If three resistors of resistance `2Omega, 4Omega` and `5 Omega` are connected in parallel then the total resistance of the combination will be

Options

A. `(20)/(19)Omega`
B. `(19)/(20)Omega`
C. `(19)/(10)Omega`
D. `(10)/(19)Omega`

Answer

To find the total resistance of three resistors connected in parallel, we can use the formula for equivalent resistance in parallel circuits. The formula is given by:

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

Where:
- \( R_{\text{eq}} \) is the equivalent resistance,
- \( R_1, R_2, R_3 \) are the individual resistances.

### Step-by-Step Solution:

1. **Identify the resistances**:
- \( R_1 = 2 \, \Omega \)
- \( R_2 = 4 \, \Omega \)
- \( R_3 = 5 \, \Omega \)

2. **Apply the formula for parallel resistors**:
\[ \frac{1}{R_{\text{eq}}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} \]
Substituting the values:
\[ \frac{1}{R_{\text{eq}}} = \frac{1}{2} + \frac{1}{4} + \frac{1}{5} \]

3. **Calculate each term**:
- The first term: \( \frac{1}{2} = 0.5 \)
- The second term: \( \frac{1}{4} = 0.25 \)
- The third term: \( \frac{1}{5} = 0.2 \)

4. **Add the fractions**:
To add these fractions, we need a common denominator. The least common multiple of 2, 4, and 5 is 20.
- Convert each term:
\[ \frac{1}{2} = \frac{10}{20}, \quad \frac{1}{4} = \frac{5}{20}, \quad \frac{1}{5} = \frac{4}{20} \]
- Now add them:
\[ \frac{10}{20} + \frac{5}{20} + \frac{4}{20} = \frac{19}{20} \]

5. **Find the equivalent resistance**:
Now we can find \( R_{\text{eq}} \):
\[ \frac{1}{R_{\text{eq}}} = \frac{19}{20} \]
Taking the reciprocal gives:
\[ R_{\text{eq}} = \frac{20}{19} \, \Omega \]

### Final Answer:
The total resistance of the combination is:
\[ R_{\text{eq}} = \frac{20}{19} \, \Omega \approx 1.05 \, \Omega \]