In basic circuit theory, which statement about the equivalent resistance of a parallel network is correct? Select the one that best describes how total resistance compares with the individual branch resistances in a parallel circuit.

Introduction / Context:
Understanding how resistors combine is a core skill in electrical engineering and electronics. In parallel circuits, multiple resistive paths connect across the same two nodes. The question asks you to identify the correct relationship between the total (equivalent) resistance and the individual branch resistances in such a network.

Given Data / Assumptions:

• A resistive network with two or more branches in parallel.
• All components are ideal resistors.
• Direct current or the resistive part of an AC circuit is considered (no reactive elements).

Concept / Approach:
For resistors in parallel, conductances (the reciprocals of resistances) add. If R_eq is the equivalent resistance, then 1 / R_eq = 1 / R1 + 1 / R2 + ... + 1 / Rn. Because each term on the right is positive, the sum of reciprocals is greater than the reciprocal of any single branch. Consequently, R_eq is less than the smallest branch resistance. This is the intuitive result of adding extra current paths that reduce overall opposition to current.

Step-by-Step Solution:
Write the parallel formula: 1 / R_eq = Σ (1 / Ri).Observe that Σ (1 / Ri) > 1 / R_min, where R_min is the smallest Ri.Take reciprocals: R_eq < R_min.Therefore, the equivalent resistance is always less than the smallest branch resistance.

Verification / Alternative check:
Consider two equal resistors R in parallel: 1 / R_eq = 1 / R + 1 / R = 2 / R ⇒ R_eq = R / 2, clearly less than R. Adding more parallel branches continues to decrease R_eq.

Why Other Options Are Wrong:
Equal to the sum: that rule applies to series, not parallel.Equal to the total voltage: resistance is not equal to voltage; they are different quantities.Equal to the average: there is no averaging rule for parallel resistance; the value is always below the smallest branch.None of the above: incorrect because the correct statement is provided.

Common Pitfalls:
Confusing series and parallel rules, mixing up units (ohms vs volts), or forgetting that adding parallel branches decreases total resistance.

Final Answer:
The total resistance of a parallel circuit is always less than the value of the smallest resistance