OHM'S LAW

Ohm's law states that for a given current, the Potential difference (P.d) between two points is directly proportional to the resistance of the circuit between those points.

V = IR

Calculating Resistances in a Circuit

If current is to flow through a circuit, it must go against resistance with majority of it taking paths of least resistance and if resistance is great (in a certain path), current will not flow.
Circuits are designed with resistances both in parallel and in series.

Series Connection of Resistances
The figure below shows three resistors having resistances of R1, R2, R3, connected in series across the supply voltage labelled Vs.

Note. When resistors are connected in series the total current flowing through the circuit is the same (current supplied (Is)) while voltage drop depends on the value of the resistance of the individual resistor and total voltage of the circuit is the sum of the voltage drops of the individual resistors.
Vs = V1+V2+V3
From ohm's law
V=IR
Substitute V for IR in the equation Vs = V1+V2+V3
IsRT=IsR1+IsR2+IsR3 (Is being current supplied and RT being total resistance)
Divide both sides by Is
IsRT/Is=Is(R1+R2+R3)/Is (Is dies on both sides)
RT=R1+R2+R3

Parallel Connection of Resistances
For resistors in parallel connection, the supply voltage is the same while current flowing depends on the value of the resistance of the individual resistor. The total supply current is the sum of all the currents flowing  through the circuit.

In the figure above three resistors having resistances of R1, R2 and R3 respectively, are connected across supply voltage Vs.
From total current
Is = I1+I2+I3
And from Ohm's law
I = V/R
Substituting I for V/R in the equation Is = I1+I2+I3
Vs/RT = Vs/R1+Vs/R2+Vs/R3 (Vs being supply voltage and RT being total resistance)
Vs/RT *Vs = Vs(1/R1+1/R2+1/R3)/Vs
1/RT = 1/R1+1/R2+1/R3
If there are two resistors connected in parallel, the total resistance RT is given by product/sum (Product over Sum)
1/RT= 1/R1+1/R2
1/RT = R1+R2/R1*R2
RT = R1*R2 /R1+R2