What value of p satisfies the equation 5p + 180 = 250?

1. INFER the solving strategy

• Given equation: \(\mathrm{5p + 180 = 250}\)
• Goal: Isolate the variable p on one side
• Strategy: Use inverse operations to "undo" what's been done to p
• Since p is multiplied by 5 and then has 180 added, we'll reverse this: first subtract 180, then divide by 5

2. SIMPLIFY by removing the constant term

• Subtract 180 from both sides:

\(\mathrm{5p + 180 - 180 = 250 - 180}\)
\(\mathrm{5p = 70}\)

3. SIMPLIFY by removing the coefficient

• Divide both sides by 5:

\(\mathrm{5p ÷ 5 = 70 ÷ 5}\)
\(\mathrm{p = 14}\)

4. Verify the solution

• Substitute \(\mathrm{p = 14}\) back into original equation:

\(\mathrm{5(14) + 180 = 70 + 180 = 250}\) ✓

Answer: A. 14

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Making arithmetic errors during the calculation steps

Students might incorrectly compute \(\mathrm{250 - 180 = 80}\) instead of 70, or miscalculate \(\mathrm{70 ÷ 5 = 15}\) instead of 14. These calculation errors lead to wrong final answers that don't match any of the given choices, causing confusion and potentially leading to guessing.

Second Most Common Error:

Poor INFER reasoning: Attempting to solve by working with the wrong side of the equation first

Some students might try to work backwards from the answer choices or get confused about the order of operations needed to isolate p. They might divide by 5 before subtracting 180, leading to fractional intermediate steps that complicate the solution unnecessarily.

The Bottom Line:

This problem tests fundamental equation-solving skills that serve as building blocks for all of algebra. Success requires both strategic thinking about the sequence of inverse operations and careful arithmetic execution.