Question:A rectangular plot of land has a length of p-2 meters. The total cost, in dollars, to build a fence...

1. TRANSLATE the problem information

• Given information:
  • First company cost: \(12(\mathrm{p}-2)\) dollars
  • Second company cost: \(9(\mathrm{p}-2) + 75\) dollars
  • Both costs are equal
  • Find the value of \(\mathrm{p}-2\) (not p itself!)

2. TRANSLATE the equal costs condition

Since both companies charge the same amount for the identical fence:
\(12(\mathrm{p}-2) = 9(\mathrm{p}-2) + 75\)

3. INFER the most efficient solution strategy

• Notice the question asks for \(\mathrm{p}-2\), not p
• Rather than expanding everything and solving for p, we can substitute a single variable for the entire expression \(\mathrm{p}-2\)
• Let \(\mathrm{u} = \mathrm{p}-2\), so our equation becomes: \(12\mathrm{u} = 9\mathrm{u} + 75\)

4. SIMPLIFY to solve for u

Subtract \(9\mathrm{u}\) from both sides:
\(12\mathrm{u} - 9\mathrm{u} = 75\)
\(3\mathrm{u} = 75\)

Divide both sides by 3:
\(\mathrm{u} = 25\)

5. INFER the final answer

Since we let \(\mathrm{u} = \mathrm{p}-2\), the value of \(\mathrm{p}-2\) is 25.

Answer: C (25)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the substitution shortcut and instead expand both expressions completely, leading to unnecessarily complex algebra.

They might expand to get: \(12\mathrm{p} - 24 = 9\mathrm{p} - 18 + 75\), then solve \(12\mathrm{p} - 24 = 9\mathrm{p} + 57\) to find \(\mathrm{p} = 27\). However, they then forget the question asks for \(\mathrm{p}-2\), not p, and incorrectly select Choice D (27).

Second Most Common Error:

Poor SIMPLIFY execution: Students set up the equation correctly but make arithmetic errors when solving \(3\mathrm{u} = 75\).

Common calculation mistakes include dividing incorrectly or losing track of the variable substitution, leading them to guess among the remaining choices.

The Bottom Line:

This problem rewards students who read carefully (noticing it asks for \(\mathrm{p}-2\), not p) and think strategically about the most efficient solution path. The key insight is recognizing that substitution can dramatically simplify the algebra.