A} = 2 \text{ A} \]

### Step 5: Calculate the Value of the Third Resistor (R3)
Now that we know the current through the third resistor (I3 = 2 A), we can use Ohm's Law again to find R3:
\[ V = I_3 \times R_3 \implies R_3 = \frac{V}{I_3} = \frac{30 \text{ V}}{2 \text{ A}} = 15 \text{ Ω} \]

### Conclusion
The value of the third resistor is **15 Ω**.

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