A small manufacturing company produces custom widgets. In January, when they produced 200 widgets, they had a loss of $1,500....

1. TRANSLATE the problem information

• Given information:
  • January: 200 widgets produced, loss of $1,500
  • July: 500 widgets produced, profit of $900
  • Profit changes linearly with widgets produced

• What this tells us: "Loss of $1,500" means profit = -$1,500, so our coordinate points are \(\mathrm{(200, -1500)}\) and \(\mathrm{(500, 900)}\)

2. INFER the mathematical approach

• Since profit changes linearly, we need to find a function \(\mathrm{P(w) = mw + b}\)
• Strategy: Find the slope first, then use point-slope form to get the equation
• We have two points, which is exactly what we need for a linear function

3. SIMPLIFY to find the slope

• Using slope formula: \(\mathrm{m = \frac{y_2 - y_1}{x_2 - x_1}}\)
• \(\mathrm{m = \frac{900 - (-1500)}{500 - 200}}\)
• \(\mathrm{= \frac{900 + 1500}{300}}\)
• \(\mathrm{= \frac{2400}{300} = 8}\)

4. INFER which form to use and SIMPLIFY the equation

• Use point-slope form with point \(\mathrm{(200, -1500)}\): \(\mathrm{P(w) - (-1500) = 8(w - 200)}\)
• Expand: \(\mathrm{P(w) + 1500 = 8w - 1600}\)
• Solve for \(\mathrm{P(w)}\):
  \(\mathrm{P(w) = 8w - 1600 - 1500}\)
  \(\mathrm{P(w) = 8w - 3100}\)

5. Verify the solution

• Check with second point:
  \(\mathrm{P(500) = 8(500) - 3100}\)
  \(\mathrm{= 4000 - 3100}\)
  \(\mathrm{= 900}\) ✓

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret "loss of $1,500" as a positive value instead of recognizing it means profit = -$1,500.

When they use \(\mathrm{(200, 1500)}\) instead of \(\mathrm{(200, -1500)}\), their slope calculation becomes:
\(\mathrm{m = \frac{900 - 1500}{500 - 200}}\)
\(\mathrm{= \frac{-600}{300} = -2}\)

This leads them toward Choice B (\(\mathrm{P(w) = -8w + 1100}\)) since it has a negative slope, even though the actual slope and y-intercept are wrong.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly find slope = 8 and set up point-slope form, but make algebraic errors when expanding and combining terms.

Common mistake: \(\mathrm{P(w) = 8(w - 200) - 1500}\) becomes \(\mathrm{P(w) = 8w - 1600 + 1500 = 8w - 100}\) (wrong sign)

This doesn't match any answer choice exactly, leading to confusion and guessing or incorrectly selecting Choice D (\(\mathrm{P(w) = 8w - 2500}\)) as the "closest" option.

The Bottom Line:

This problem tests whether students can accurately translate business language into mathematical coordinates and then execute the linear function process without algebraic errors. The key insight is recognizing that "loss" means negative profit.