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

`r_(1) = 4` ω

`r_(2) = 12` ω

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) = (1)/((1)/(r1) + (1)/(r2) )` or
`(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) = (1)/((1)/(4) + (1)/(12))`

`r_(parallel) = (1)/((1)/(3))`

`r_(parallel) = 3` ω

the second way:

`r_(parallel) = (4 * 12)/(4 + 12)`

`r_(parallel) = (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.