A custom phone case company determines the price for a bulk order based on a set-up fee and a price...

1. TRANSLATE the problem information

• Given information:
  • Equation: \(\mathrm{P = 25 + 6n}\) (where \(\mathrm{P}\) = total price, \(\mathrm{n}\) = number of cases)
  • Need to find total price for 10 cases

• This tells us we need to substitute \(\mathrm{n = 10}\) into the equation

2. SIMPLIFY using substitution and order of operations

• Substitute \(\mathrm{n = 10}\) into \(\mathrm{P = 25 + 6n}\):
  \(\mathrm{P = 25 + 6(10)}\)

• Apply order of operations (PEMDAS):
  • First, perform multiplication: \(\mathrm{6 \times 10 = 60}\)
  • Then, perform addition: \(\mathrm{P = 25 + 60 = 85}\)

Answer: $85 (Choice C)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students incorrectly apply order of operations by adding before multiplying.

They might calculate:
\(\mathrm{P = 25 + 6(10)}\)
\(\mathrm{P = (25 + 6) \times 10}\)
\(\mathrm{P = 31 \times 10 = 310}\)

This may lead them to select Choice D (310).

Second Most Common Error:

Poor TRANSLATE reasoning: Students misunderstand what the equation represents and think they only need part of the calculation.

They might only calculate the "per case" portion: \(\mathrm{6 \times 10 = 60}\), forgetting about the setup fee of $25.

This may lead them to select Choice B (60).

The Bottom Line:

This problem tests whether students can correctly substitute values into linear equations and apply order of operations. The key insight is recognizing that the $25 represents a fixed setup fee that gets added to the variable cost of $6 per case.