Q.In the series circuit shown, a $2.5\ \Omega$ resistor, a $3\ \Omega$ resistor and an unknown resistor P are all connected in series to a 9 V battery, and there is a current of 1.0 A in the circuit. What is the resistance of P?

Series combination. When resistors are connected end to end so the same current I is forced through every one of them in turn (since charge cannot accumulate anywhere along the chain), the voltage across each one differs according to Ohm's law ($V_1=IR_1$, etc.), and the supply voltage V is the SUM of these individual drops: $V=V_1+V_2+V_3=I(R_1+R_2+R_3)$. This defines the equivalent series resistance,

which is always GREATER than even the largest individual resistor in the chain.

Parallel combination. When resistors are instead all connected directly across the same two points, each one experiences the identical voltage V, but the total current I splits into separate branch currents ($I_1=V/R_1$, etc.) that must add back up to the total, $I=I_1+I_2+I_3$. This gives the equivalent parallel resistance through the reciprocal-sum rule,

which is always LESS than even the smallest individual resistor -- adding another parallel branch can only open up additional room for current to flow, never restrict it.

Why household wiring is parallel. Every appliance in a house is deliberately connected in parallel across the mains supply, precisely because a parallel connection guarantees each device gets the full supply voltage and operates completely independently of the others -- switching any one appliance off leaves the voltage across, and the current through, every other appliance completely unaffected. A series connection, by contrast, would mean switching off any single appliance breaks the circuit for every appliance downstream of it. …