Question

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

Answer 1

Let the resistances be $R_{1}, \, R_{2}$ and $R_{3}$
$\therefore \frac{R_{1}}{R_{2}}=\frac{1}{2}\Longrightarrow R_{1}=k, \, R_{2}=2k$
In parallel $\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}}+\frac{1}{R_{3}}$
$\frac{1}{1}=\frac{1}{k}+\frac{1}{2 k}+\frac{1}{R_{3}}$
$\frac{1}{R_{3}}=1-\frac{3}{2 k}$
$ \, \, \, =\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\Omega$
$\therefore $ Largest resistance $=6\Omega \, $