Two printers, X and Y, produce 18 and 24 pages per minute, respectively. During a print job, Printer X ran...

1. TRANSLATE the problem information

• Given information:
  • Printer X: 18 pages per minute, runs for p minutes
  • Printer Y: 24 pages per minute, runs for q minutes
  • Combined: more than 1,000 pages total

2. INFER the approach

• Total output strategy: Each printer's contribution = \(\mathrm{rate \times time}\)
• We need to add both printers' outputs and compare to 1,000
• "More than" means we use the > inequality symbol

3. TRANSLATE each printer's contribution

• Printer X contribution: \(\mathrm{18\ pages/minute \times p\ minutes = 18p\ pages}\)
• Printer Y contribution: \(\mathrm{24\ pages/minute \times q\ minutes = 24q\ pages}\)

4. INFER the total and set up inequality

• Total pages = \(\mathrm{18p + 24q}\)
• Since total is "more than 1,000 pages": \(\mathrm{18p + 24q \gt 1000}\)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skills: Mixing up which printer has which rate when setting up the mathematical expressions.

Students might incorrectly assign 24 to Printer X and 18 to Printer Y, leading to \(\mathrm{24p + 18q \gt 1000}\). They scan the problem quickly, see both numbers (18 and 24), and don't carefully match each rate to the correct printer variable.

This may lead them to select Choice C (\(\mathrm{24p + 18q \gt 1000}\))

Second Most Common Error:

Poor TRANSLATE reasoning: Using the wrong inequality direction by misinterpreting "more than."

Some students might think "more than 1,000" means the expression should be less than 1,000, perhaps confusing the constraint direction or thinking about remaining capacity instead of total output.

This may lead them to select Choice B (\(\mathrm{18p + 24q \lt 1000}\))

The Bottom Line:

This problem tests your ability to carefully match rates with their corresponding variables and correctly translate inequality language. Success depends on methodical translation rather than rushing through the setup.