The equation 12t + b = c relates the variables t, b, and c. Which of the following correctly expresses...

1. TRANSLATE the problem requirements

• Given equation: \(12\mathrm{t} + \mathrm{b} = \mathrm{c}\)
• Need to find: An expression for \(\mathrm{c} - \mathrm{b}\) in terms of \(\mathrm{t}\)
• This means we need to rearrange the equation so that \(\mathrm{c} - \mathrm{b}\) appears by itself on one side

2. INFER the algebraic strategy

• Looking at our equation \(12\mathrm{t} + \mathrm{b} = \mathrm{c}\), we can see that \(\mathrm{c} - \mathrm{b}\) would give us just the \(12\mathrm{t}\) term
• Strategy: Subtract \(\mathrm{b}\) from both sides to isolate \(\mathrm{c} - \mathrm{b}\)
• This will eliminate \(\mathrm{b}\) from the right side and move it to the left

3. SIMPLIFY through algebraic manipulation

• Start with: \(12\mathrm{t} + \mathrm{b} = \mathrm{c}\)
• Subtract \(\mathrm{b}\) from both sides: \(12\mathrm{t} + \mathrm{b} - \mathrm{b} = \mathrm{c} - \mathrm{b}\)
• Left side simplifies: \(12\mathrm{t} = \mathrm{c} - \mathrm{b}\)
• Therefore: \(\mathrm{c} - \mathrm{b} = 12\mathrm{t}\)

Answer: D. \(12\mathrm{t}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the direct path to isolating \(\mathrm{c} - \mathrm{b}\) and instead try to solve for individual variables first.

They might attempt to solve for \(\mathrm{c}\) first (getting \(\mathrm{c} = 12\mathrm{t} + \mathrm{b}\)), then try to find \(\mathrm{b}\) separately, making the problem unnecessarily complex. Or they might try to solve for \(\mathrm{t}\) in terms of other variables, missing that the question asks for \(\mathrm{c} - \mathrm{b}\) as a unit.

This leads to confusion and potentially guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify the strategy but make algebraic errors.

They might subtract \(\mathrm{b}\) incorrectly, forgetting to subtract it from both sides, or make sign errors during the manipulation. Some students might correctly get to \(12\mathrm{t} = \mathrm{c} - \mathrm{b}\) but fail to recognize this directly answers the question.

This may lead them to select Choice A (\(\mathrm{t}/12\)) if they incorrectly divide, or Choice B (\(\mathrm{t}\)) if they drop the coefficient.

The Bottom Line:

This problem tests whether students can recognize that they don't need to solve for individual variables - they can directly manipulate the equation to isolate the target expression \(\mathrm{c} - \mathrm{b}\). The key insight is seeing \(\mathrm{c} - \mathrm{b}\) as a single unit to isolate, not as two separate variables.