Question: If 15 - 3p = 27, what is the value of p - 5?-14-9-41

1. SIMPLIFY the equation to solve for p

• Given equation: \(15 - 3\mathrm{p} = 27\)
• Move the constant to the right side:
  • Subtract 15 from both sides: \(-3\mathrm{p} = 12\)
• Isolate p:
  • Divide both sides by -3: \(\mathrm{p} = -4\)

2. SIMPLIFY to find the value of p - 5

• Substitute \(\mathrm{p} = -4\) into the expression \(\mathrm{p} - 5\):

\(\mathrm{p} - 5 = (-4) - 5 = -9\)

Answer: B. -9

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Sign errors when working with negative numbers

Students often make mistakes like:

• When dividing \(-3\mathrm{p} = 12\) by -3, getting \(\mathrm{p} = +4\) instead of \(\mathrm{p} = -4\)
• When calculating \(\mathrm{p} - 5 = (-4) - 5\), getting -1 instead of -9 by treating it as \((-4) + 5\)

This may lead them to select Choice (D) (1) if they get \(\mathrm{p} = 4\), then \(\mathrm{p} - 5 = -1\), or other incorrect choices.

Second Most Common Error:

Poor INFER reasoning: Not recognizing the direct manipulation approach

Some students might attempt to expand or manipulate the original equation incorrectly, missing that they can solve systematically for p first. They might try to work directly with \(\mathrm{p} - 5\) without a clear strategy, leading to confusion and guessing.

The Bottom Line:

This problem tests careful algebraic manipulation with negative numbers. The key is systematic step-by-step solving while being extra careful with signs throughout the process.