Three resistors having resistances r₁, r₂ and r₃ are connected as shown in the given circuit. The ratio i₃/i₁ of currents in terms of resistances used in the circuit is :

The circuit shown in the image consists of resistor $r$₁ connected in series with a junction that splits into two parallel branches. One branch contains resistor $r$₂ and the other branch contains resistor $r$₃. The currents in the three resistors are denoted as $i$₁ (through $r$₁), $i$₂ (through $r$₂) and $i$₃ (through $r$₃).

Because $r$₂ and $r$₃ are in parallel, they share the same voltage across them. Let the total current entering the parallel network be $i$₁. By the current‑division rule, the current flowing through a particular branch of a parallel network is proportional to the conductance (reciprocal of resistance) of the other branch:

$i$₃ = i₁ \frac{G₂}{G₂+G₃}

where $G$₂ = 1/r₂ and $G$₃ = 1/r₃. Substituting the conductances gives:

$i$₃ = i₁ \frac{1/r₂}{1/r₂ + 1/r₃}

Multiplying numerator and denominator by $r$₂ r₃ simplifies the expression:

$i$₃ = i₁ \frac{r₃}{r₃ + r₂}

Taking the ratio $i$₃/i₁:

$\backslash frac\{i$₃}{i₁} = \frac{r₂}{r₂ + r₃ }

Thus the required ratio of currents expressed in terms of the resistances is:

$\backslash displaystyle\; \backslash frac\{i\_\{3\}\}\{i\_\{1\}\}\; =\; \backslash frac\{r\_\{2\}\}\{\backslash ,r\_\{2\}+r\_\{3\}\backslash ,\}$