Well, You know that Watts = V X A. And you know that Amps = V/R If you half the voltage on a fixed resistance, you also cut the current flow in half. So, you have half the voltage and half the current which is equal to 1/4 the wattage.
To prove this mathematically without creating voltage, resistance and current values, and to eliminate variables, use the formula Power = the square of the Voltage divided by the Resistance.
Equation 1 P = E Square/R
Equation 1A P = E X E / R This is the same as the following equation because of the understood coefficients of the components.
Equation 1B 1P = 1E X 1E / 1R
Equation 2 1P = 0.5E X 0.5E /1R
Equation 2A P = 0.25E / R
Both equations 1 and 2 are valid and the value for E and R in the equations have not changed. Just the coefficients of E has changed. The original voltage value is halved. By halving the coefficients of V the value of Equation 2 is 0.25 the value of Equation 1.
Since equation 2A is proven mathematically it can be used in the following form: P = (New E/Original E) X (New E/Original E) X Original E / R
If the value for E is unchanged the ratio of new to original is one and there is no change in power. If new is larger the ratio is greater than 1 and the power is increased; if original is larger the ratio will be less than 1 and the power is decreased
As an apprenticeship instructor I only have one comment .... I wish my students were resourceful enough to ask for help in here. Heck I'd answer their questions if I saw them posted here. After all, electricity is an open book career.
Figure a certain Heater [AKA Resistor] having a Resistance of 200 Ohms will draw a True Power of 200 Watts [dissipate that much heat energy], when connected to a power supply with an EMF [Voltage] of 200 Volts - thus driving one Ampere through the Resistor.
Looks like this math-wise:
(E= Voltage / EMF, I= Amperes / Intensity, R= Resistance -in Ohms, P= True Power -in Watts)
I = E / R [or E x R = I] R=200 Ohms E=200 Volts I=1 Ampere ( 200 / 200 = 1 )
E x I = P [True Power across a linear or pure resistance load] E=200 Volts I=1 Amperes P=200 Watts