An economist modeled the demand Q for a certain product as a linear function of the selling price P. The...

1. TRANSLATE the problem information

• Given information:
  • When price \(\mathrm{P = \$40}\), demand \(\mathrm{Q = 20{,}000}\) units → coordinate point \(\mathrm{(40, 20000)}\)
  • When price \(\mathrm{P = \$60}\), demand \(\mathrm{Q = 15{,}000}\) units → coordinate point \(\mathrm{(60, 15000)}\)
  • Need to find Q when \(\mathrm{P = \$55}\)

2. INFER the solution approach

• Since this is a linear function, I need to find the equation \(\mathrm{Q = mP + b}\)
• Strategy: Use the two points to find slope m, then find the y-intercept b
• Once I have the equation, substitute \(\mathrm{P = 55}\)

3. SIMPLIFY to find the slope

• Using slope formula: \(\mathrm{m = \frac{Q_2 - Q_1}{P_2 - P_1}}\)
• \(\mathrm{m = \frac{15000 - 20000}{60 - 40}}\)
  \(\mathrm{= \frac{-5000}{20}}\)
  \(\mathrm{= -250}\)

4. SIMPLIFY to find the complete equation

• Using point-slope form with \(\mathrm{(40, 20000)}\):
• \(\mathrm{Q - 20000 = -250(P - 40)}\)
• \(\mathrm{Q - 20000 = -250P + 10000}\)
• \(\mathrm{Q = -250P + 30000}\)

5. SIMPLIFY to find the final answer

• Substitute \(\mathrm{P = 55}\):
  \(\mathrm{Q = -250(55) + 30000}\)
• \(\mathrm{Q = -13750 + 30000}\)
  \(\mathrm{= 16250}\)

Answer: A. 16,250

Why Students Usually Falter on This Problem

Most Common Error Path:

Poor TRANSLATE reasoning: Students mix up which variable goes with which coordinate, setting up points as \(\mathrm{(Q, P)}\) instead of \(\mathrm{(P, Q)}\). This leads to calculating slope as \(\mathrm{\frac{60 - 40}{15000 - 20000} = \frac{20}{-5000} = -0.004}\), which produces completely wrong equations and final answers that don't match any choice. This leads to confusion and guessing.

Second Most Common Error:

Weak SIMPLIFY execution: Students make sign errors when working with the negative slope, particularly when distributing \(\mathrm{-250(P - 40)}\) or when performing the final arithmetic. For example, calculating \(\mathrm{-250(55)}\) as \(\mathrm{+13750}\) instead of \(\mathrm{-13750}\), which gives \(\mathrm{Q = 13750 + 30000 = 43750}\). This leads to an answer far outside the given choices and causes them to get stuck and randomly select an answer.

The Bottom Line:

This problem tests whether students can systematically work with linear functions in a real-world context, requiring careful coordinate setup and consistent algebraic manipulation with negative slopes.