QUESTION STEM:A parking garage charges a flat entry fee of $10 plus an additional $3 for each hour parked. If...

1. TRANSLATE the problem information

• Given information:
  • Entry fee = $10 (one-time charge)
  • Hourly rate = $3 per hour
  • Total paid = $46
  • Need to find: number of hours parked

• What this tells us: We have a linear cost structure with a fixed cost plus variable cost

2. INFER the approach

• We need to set up an equation where total cost equals fixed cost plus variable cost
• The structure should be: Total = Fixed + (Rate × Time)
• Let \(\mathrm{h}\) = number of hours parked

3. Set up and solve the equation

• TRANSLATE the cost structure: \(10 + 3\mathrm{h} = 46\)
• SIMPLIFY by subtracting 10 from both sides: \(3\mathrm{h} = 36\)
• SIMPLIFY by dividing both sides by 3: \(\mathrm{h} = 12\)

Answer: 12

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret the cost structure and set up incorrect equations like \(10 \times 3\mathrm{h} = 46\) or \(10 + 3 = 46 - \mathrm{h}\)

They may confuse the flat fee with a multiplier or not properly understand that the $3 applies to each hour. This leads to completely wrong numerical answers like 1.53 or other values that don't make sense in context.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(10 + 3\mathrm{h} = 46\) but make arithmetic errors in solving

For example, they might subtract incorrectly (\(3\mathrm{h} = 44\) instead of \(3\mathrm{h} = 36\)) or divide incorrectly (\(\mathrm{h} = 13\) instead of \(\mathrm{h} = 12\)). These calculation mistakes lead to answers like 13, 14.67, or other close-but-wrong values.

The Bottom Line:

This problem requires students to recognize the linear cost function pattern in real-world language and then execute multi-step algebra accurately. The combination of word-problem interpretation and algebraic manipulation creates multiple opportunities for error.