Out of five resistance of `ROmega` each 3 are connected in parallel and are joined to the rest 2 in series. Find the resultant resistance.

Options

A. `(3//7)ROmega`
B. `(7//3)ROmega`
C. `(7//8)ROmega`
D. `(8//7)ROmega`

Answer

To solve the problem of finding the resultant resistance when three resistors are connected in parallel and then combined in series with two other resistors, we can follow these steps:

### Step 1: Identify the Configuration
We have 5 resistors, each of resistance \( R \) ohms. Three of these resistors are connected in parallel, and the other two are connected in series.

### Step 2: Calculate the Equivalent Resistance of the Parallel Resistors
For resistors in parallel, the formula for equivalent resistance \( R_{eq} \) is given by:

\[ \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} \]

Since all three resistors have the same resistance \( R \):

\[ \frac{1}{R_{eq}} = \frac{1}{R} + \frac{1}{R} + \frac{1}{R} = \frac{3}{R} \]

Thus, the equivalent resistance for the three resistors in parallel is:

\[ R_{parallel} = \frac{R}{3} \]

### Step 3: Add the Series Resistors
Now, we have the equivalent resistance of the three parallel resistors, which is \( R_{parallel} = \frac{R}{3} \). This is in series with the other two resistors, each of resistance \( R \).

The total resistance \( R_{total} \) for resistors in series is simply the sum of their resistances:

\[ R_{total} = R_{parallel} + R + R \]

Substituting the value of \( R_{parallel} \):

\[ R_{total} = \frac{R}{3} + R + R = \frac{R}{3} + 2R \]

### Step 4: Simplify the Expression
To combine these terms, we need a common denominator:

\[ R_{total} = \frac{R}{3} + \frac{6R}{3} = \frac{R + 6R}{3} = \frac{7R}{3} \]

### Conclusion
The resultant resistance of the entire circuit is:

\[ R_{total} = \frac{7R}{3} \text{ ohms} \]