In a simple electric circuit, if you want to decrease the overall or effective resistance of the circuit by adding more resistors, in which way should these additional resistances be connected?

Introduction / Context:
In basic electrical circuits, resistors can be combined in series or in parallel to obtain different effective resistance values. Understanding how these combinations change the total resistance is crucial for designing circuits that work safely and efficiently. This question asks you to identify the arrangement that decreases the overall resistance when additional resistors are added to a circuit. It tests your grasp of the formulas for series and parallel combinations of resistances.

Given Data / Assumptions:

• We have more than one resistor that can be connected in a circuit.
• The goal is specifically to decrease the effective or total resistance.
• Possible arrangements are series, parallel, mixed, or unspecified.
• We assume all resistors have finite, positive resistance values.

Concept / Approach:
When resistors are connected in series, their resistances simply add, so the total resistance increases as more resistors are added. The formula is R_total = R1 + R2 + R3 and so on. In contrast, when resistors are connected in parallel, the reciprocal of the total resistance equals the sum of the reciprocals of individual resistances: 1 / R_total = 1 / R1 + 1 / R2 + 1 / R3 and so on. For any two or more positive resistances in parallel, the effective resistance is always less than the smallest individual resistance. Therefore, to reduce the overall resistance of a circuit, you should connect additional resistors in parallel across the same two points of the circuit.

Step-by-Step Solution:
Step 1: Recall the formula for series combination: R_series = R1 + R2 + R3, which always makes the total resistance larger than any single resistance.Step 2: Recall the formula for parallel combination: 1 / R_parallel = 1 / R1 + 1 / R2 + 1 / R3.Step 3: Observe that for positive resistances, adding more terms to the right side increases the sum of reciprocals, making R_parallel smaller.Step 4: Conclude that the effective resistance in parallel is always less than each individual resistor in that branch.Step 5: Therefore, to decrease overall resistance, you must connect additional resistors in parallel, not in series.

Verification / Alternative check:
Take a simple numerical example. Suppose you have a single 10 ohm resistor. Its resistance is 10 ohm. If you add another 10 ohm resistor in series, the total becomes 20 ohm, which is larger. But if you add the same second resistor in parallel, the effective resistance is given by 1 / R_total = 1 / 10 + 1 / 10 = 2 / 10, so R_total = 5 ohm, which is smaller. This direct numerical check confirms that series connections increase resistance, while parallel connections reduce it. Mixed arrangements can produce intermediate values but are not guaranteed to reduce the resistance below the smallest component.

Why Other Options Are Wrong:
Connecting resistors in series, option A, always increases the total resistance and therefore does the opposite of what the question requires. A mixed series parallel arrangement, option C, may either increase or decrease resistance depending on the details but does not guarantee a reduction and is not the simplest answer. Claiming that the connection method does not matter, option D, is incorrect because the formulas for series and parallel combinations give clearly different results. Only option B, connecting resistors in parallel across the same two points, reliably decreases the total resistance.

Common Pitfalls:
Students sometimes memorise the formulas but forget their physical meaning. It is easy to think that adding more resistors must always increase resistance, like adding more obstacles in a path. However, a parallel branch offers extra pathways for current, which effectively reduces the opposition to current flow. To avoid confusion, remember the guiding idea: series narrows the path and increases resistance, while parallel opens new paths and decreases resistance. Combining this intuition with the formulas helps in solving many circuit questions quickly.

Final Answer:
Connected in parallel across the same two points