Question

A student has two resistors. He can obtain resistances of $3$, $4$, $12$ and $16$ ohm using them separately or together. What are the values of the resistors?

Answer 1

To solve these types of questions we need to take help of the hit and trial method. We need to think of any such two resistors with the help of which we can obtain resistances of $3$, $4$, $12$ and $16$ ohm by using the resistors individually or by connecting them in series or in parallel combination.

Complete step-by-step solution: -
Let us assume the two resistors to be of $4$ and $12$ ohm resistance. If we use these resistors then we can obtain a resistance of $4$ and $12$ ohm. Now we know that when resistors are connected in parallel combination, the equivalent resistance comes out to be:
$\dfrac{1}{{{R}_{eq}}}=\dfrac{1}{{{R}_{1}}}+\dfrac{1}{{{R}_{2}}}$
If the resistors of resistance $4$ and $12$ ohm are connected in parallel combination, then their equivalent resistance would be obtained by substituting the values in the formula as follows:
$\begin{aligned} & \dfrac{1}{{{R}_{eq}}}=\dfrac{1}{4}+\dfrac{1}{12} \\\ & \Rightarrow \dfrac{1}{{{R}_{eq}}}=\dfrac{3+1}{12} \\\ & \Rightarrow \dfrac{1}{{{R}_{eq}}}=\dfrac{4}{12} \\\ & \Rightarrow \dfrac{1}{{{R}_{eq}}}=\dfrac{1}{3} \\\ & \Rightarrow {{R}_{eq}}=3\Omega \\\ \end{aligned}$
Now we know that when resistors are connected in series combination, the equivalent resistance comes out to be:
${{R}_{eq}}={{R}_{1}}+{{R}_{2}}$
If the resistors of resistance $4$ and $12$ ohm are connected in series combination, then their equivalent resistance would be obtained by substituting the values in the formula as follows:
$\begin{aligned} & {{R}_{eq}}=4+12 \\\ & \Rightarrow {{R}_{eq}}=16\Omega \\\ \end{aligned}$
Hence by using resistors of resistance $4$ and $12$ ohm, the student can obtain resistances of $3$, $4$, $12$ and $16$ ohm by either using them separately or connecting them in series or parallel combination.

Note: We must remember that when resistances are connected in parallel, the inverse of equivalent resistance of the combination is given by the summation of inverse of the individual resistances while in a series combination the equivalent resistance of the combination is simply given by the summation of the resistances.