Math Problem Statement

A circuit consists of two resistors in parallel. The total resistance of the parallel arrangement is 4Ω. If one of the resistors has a resistance of 12Ω, what will be the resistance of the other one?

Solution

To solve this problem, we need to use the formula for the total resistance of resistors in parallel:

1Rtotal=1R1+1R2\frac{1}{R_{\text{total}}} = \frac{1}{R_1} + \frac{1}{R_2}

Given:

  • Rtotal=4 ΩR_{\text{total}} = 4\ \Omega
  • R1=12 ΩR_1 = 12\ \Omega

We need to find R2R_2.

Step-by-step solution:

  1. Substituting the values into the formula: 14=112+1R2\frac{1}{4} = \frac{1}{12} + \frac{1}{R_2}

  2. Simplify 112\frac{1}{12}:

\implies \frac{1}{R_2} = \frac{1}{4} - \frac{1}{12}$$ 3. Calculate the right-hand side: $$\frac{1}{4} - \frac{1}{12} = \frac{3}{12} - \frac{1}{12} = \frac{2}{12} = \frac{1}{6}$$ 4. Therefore: $$\frac{1}{R_2} = \frac{1}{6} \implies R_2 = 6\ \Omega$$ ### Final Answer: The resistance of the other resistor is: $$\boxed{6\ \Omega}$$ --- Let me know if you'd like a more detailed explanation or further examples! --- ### Related Questions: 1. How do you calculate total resistance for three resistors in parallel? 2. What is the formula for resistors connected in series, and how does it differ from parallel? 3. What happens to the total resistance when more resistors are added in parallel? 4. Can the total resistance of a parallel circuit ever be higher than the largest individual resistor? 5. How does the current divide between resistors in a parallel circuit? --- ### Tip: For parallel resistors, the total resistance is always less than the smallest individual resistance.

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Math Problem Analysis

Mathematical Concepts

Electrical Circuits
Parallel Resistance
Reciprocal Calculations

Formulas

1 / R_total = 1 / R1 + 1 / R2

Theorems

Reciprocal Rule for Parallel Resistances

Suitable Grade Level

Grades 9-12

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