A technician laminates posters and ID cards. Laminating one poster takes 40 seconds, and laminating one ID card takes 18...

1. TRANSLATE the problem information

• Given information:
  • One poster takes 40 seconds to laminate
  • One ID card takes 18 seconds to laminate
  • Total time period: 31 minutes
  • Number of posters laminated: x
  • Number of ID cards laminated: y

2. TRANSLATE time units for consistency

• The individual times are given in seconds, but the total time is in minutes
• Convert: \(\mathrm{31\ minutes \times 60\ seconds/minute = 1860\ seconds}\)

3. INFER the relationship structure

• Total time used = time for posters + time for ID cards
• Time for x posters = \(\mathrm{x \times 40\ seconds = 40x\ seconds}\)
• Time for y ID cards = \(\mathrm{y \times 18\ seconds = 18y\ seconds}\)

4. TRANSLATE into equation form

• Total equation: \(\mathrm{40x + 18y = 1860}\)
• This matches choice (D) exactly

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students set up the equation without converting time units consistently.

They see "31 minutes" and directly write \(\mathrm{40x + 18y = 31}\), forgetting that the individual times (40 seconds, 18 seconds) are in seconds, not minutes. They don't recognize the unit mismatch.

This leads them to select Choice A (\(\mathrm{40x + 18y = 31}\)).

Second Most Common Error:

Poor TRANSLATE reasoning: Students mix up which coefficient goes with which variable.

They might think "18 seconds for posters, 40 seconds for ID cards" instead of carefully tracking which time belongs to which item. This reversal of coefficients happens when students rush through the setup.

This leads them to select Choice B (\(\mathrm{18x + 40y = 1860}\)).

The Bottom Line:

This problem tests your ability to maintain unit consistency and carefully match coefficients with their corresponding variables. The key insight is recognizing that all time measurements must use the same units before creating the equation.