Compare resistance before and after a parallel branch is added
Explain why two ordinary positive resistors in parallel have less combined resistance than either alone.
Section 4.3.2 · Series and parallel circuits
Core and Extended; Physics 0625, 2026–2028. Use the stated source, operating conditions and meter ranges. Written practice is self-checked.
Before you begin
- Recognize the same pair of branch endpoints.
- Compare resistance and current at fixed p.d.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
At fixed p.d., what does a greater total current imply for equivalent resistance?
What you will learn
- State the qualitative parallel-resistance comparison.
- Distinguish an extra branch from an extra series resistor.
Provide another path
Two finite positive resistors in parallel have a combined resistance smaller than either resistor by itself. At the same p.d., each branch permits current and the source supplies both flows. This greater total current corresponds to a smaller equivalent resistance across that endpoint pair.
If the resistors are unequal, the equivalent is still below the smaller individual value. It is not a value between them and not their average. The numerical reciprocal and product-over-sum rules are taught in Supplement; this Core lesson establishes and applies the qualitative comparison.
Check what changed
Adding a positive series resistor increases an ordinary series resistance total. Adding a finite positive resistor in parallel with an existing branch decreases the equivalent across the same endpoints. First classify the connections; the word add does not determine the effect.
The strict below-either comparison assumes two finite positive resistances. An open branch is not a finite positive path and adds no current; the remaining equivalent is unchanged in the ideal model. A plain-wire short is a different limiting case and should not be treated as a useful ordinary resistor branch.
Pause and explain
Two ordinary resistors of 4 Ω and 10 Ω are in parallel. Which qualitative result is valid?
Worked example
A 4 Ω resistor and a 10 Ω resistor are connected in parallel. Without calculating an exact value, compare their equivalent with 4 Ω.
Show the worked solution
- Both branch resistances are positive and finite.
- Parallel provides an additional conducting path.
- The equivalent is below the smaller value, 4 Ω.
Answer Less than 4 Ω.
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Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Compare the equivalent of 4 Ω and 10 Ω in parallel with each individual resistance.
Give me a hint
Apply the below-either qualitative rule.
Compare my reasoning
- Both values are finite and positive.
- The extra path increases total current at the same p.d.
- The equivalent is below 4 Ω and therefore below 10 Ω.
Less than 4 Ω and less than 10 Ω.
Look for these in your work
- Compare both values.
- Do not invent an exact Core calculation.
A finite positive resistor is added first in series, then in a separate comparison in parallel across the same endpoints. State each resistance trend.
Give me a hint
Classify the two connection patterns independently.
Compare my reasoning
- A series addition increases the series total.
- A parallel addition provides another conducting path.
- The parallel equivalent decreases.
Series addition increases resistance; parallel addition decreases the endpoint equivalent.
Look for these in your work
- State both arrangements.
- Keep the endpoint comparison fixed.
An open extra branch is claimed to halve the existing resistance. Assess the claim.
Give me a hint
An open path does not carry an additional branch current.
Compare my reasoning
- The extra branch has no finite conducting path.
- It contributes zero ideal current.
- The original resistance is unchanged, not halved.
No; an ideal open branch leaves the existing equivalent unchanged.
Look for these in your work
- Identify the excluded finite-positive condition.
- Do not assume equal conducting branches.
Common mistakes
- The parallel equivalent is not an average.
- An open branch does not supply another conducting path.
- Compare across the same endpoint pair.
What can you explain now?
How does the equivalent of two finite positive parallel resistors compare with either alone?
Compare with the explanation
It is smaller than either.
Both branch currents contribute at the same endpoint p.d.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Use the below-either rule for finite positive parallel resistors and distinguish series additions and open branches.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.