\(\mathrm{T = 0.01(P - 40,000)}\)In a city, the property tax T, in dollars, is calculated using the formula above, where...

1. INFER the goal

• Given: \(\mathrm{T = 0.01(P - 40,000)}\)
• Need: Express P in terms of T (solve for P)
• Strategy: Use inverse operations to isolate P on one side

2. SIMPLIFY by eliminating the decimal

• Multiply both sides by 100 to clear the decimal:
  \(\mathrm{100T = 100 \times 0.01(P - 40,000)}\)
  \(\mathrm{100T = P - 40,000}\)

3. SIMPLIFY to isolate P

• Add 40,000 to both sides:
  \(\mathrm{100T + 40,000 = P - 40,000 + 40,000}\)
  \(\mathrm{100T + 40,000 = P}\)

Answer: D. P = 100T + 40,000

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students incorrectly handle the decimal conversion and the constant term simultaneously.

Instead of multiplying the entire equation by 100, they might multiply 40,000 by 0.01 to get 400, thinking they need to "convert everything to the same decimal form." Then they might also make a sign error, writing \(\mathrm{P = 100T - 400}\).

This leads them to select Choice A (P = 100T - 400)

Second Most Common Error:

Incomplete SIMPLIFY process: Students correctly eliminate the decimal but make a sign error when moving the constant term.

They get to \(\mathrm{100T = P - 40,000}\) but then incorrectly subtract 40,000 from both sides instead of adding it, arriving at \(\mathrm{100T - 40,000 = P}\).

This leads them to select Choice C (P = 100T - 40,000)

The Bottom Line:

This problem requires careful attention to sign changes and systematic application of inverse operations. The decimal factor adds complexity that can cause students to lose track of proper algebraic steps.