Three unequal resistors in parallel are equivalent to a resistance $1$ ohm. If two of them are in the ratio $1 : 2$ and if no resistance value is fractional, the largest of the three resistance in ohm is?

4

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Solution

The correct option is B $6$
Let the resistance be $R_{1}, R_{2}$ and $R_{3}$

$\frac{R_{1}}{R_{2}} = \frac{1}{2}$ => $R_{1} = k, R_{2} = 2 k$

In parallel $\frac{1}{R} = \frac{1}{R_{1}} + \frac{1}{R_{2}} + \frac{1}{R_{3}}$

$\frac{1}{1} = \frac{1}{k} + \frac{1}{2 k} + \frac{1}{R_{2}}$

$\frac{1}{R_{2}} = 1 - \frac{1}{k} - \frac{1}{2 k}$

= $\frac{2 k - 2 - 1}{2 k} = \frac{2 k - 3}{2 k}$

$R_{3} = \frac{2 k}{2 k - 3}$

if k = 1, then $R_{3}$ is found to be negative, which is impossible.

if k = 2, then $R_{1} = 2, R_{2} = 4, R_{3} = 4$

$R_{2} = R_{3}$ , not satisfying the condition of the question that all resistance are unequal.
If k = 3 then $R_{1} = 3, R_{2} = 6$

$R_{3} = 2 Ω$

Largest resistance = 6 $Ω$

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