Given Circuit:

To calculate the current in either branch and the value of the unknown resistance, we need to analyze the given circuit. The circuit consists of a voltage source (denoted by V), two resistors (R1 and R2), and an unknown resistance (Rx). The current flowing through the resistance Rx is denoted as Ix.

Using Ohm's Law:

According to Ohm's Law, the current flowing through a resistor is directly proportional to the voltage across it and inversely proportional to its resistance. Mathematically, Ohm's Law can be expressed as:

V = I * R

where V is the voltage, I is the current, and R is the resistance.

Calculating the Current:

1. Let's assume the current flowing through resistor R1 is I1 and the current flowing through resistor R2 is I2.

2. Since the current flowing through resistors R1 and R2 is the same, we can write the following equation:

I1 = I2

3. Using Ohm's Law, we can express the voltage across resistor R1 as:

V1 = I1 * R1

4. Similarly, the voltage across resistor R2 can be expressed as:

V2 = I2 * R2

5. We know that the total voltage provided by the voltage source is equal to the sum of the voltages across resistors R1 and R2:

V = V1 + V2

6. Substituting the values of V1 and V2 in terms of I1 and I2, we get:

V = I1 * R1 + I2 * R2

Finding the Value of Rx:

7. Now, let's consider the current flowing through the unknown resistance Rx as Ix.

8. Since the current flowing through the entire circuit is the same, we can write the following equation:

I1 + Ix = I2

9. Rearranging the equation, we get:

Ix = I2 - I1

10. Substituting the values of I1 and I2 from step 2, we get:

Ix = I2 - I2 = 0

This means that the current flowing through the unknown resistance Rx is 0A. Therefore, the value of the unknown resistance Rx is irrelevant when the current taken by the resistance is 2.25A.

Summary:

In summary, when the current taken by the resistance is 2.25A, the current flowing through the unknown resistance Rx is 0A, and the value of the unknown resistance is irrelevant. This conclusion is based on the analysis of the given circuit using Ohm's Law and the principle of current conservation.