Question

A piece of wire of resistance R is cut into five equal parts. These parts are then connected in parallel. If the equivalent resistance of these of this combination is $R^{\prime }$ then the ratio ${\displaystyle \frac{R}{R^{\prime }}}$ is

Answer 1

Hint: The question talks about the connection of resistors together to fix the end (parallel connection) with a certain length of wire cut in bits into five portions of basically the same length, cross sectional area and resistivity. When the resistors are connected in parallel, then the reciprocal of the equivalent resistance is the sum of the reciprocals of all the resistances.

Complete step by step answer:
We are given that a piece of wire of resistance R is cut into five equal parts and these parts are then connected in parallel. The equivalent resistance of these of this combination is $R^{\prime }$
Since the resistors are equal in values then, we might decide to name the first resistor as $R_1$, the second resistor as $R_2$, the third resistor as $R_3$, the fourth resistor as $R_4$ and the fifth resistor as $R_5$.
So for resistors in parallel, the equivalent resistor becomes ${\displaystyle \frac{1}{R^{\prime }}}={\displaystyle \frac{1}{R_1}}+{\displaystyle \frac{1}{R_2}}+{\displaystyle \frac{1}{R_3}}+{\displaystyle \frac{1}{R_4}}+{\displaystyle \frac{1}{R_5}}$
Also since the resistor (R) is divided into $R_1=R_2=R_3=R_4=R_5$
$$\begin{gathered} R=R_1+R_2+R_3+R_4+R_5 \\ ⟹5R_1=R \\ ⟹R_1={\displaystyle \frac{R}{5}} \\ ⟹R_1=R_2=R_3=R_4=R_5={\displaystyle \frac{R}{5}} \end{gathered}$$
We can then simplify
 $$\begin{gathered} {\displaystyle \frac{1}{R^{\prime }}}={\displaystyle \frac{1}{\left({\displaystyle \frac{R}{5}}\right)}}+{\displaystyle \frac{1}{\left({\displaystyle \frac{R}{5}}\right)}}+{\displaystyle \frac{1}{\left({\displaystyle \frac{R}{5}}\right)}}+{\displaystyle \frac{1}{\left({\displaystyle \frac{R}{5}}\right)}}+{\displaystyle \frac{1}{\left({\displaystyle \frac{R}{5}}\right)}} \\ ⟹{\displaystyle \frac{1}{R^{\prime }}}={\displaystyle \frac{5}{R}}+{\displaystyle \frac{5}{R}}+{\displaystyle \frac{5}{R}}+{\displaystyle \frac{5}{R}}+{\displaystyle \frac{5}{R}} \\ ⟹{\displaystyle \frac{1}{R^{\prime }}}={\displaystyle \frac{25}{R}} \\ ⟹{\displaystyle \frac{R}{R^{\prime }}}={\displaystyle \frac{25}{1}} \end{gathered}$$
Therefore, the ratio ${\displaystyle \frac{R}{R^{\prime }}}$ is ${\displaystyle \frac{25}{1}}$, $R:R^{\prime }=25:1$

Note:
Same applies to inductors connected in parallel too but differs in a capacitor. The voltage of resistors connected in series is different for each resistor and current is the same for all the resistors in series, whereas the voltage of resistors connected in parallel is the same for each resistor and current is different for all the resistors. The same resistance of a wire is dependent on three factors or parameters namely resistivity, cross sectional area and length.