At a fitness center, members can purchase day passes and class passes, each with a different fixed price. A member...

1. TRANSLATE the equation information

• Given equation: \(12\mathrm{D} + 28\mathrm{C} = 268\)
• This represents: (cost per day pass)(number of day passes) + (cost per class pass)(number of class passes) = total cost
• What this tells us:
  • 12 = cost of one day pass in dollars
  • 28 = cost of one class pass in dollars

2. INFER what the question is asking

• "How many more dollars does a class pass cost than a day pass?"
• This means: (cost of class pass) - (cost of day pass)
• We need to find: \(28 - 12\)

3. Calculate the difference

• Difference = \(28 - 12 = 16\)

Answer: 16

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misunderstand what the coefficients represent in the context equation.

They might think the coefficients represent the actual number of passes purchased rather than the unit cost per pass. This leads them to try solving for D and C values instead of recognizing that the coefficients directly give the answer. This leads to confusion and overcomplicating the problem when they realize they can't uniquely solve for D and C.

Second Most Common Error:

Poor INFER reasoning: Students correctly identify the unit costs but get confused about what "how many more" means.

They might add the costs (\(12 + 28 = 40\)) instead of finding the difference, or they might subtract in the wrong order (\(12 - 28 = -16\)) and then get confused by the negative result. This may lead them to select incorrect numerical answers or abandon systematic solution and guess.

The Bottom Line:

This problem tests whether students can extract meaning from equation coefficients in real-world contexts rather than just solve algebraically. The key insight is recognizing that sometimes the setup gives you the answer directly without needing to solve for variables.