Two resistance wires on joining in parallel the resultant resistance is ohms `(6)/(5)` ohms. One of the wire breaks, the effective resistance is 2 ohms . The resistance of the broken wire is

Options

A. `(3)/(5)` ohm
B. 2 ohm
C. `(6)/(5)` ohm
D. 3 ohm

Answer

To find the resistance of the broken wire, we can follow these steps:

### Step 1: Understand the Problem
We have two resistances \( R_1 \) and \( R_2 \) connected in parallel. The equivalent resistance \( R_{eq} \) of the two wires is given as \( \frac{6}{5} \) ohms. When one wire (let's say \( R_1 \)) breaks, the effective resistance becomes \( 2 \) ohms.

### Step 2: Write the Formula for Parallel Resistance
The formula for the equivalent resistance \( R_{eq} \) of two resistors \( R_1 \) and \( R_2 \) in parallel is given by:
\[ \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} \]

### Step 3: Substitute the Known Values
Substituting the known value of \( R_{eq} = \frac{6}{5} \) ohms into the formula:
\[ \frac{1}{\frac{6}{5}} = \frac{1}{R_1} + \frac{1}{R_2} \]
This simplifies to:
\[ \frac{5}{6} = \frac{1}{R_1} + \frac{1}{R_2} \]

### Step 4: Use the Value of \( R_2 \)
When one wire breaks, we know that the remaining wire \( R_2 \) has a resistance of \( 2 \) ohms. Therefore, we can substitute \( R_2 = 2 \) ohms into the equation:
\[ \frac{5}{6} = \frac{1}{R_1} + \frac{1}{2} \]

### Step 5: Solve for \( \frac{1}{R_1} \)
Rearranging the equation to isolate \( \frac{1}{R_1} \):
\[ \frac{1}{R_1} = \frac{5}{6} - \frac{1}{2} \]

### Step 6: Find a Common Denominator
To subtract the fractions, we need a common denominator. The least common multiple of \( 6 \) and \( 2 \) is \( 6 \):
\[ \frac{1}{R_1} = \frac{5}{6} - \frac{3}{6} = \frac{2}{6} \]

### Step 7: Simplify the Result
Now, simplifying \( \frac{2}{6} \):
\[ \frac{1}{R_1} = \frac{1}{3} \]
Thus, taking the reciprocal gives:
\[ R_1 = 3 \text{ ohms} \]

### Conclusion
The resistance of the broken wire \( R_1 \) is \( 3 \) ohms.

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