Just quoting in Danny Dorito so she can move the thread if needed
Well the open circuit potential is 12V so the EMF (curly E) is 12 V
because no current is flowing in the internal resistance no voltage is present across the internal resistance and the voltmeter measures the full EMF
with the switch closed the EMF is the same 12.0V as before
but now a current is flowing in the internal resistance and in the external 4.0ohm resistance - the current in both must be the same because they are in series.
The PD across the internal resistance plus the PD across the external 4.0 ohm resistance must equal the EMF of 12.0V so the PD across the internal resistance must be 12.0 minus 8.0 Volts
The PD across the external resistance of 4.0 ohms is 8.0V so we can work out the current using I=V/R
The same current is flowing in the internal resistance and we need to find the value of r that gives us the correct PD across the internal resistance.