In electrical circuits, when identical resistors are connected in parallel, the total resistance R_total is given by the formula {1}{R_total}...

1. TRANSLATE the problem information

• Given information:
  • Each resistor: \(\mathrm{R = 60}\) ohms
  • Formula for parallel circuits: \(\frac{1}{\mathrm{R_{total}}} = \frac{\mathrm{n}}{\mathrm{R}}\)
  • Constraint: "at least 4 ohms" means \(\mathrm{R_{total} \geq 4}\)
  • Find: maximum number of resistors n

2. INFER the approach needed

• Since we want the "maximum number" of resistors with a constraint, this becomes an inequality problem
• We'll use the parallel resistance formula, then solve an inequality

3. Apply the parallel resistance formula

Start with: \(\frac{1}{\mathrm{R_{total}}} = \frac{\mathrm{n}}{\mathrm{R}}\)

Substitute \(\mathrm{R = 60}\): \(\frac{1}{\mathrm{R_{total}}} = \frac{\mathrm{n}}{60}\)

4. SIMPLIFY to find R_total in terms of n

• Flip both sides: \(\mathrm{R_{total} = \frac{60}{n}}\)
• Now we can work with the constraint

5. TRANSLATE and apply the constraint

• "At least 4 ohms" means: \(\mathrm{R_{total} \geq 4}\)
• Substitute our expression: \(\frac{60}{\mathrm{n}} \geq 4\)

6. SIMPLIFY the inequality

• Multiply both sides by n (n is positive, so inequality direction stays the same)
• \(60 \geq 4\mathrm{n}\)
• Divide by 4: \(15 \geq \mathrm{n}\), or \(\mathrm{n \leq 15}\)

7. APPLY CONSTRAINTS for the final answer

• Since n represents number of resistors, it must be a positive integer
• The maximum whole number satisfying \(\mathrm{n \leq 15}\) is \(\mathrm{n = 15}\)

Answer: 15

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may misinterpret "at least 4 ohms" as meaning \(\mathrm{R_{total} = 4}\) exactly, rather than \(\mathrm{R_{total} \geq 4}\). They solve \(\frac{60}{\mathrm{n}} = 4\) to get \(\mathrm{n = 15}\), which happens to be correct by coincidence, but they miss the underlying inequality reasoning. This could lead to confusion on similar problems where the constraint creates a different boundary.

Second Most Common Error:

Poor INFER reasoning: Students may not recognize this as a constraint optimization problem. They might try to work backwards from common circuit values or guess-and-check with small numbers of resistors, never systematically finding the maximum. This leads to confusion and guessing among the answer choices.

The Bottom Line:

This problem requires students to bridge electrical circuit knowledge with algebraic inequality solving. The key insight is recognizing that "maximum number with constraint" signals an inequality problem, not a simple equation.