Question

By using only two resistors, r1 and r2, a student is able to obtain resistances of 3 ω, 4 ω, 12 ω, and 16 ω. the values of r1 and r2 (in ohms) are:

Answer 1

The values of r1 and r2 are 4 ohms and 12 ohms.
Putting 4 ohms and 12 ohms in parallel gives 3 ohms.
Putting 4 ohms and 12 ohms in series gives 16 ohms.

Answer 2

Answer:

$r_{1} = 4$ ω

$r_{2} = 12$ ω

Step-by-step explanation:

The Student is able to obtain four values of resistances using only two resistors $r_{1}$ and $r_{2}$, like that:

1. Resistance with a value of $r_{1}$

2. Resistance with a value of $r_{2}$

3. Resistance making series with $r_{1}$ and $r_{2}$

4. Resistance making a parallel with $r_{1}$ and $r_{2}$

You know that the value of resistance in series is defined by the sum of each one of them, in this case as shown below:

$r_{Series} = r_{1} + r_{2}$

Now, we´ll combine in pair every value of resistance given in the statement using the formula for resistance in series and in this way to know the values for $r_{1}$ and $r_{2}$.

• $r_{a}$ = 3 ω + 4 ω => $r_{a}$ = 7 ω

• $r_{b}$ = 3 ω + 12 ω => $r_{b}$ = 15 ω

• $r_{c}$ = 3 ω + 16 ω => $r_{c}$ = 19 ω

• $r_{d}$ = 4 ω + 12 ω => $r_{d}$ = 16 ω

• $r_{e}$ = 12 ω + 16 ω => $r_{e}$ = 28 ω

If you realize, the results of resistance from what we called $r_{a}$, $r_{b}$, $r_{c}$ and $r_{e}$  not corresponding with the values in the statement, but $r_{d}$ does.

So you guess that $r_{1} = 4$ ω and $r_{2} = 12$ ω, but you must meet the condition, resistance in parallel be equal to 3 ω. So you use the formula for resistance in parallel:

$r_{Parallel} = \frac{1}{\frac{1}{r1} + \frac{1}{r2} }$ or $\frac{r1*r2}{r1 + r2}$

the first way:

you replaces the values for $r_{1} = 4$ ω and $r_{2} = 12$ ω in the formula for resistance in parallel and solve the operation:

$r_{parallel} = \frac{1}{\frac{1}{4} + \frac{1}{12}}$

$r_{parallel} = \frac{1}{\frac{1}{3}}$

$r_{parallel} = 3$ ω

the second way:

$r_{parallel} = \frac{4 * 12}{4 + 12}$

$r_{parallel} = \frac{48}{16}$

$r_{parallel} = 3$ ω

and you got it, the student has getting four values of resistance by using  $r_{1} = 4$ ω and $r_{2} = 12$ ω and making series and parallel.