Combine two parallel resistances

Supplied materials

• All measurements and observations below are supplied classroom examples. They are not data from actual learners or a claim about a particular real site.
• Paper or browser document; an ordinary calculator is optional.
• Rule: Two-resistor parallel equivalent resistance equals their product divided by their sum. Formula: a*b/(a+b). Variable key: a = first resistance (Ω), b = second resistance (Ω)
• Distinct worked inputs and result: A distinct parallel model has 2 Ω and 2 Ω: (2×2)/(2+2)=1 Ω.
• Assessment case 1: Supplied case 1: first resistance (Ω)=4; second resistance (Ω)=4, using the explicitly named quantities. Two-resistor parallel equivalent resistance equals their product divided by their sum. What result follows?
• Assessment case 2: Supplied case 2: first resistance (Ω)=6; second resistance (Ω)=3, using the explicitly named quantities. Two-resistor parallel equivalent resistance equals their product divided by their sum. What result follows?
• Assessment case 3: Supplied case 3: first resistance (Ω)=8; second resistance (Ω)=8, using the explicitly named quantities. Two-resistor parallel equivalent resistance equals their product divided by their sum. What result follows?
• Assessment case 4: Supplied case 4: first resistance (Ω)=12; second resistance (Ω)=6, using the explicitly named quantities. Two-resistor parallel equivalent resistance equals their product divided by their sum. What result follows?
• Assessment case 5: Supplied case 5: first resistance (Ω)=10; second resistance (Ω)=10, using the explicitly named quantities. Two-resistor parallel equivalent resistance equals their product divided by their sum. What result follows?
• Assessment case 6: Supplied case 6: first resistance (Ω)=9; second resistance (Ω)=18, using the explicitly named quantities. Two-resistor parallel equivalent resistance equals their product divided by their sum. What result follows?

Learn the idea

Parallel branches share voltage. For two ideal resistors, reciprocal resistances add, yielding equivalent resistance smaller than either positive branch resistance. This is an idealized model.

Two-resistor parallel equivalent resistance equals their product divided by their sum. The formula in the supplied model is a*b/(a+b). Variable key: a = first resistance (Ω), b = second resistance (Ω). Do not apply the series sum. The result should be below the smaller branch resistance.

A distinct parallel model has 2 Ω and 2 Ω: (2×2)/(2+2)=1 Ω. Parallel branches share voltage. Track the quantity and unit before calculating. Explain the meaning of a positive, negative or fractional result when relevant. A numerical answer supports the stated model calculation; broader claims require the additional measurements and assumptions described here.

1. Name the measured quantities and their units before substituting values.
2. Write the stated rule and use the variable key to match each symbol to its explicitly named quantity.
3. Calculate each supplied case separately, keeping intermediate arithmetic and the result unit.
4. Explain the result in context and identify the assumption that would invalidate direct use of the rule.

Check your reasoning

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Take it further

Changed-input task: use the same stated model with first resistance (Ω)=5; second resistance (Ω)=4. Expected result: approximately 2.2222222 Ω. Explain what changes and keep this assumption: Parallel branches share voltage.

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