An engineer uses the equation I = V/4R to model the current I, in amperes, through a resistor, where V...

1. TRANSLATE the problem information

• Given equation: \(\mathrm{I = \frac{V}{4R}}\)
• Goal: Solve for \(\mathrm{R}\) in terms of \(\mathrm{I}\) and \(\mathrm{V}\) (isolate \(\mathrm{R}\))

2. INFER the solution strategy

• Since \(\mathrm{R}\) is in the denominator and multiplied by 4, we need to:
  • First eliminate the fraction by multiplying both sides by \(\mathrm{4R}\)
  • Then isolate \(\mathrm{R}\) by dividing appropriately

3. SIMPLIFY through algebraic manipulation

• Start with: \(\mathrm{I = \frac{V}{4R}}\)
• Multiply both sides by \(\mathrm{4R}\): \(\mathrm{I \times 4R = V \times 1}\)
• This gives us: \(\mathrm{4RI = V}\)
• Divide both sides by \(\mathrm{4I}\): \(\mathrm{R = \frac{V}{4I}}\)

4. Verify the solution

• Substitute \(\mathrm{R = \frac{V}{4I}}\) back into the original equation:
• \(\mathrm{I = \frac{V}{4(\frac{V}{4I})} = \frac{V}{(\frac{V}{I})} = I}\) ✓

Answer: (A) \(\mathrm{R = \frac{V}{4I}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the strategic approach needed to isolate a variable that appears in a denominator. Instead of multiplying by \(\mathrm{4R}\) to clear the fraction, they might try to "flip" the equation or make random algebraic moves.

This leads to confusion and incorrect manipulations, causing them to select wrong answer choices or abandon the systematic approach and guess.

Second Most Common Error:

Poor SIMPLIFY execution: Students understand the strategy but make algebraic errors during execution. Common mistakes include:

• Forgetting to multiply BOTH sides by \(\mathrm{4R}\)
• Confusing the order of operations when dealing with fractions
• Making arithmetic errors when combining terms

This may lead them to select Choice (B) (\(\mathrm{R = \frac{4V}{I}}\)) if they incorrectly manipulate the algebra and get the 4 in the wrong position.

The Bottom Line:

This problem tests whether students can systematically approach equation solving when the target variable is "buried" in a fraction. Success requires both strategic thinking about the sequence of operations and careful algebraic execution.