Benchmark rule for parallels: evaluate the statement that the equivalent resistance of any nontrivial parallel network is always less than the smallest individual resistor value present.

Introduction / Context:This benchmark rule is a quick mental check when simplifying circuits: the equivalent resistance of resistors in parallel should never exceed the smallest branch value. Recognizing this prevents calculation mistakes and helps validate calculator results before committing designs.

Given Data / Assumptions:

• At least two resistors in parallel.
• Resistors are linear and time invariant.
• We ignore parasitic effects and measurement errors.

Concept / Approach:For N parallel resistors, 1 / R_eq = sum(1 / R_k) with k from 1 to N. Because each term 1 / R_k is positive, the sum is greater than 1 / R_min, where R_min is the smallest resistor. Taking reciprocals, R_eq is less than R_min. This holds for any finite set of positive resistances and becomes a strict inequality when there are two or more branches.

Step-by-Step Solution:

Verification / Alternative check:Numerical sanity check: 6 ohm in parallel with 3 ohm gives R_eq = 2 ohm, which is less than 3 ohm, the smallest branch.

Why Other Options Are Wrong:Incorrect: contradicts the reciprocal property. Only for two branches or small ratio spreads: the inequality is general for any count and ratios.

Common Pitfalls:Applying series formula by mistake; forgetting that adding a parallel path lowers R_eq.

Final Answer:Correct