A small business operates under the budget constraint 12h - 8s = -24, where h represents hours of consulting services...

1. TRANSLATE the problem information

• Given information:
  • Budget constraint equation: \(12\mathrm{h} - 8\mathrm{s} = -24\)
  • Number of software licenses purchased: \(\mathrm{s} = 15\)
  • Need to find: hours of consulting services (h)

• What this tells us: We need to substitute s = 15 into the equation and solve for h

2. SIMPLIFY through substitution and algebra

• Substitute s = 15 into \(12\mathrm{h} - 8\mathrm{s} = -24\):
  \(12\mathrm{h} - 8(15) = -24\)

• Calculate \(8 \times 15\):
  \(12\mathrm{h} - 120 = -24\)

• Add 120 to both sides:
  \(12\mathrm{h} = -24 + 120\)
  \(12\mathrm{h} = 96\)

• Divide both sides by 12:
  \(\mathrm{h} = 8\)

Answer: B (8)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Sign errors when moving terms across the equals sign
Students might incorrectly write: \(12\mathrm{h} = -24 - 120 = -144\), leading to \(\mathrm{h} = -12\)
Since this isn't an answer choice, this leads to confusion and guessing.

Second Most Common Error:

Poor TRANSLATE reasoning: Confusing what the problem is asking for
Students might see s = 15 and think the answer is 15 hours, not recognizing they need to substitute this value to find h.
This may lead them to select Choice C (15).

The Bottom Line:

This problem tests your ability to systematically substitute known values and carefully track positive/negative signs through algebraic manipulation. The business context is just window dressing - the core skill is solving a linear equation.