When resistors are connected in series, the supply voltage is divided between them. Across resistors with a higher value there is more voltage and across lower value ones there is less. The ratio of voltages is the same as the ratio of resistances, and with some rearrangement, a general formula for voltage dividers is made.
$ \dfrac{V_{R1}}{V_{R2}} = \dfrac{R_1}{R_2} $
$ \dfrac{V_{R1}}{V_0} = \dfrac{R_1}{R_1 + R_2} $
$ \dfrac{V_{R2}}{V_0} = \dfrac{R_2}{R_1 + R_2} $
$ V_{R1} = V_{0} ⋅ \dfrac{R_1}{R_1 + R_2} $
$ V_{R2} = V_{0} ⋅ \dfrac{R_2}{R_1 + R_2} $
$ V_{out} = V_{in} ⋅ \dfrac{R_{meas.}}{R_{total}} $
A variable resistor with three connections operating on the principle of voltage dividers. It may be rotary or slide type, linear or logarithmic, single- or multi-turn. Between its two terminals it can take any value from zero to the maximum value specified by the manufacturer. A similar but very different component is the rotary encoder, which is not a resistor, has no end positions and usually rotates in little steps.
$ V_{out} = V_0 ⋅ \dfrac{R_1}{R_1 + R_2 + (R_3 \times R_4)} $
$ V_{out} = V_0 ⋅ \dfrac{R_2 \times R_3}{R_1+ (R_2 \times R_3)} $
There are cases when it is not possible to present the present the divider formula in a single fraction, but only in multiple steps. Taking advantage of the fact that in the case of parallel connection, the voltage is the same across all branches, the formula must first be written up considering the first pair of nodes, and then a successive fraction for the following branch.
$ V_{out} = V_0 ⋅ \dfrac{R_2 \times (R_3 + R_4)}{R_1 + R_2 \times (R_3 + R_4)} ⋅ \dfrac{R_4}{R_3 + R_4} $