Question

A parallel combination of an 4 resistor and an unknown resistor R is connected in series with a 19 resistor and a battery. This circuit is then disassembled and the three resistors are then connected in series with each other and the same battery. In both arrangements, the current through the 4 resistor is the same. What is the unknown resistance R?

Answer 1

So in the series circuit the voltage of the battery is R is the unknown resistor):
I ( R + 23)
And in the first circuit the current through R is 4I/R (since the voltage across this resistor is 4I (same as in the 4 ohm resistor).
Then the voltage of the battery in this first circuit is given by:
4I + 19(I + 4I/R)
If you equal both equations given the battery voltage you can eliminate I and solving -

IR+23I= 4I+19I+76I/R

R+23=4+19+76/R

R.R=76

you will find that the unknown resistor is sqrt(76) ohms= 8.71 ohm

Ans- R= 8.71 ohm