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Question
The equivalent resistance of the circuit is 5 Ω. Calculate the value of x.
Numerical
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Solution
\[\text{Equivalent resistance of circuit }R = 5\Omega\]
\[\text{Let }R_p\text{ be the equivalent resistance of }6\Omega\text{, }x\text{ and }2\Omega\text{ in parallel}\]
\[R_p\text{ is in series with }4\Omega\]
\[R = 4 + R_p\]
\[\Rightarrow 5 = 4 + R_p\]
\[\therefore R_p = 1\Omega\]
\[\text{As }6\Omega\text{, }x\text{ and }2\Omega\text{ are in parallel}\]
\[\therefore \frac{1}{R_p} = \frac{1}{6} + \frac{1}{x} + \frac{1}{2}\]
\[ \frac{1}{1} = \frac{1}{6} + \frac{1}{x} + \frac{1}{2}\]
\[\frac{1}{x} = 1 - \frac{1}{6} - \frac{1}{2}\]
\[\frac{1}{x} = \frac{6 - 1 - 3}{6}\]
\[\frac{1}{x} = \frac{2}{6}\]
\[\frac{1}{x} = \frac{1}{3}\]
\[\therefore x = 3\Omega\]
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