A school club purchased a set of T-shirts for its members, with a total cost of $165. The price of...

1. TRANSLATE the problem information

• Given information:
  • Total cost of T-shirts = \(\$165\)
  • Price per T-shirt = (number of T-shirts) - 4

• What this tells us: We need to find both the number of T-shirts and the price, using the relationship between them.

2. INFER the approach

• Since we have two unknowns but clear relationships, we should set up equations
• Let n = number of T-shirts and p = price per T-shirt
• The key insight is that total cost = number × price per item

3. TRANSLATE into mathematical equations

• From total cost: \(\mathrm{n} \times \mathrm{p} = 165\)
• From the price relationship: \(\mathrm{p} = \mathrm{n} - 4\)

4. SIMPLIFY by substitution

• Substitute the second equation into the first:
  \(\mathrm{n}(\mathrm{n} - 4) = 165\)

• Expand: \(\mathrm{n}^2 - 4\mathrm{n} = 165\)
• Rearrange: \(\mathrm{n}^2 - 4\mathrm{n} - 165 = 0\)

5. SIMPLIFY through factoring

• We need two numbers that multiply to -165 and add to -4
• Since \(-15 \times 11 = -165\) and \(-15 + 11 = -4\):
• \((\mathrm{n} - 15)(\mathrm{n} + 11) = 0\)
• Therefore: \(\mathrm{n} = 15\) or \(\mathrm{n} = -11\)

6. APPLY CONSTRAINTS to select valid solution

• Since number of T-shirts must be positive: \(\mathrm{n} = 15\)
• Therefore: \(\mathrm{p} = \mathrm{n} - 4 = 15 - 4 = 11\)

Answer: B. \(\$11\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students struggle to set up the correct relationship equations, particularly the "price is 4 less than the number" constraint. They might write \(\mathrm{p} = 4 - \mathrm{n}\) instead of \(\mathrm{p} = \mathrm{n} - 4\), or set up incorrect total cost equations.

This leads to wrong equations and ultimately to confusion when trying to solve, causing them to guess among the answer choices.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students make errors during the quadratic factoring process. They might factor incorrectly, getting wrong values for n, or struggle to find the correct factor pairs of -165.

This may lead them to select Choice A (\(\$9\)) or Choice C (\(\$15\)) based on incorrect calculations.

The Bottom Line:

This problem requires strong algebraic translation skills to convert the word relationships into equations, followed by systematic quadratic solving. Students who rush through the setup or make computational errors in factoring will struggle to reach the correct answer.