Two resistances when connected in parallel give resultant value of 2 ohm, when connected in series the value becomes 9 ohm. Calculate the value of each resistance.

Answer

To solve the problem of finding the values of two resistances \( R_1 \) and \( R_2 \) that give a resultant resistance of 2 ohms when connected in parallel and 9 ohms when connected in series, we can follow these steps:

### Step 1: Set Up the Equations
1. **Parallel Resistance Formula**: When two resistances \( R_1 \) and \( R_2 \) are connected in parallel, the formula for the resultant resistance \( R_p \) is:
\[ R_p = \frac{R_1 R_2}{R_1 + R_2} \]
Given that \( R_p = 2 \) ohms, we can write:
\[ \frac{R_1 R_2}{R_1 + R_2} = 2 \]
Rearranging gives:
\[ R_1 R_2 = 2(R_1 + R_2) \quad \text{(Equation 1)} \]

2. **Series Resistance Formula**: When the same resistances are connected in series, the resultant resistance \( R_s \) is:
\[ R_s = R_1 + R_2 \]
Given that \( R_s = 9 \) ohms, we can write:
\[ R_1 + R_2 = 9 \quad \text{(Equation 2)} \]

### Step 2: Substitute and Rearrange
From Equation 2, we can express \( R_1 \) in terms of \( R_2 \):
\[ R_1 = 9 - R_2 \]

### Step 3: Substitute into Equation 1
Now, substitute \( R_1 \) in Equation 1:
\[ (9 - R_2) R_2 = 2((9 - R_2) + R_2) \]
This simplifies to:
\[ (9 - R_2) R_2 = 2 \cdot 9 \]
\[ (9 - R_2) R_2 = 18 \]

### Step 4: Expand and Rearrange
Expanding the left side gives:
\[ 9R_2 - R_2^2 = 18 \]
Rearranging this leads to:
\[ R_2^2 - 9R_2 + 18 = 0 \]

### Step 5: Solve the Quadratic Equation
Now we can solve the quadratic equation \( R_2^2 - 9R_2 + 18 = 0 \) using the factorization method or the quadratic formula. Factoring gives:
\[ (R_2 - 6)(R_2 - 3) = 0 \]
Thus, the solutions for \( R_2 \) are:
\[ R_2 = 6 \quad \text{or} \quad R_2 = 3 \]

### Step 6: Find Corresponding Values of \( R_1 \)
Using \( R_2 = 6 \):
\[ R_1 = 9 - 6 = 3 \]
Using \( R_2 = 3 \):
\[ R_1 = 9 - 3 = 6 \]

### Conclusion
The values of the resistances are:
\[ R_1 = 3 \, \text{ohms} \quad \text{and} \quad R_2 = 6 \, \text{ohms} \]
or vice versa.