Answer :
We are given two resistors in parallel with one resistor of resistance
[tex]$$R_1 = 20 \text{ ohms}$$[/tex]
and a total equivalent resistance of
[tex]$$R_T = 15 \text{ ohms}.$$[/tex]
For two resistors in parallel, the equivalent resistance is given by
[tex]$$
R_T = \frac{R_1 R_2}{R_1 + R_2}.
$$[/tex]
We need to find [tex]$R_2$[/tex]. To do this, we start by multiplying both sides of the equation by [tex]$(R_1 + R_2)$[/tex]:
[tex]$$
R_T (R_1 + R_2) = R_1 R_2.
$$[/tex]
Next, distribute [tex]$R_T$[/tex]:
[tex]$$
R_T R_1 + R_T R_2 = R_1 R_2.
$$[/tex]
Now, isolate terms containing [tex]$R_2$[/tex] by subtracting [tex]$R_T R_2$[/tex] from both sides:
[tex]$$
R_T R_1 = R_1 R_2 - R_T R_2.
$$[/tex]
Factor [tex]$R_2$[/tex] on the right-hand side:
[tex]$$
R_T R_1 = R_2 (R_1 - R_T).
$$[/tex]
Now, solve for [tex]$R_2$[/tex] by dividing both sides by [tex]$(R_1 - R_T)$[/tex]:
[tex]$$
R_2 = \frac{R_T R_1}{R_1 - R_T}.
$$[/tex]
Substitute the given values [tex]$R_1 = 20 \text{ ohms}$[/tex] and [tex]$R_T = 15 \text{ ohms}$[/tex]:
[tex]$$
R_2 = \frac{15 \times 20}{20 - 15} = \frac{300}{5} = 60 \text{ ohms}.
$$[/tex]
Finally, to express [tex]$R_2$[/tex] in kilo-ohms, recall that
[tex]$$1 \text{ kilo-ohm} = 1000 \text{ ohms},$$[/tex]
so
[tex]$$
R_2 = \frac{60}{1000} = 0.06 \text{ kilo-ohms}.
$$[/tex]
Thus, the resistance of [tex]$R_2$[/tex] expressed in kilo-ohms is [tex]$0.06$[/tex] kilo-ohms.
[tex]$$R_1 = 20 \text{ ohms}$$[/tex]
and a total equivalent resistance of
[tex]$$R_T = 15 \text{ ohms}.$$[/tex]
For two resistors in parallel, the equivalent resistance is given by
[tex]$$
R_T = \frac{R_1 R_2}{R_1 + R_2}.
$$[/tex]
We need to find [tex]$R_2$[/tex]. To do this, we start by multiplying both sides of the equation by [tex]$(R_1 + R_2)$[/tex]:
[tex]$$
R_T (R_1 + R_2) = R_1 R_2.
$$[/tex]
Next, distribute [tex]$R_T$[/tex]:
[tex]$$
R_T R_1 + R_T R_2 = R_1 R_2.
$$[/tex]
Now, isolate terms containing [tex]$R_2$[/tex] by subtracting [tex]$R_T R_2$[/tex] from both sides:
[tex]$$
R_T R_1 = R_1 R_2 - R_T R_2.
$$[/tex]
Factor [tex]$R_2$[/tex] on the right-hand side:
[tex]$$
R_T R_1 = R_2 (R_1 - R_T).
$$[/tex]
Now, solve for [tex]$R_2$[/tex] by dividing both sides by [tex]$(R_1 - R_T)$[/tex]:
[tex]$$
R_2 = \frac{R_T R_1}{R_1 - R_T}.
$$[/tex]
Substitute the given values [tex]$R_1 = 20 \text{ ohms}$[/tex] and [tex]$R_T = 15 \text{ ohms}$[/tex]:
[tex]$$
R_2 = \frac{15 \times 20}{20 - 15} = \frac{300}{5} = 60 \text{ ohms}.
$$[/tex]
Finally, to express [tex]$R_2$[/tex] in kilo-ohms, recall that
[tex]$$1 \text{ kilo-ohm} = 1000 \text{ ohms},$$[/tex]
so
[tex]$$
R_2 = \frac{60}{1000} = 0.06 \text{ kilo-ohms}.
$$[/tex]
Thus, the resistance of [tex]$R_2$[/tex] expressed in kilo-ohms is [tex]$0.06$[/tex] kilo-ohms.
