Simplistic but possibly useful starter:
The load current flows in both the 4k7 and the 220r resistors.
So relative voltage ratio between THE two resistors is proportional to their resistances
as V=IR and I is common to them both.
Vin aPpears Cvross the 22r and Vout across the 4k7 so
gain is ~~~= 4k7/220r ~= 21.4 or
Gain ~~~= 20.
2. Maximum possible single stage gain
Many people deny this formula is 'real'.
NB the following is a magic formula which you can either remember and use or you can choose understand it. This is true only for silicon bipolar transistors.
So Maximum possible gain is ~ 450
NB emitter reisistor must be bypassed (See below) with a suitably large cap when relying on Re alone for gain.
Available voltage swing is ~ 0 at this gain as all of Vcc needs to be dropped across Rl to maximize current to make dynamic Re small to make gain large.
Arcane mumblings re above:
The above occurs when the 220 R is bypassed with a large enough cap so that it is ~~ 0 ohms to ground at signal frequency.
The gain then becomes 4k7/Te where Re is the emitter internal resistance.
It happens as a caracteristic of silicon thet the internal Re - 26/mA. ie Re is 26 ohms at 1 mA, 13 ohms at 2 mA, 0.26 ohms at 10 mA etc.
mAx gain occurs when Re = minimum = when current = maximum. This occurs when all voltage is across load )almost) so i= 4k7/12 ~+ 2.55 mA.
From above Re = 26/2.55
So max gain = 4k7/(26/2.55) = 460.
Rearranging the above shows that Max Gain = Vsupply x Rload