A company's profit, \(\mathrm{P(n)}\), in dollars, from selling n items is modeled by the function \(\mathrm{P(n) = 17n - 459}\),...

1. TRANSLATE the problem information

• Given information:
  • Profit function: \(\mathrm{P(n) = 17n - 459}\)
  • Break-even point definition: number of items where profit = $0
  • Need to find: the value of n when profit is zero
• This tells us we need to set \(\mathrm{P(n) = 0}\)

2. TRANSLATE the break-even condition

• Break-even means profit equals zero
• Mathematically: \(\mathrm{P(n) = 0}\)
• Substitute our function: \(\mathrm{17n - 459 = 0}\)

3. SIMPLIFY the equation to solve for n

• Starting equation: \(\mathrm{17n - 459 = 0}\)
• Add 459 to both sides: \(\mathrm{17n = 459}\)
• Divide both sides by 17: \(\mathrm{n = 459 ÷ 17}\)
• Calculate: \(\mathrm{n = 27}\)

Answer: C. 27

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may misunderstand what "break-even point" means and try to find when profit equals the number of items sold, or set up \(\mathrm{P(n) = n}\) instead of \(\mathrm{P(n) = 0}\).

This conceptual confusion leads them to solve equations like \(\mathrm{17n - 459 = n}\), getting \(\mathrm{n = 28.6}\), which doesn't match any answer choice. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{P(n) = 0}\) but make arithmetic errors when dividing \(\mathrm{459 ÷ 17}\), potentially calculating incorrectly and arriving at values that might lead them to select Choice (A) -27 or Choice (D) 459.

The Bottom Line:

This problem tests whether students can connect business terminology (break-even) with mathematical conditions (profit = 0). The algebra is straightforward once the correct equation is established.