In a circuit, resistance $R_1$ is connected in series with the parallel combination of resistances $R_2$ and $R_3$. The currents passing through the resistances $R_1, R_2$ and $R_3$ are $I_1, I_2$ and $I_3$, respectively. What will be the ratio ($I_3/I_1$) of currents in terms of the resistances connected in the circuit?

$R_2/(R_2 +R_3)$

The correct answer is Option (2) → $R_2/(R_2 +R_3)$

Equivalent resistance of parallel part:

$R_{23} = \frac{R_2 R_3}{R_2 + R_3}$

Total resistance of circuit:

$R_{eq} = R_1 + R_{23}$

Current through $R_1$:

$I_1 = \frac{V}{R_{eq}}$

Voltage across parallel branch:

$V_{23} = I_1 R_{23}$

Current through $R_3$:

$I_3 = \frac{V_{23}}{R_3} = \frac{I_1 R_{23}}{R_3}$

Therefore ratio:

$\frac{I_3}{I_1} = \frac{R_{23}}{R_3}$

$\frac{I_3}{I_1} = \frac{\frac{R_2 R_3}{R_2+R_3}}{R_3} = \frac{R_2}{R_2+R_3}$

Hence, $\frac{I_3}{I_1} = \frac{R_2}{R_2 + R_3}$