A dog-walking service offers 1-hour walks for small dogs and 2-hour walks for large dogs and can work at most...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{s}\) = number of small dog walks
  • Each small dog walk takes 1 hour
  • \(\mathrm{l}\) = number of large dog walks
  • Each large dog walk takes 2 hours
  • Service can work "at most 8 hours per day"

• What this tells us: We need to express total time and compare it to the 8-hour limit.

2. INFER the mathematical relationship

• Total time = time for small dogs + time for large dogs
• Total time = (s walks × 1 hour each) + (l walks × 2 hours each)
• Total time = \(\mathrm{s + 2l}\) hours

3. TRANSLATE the constraint

• "At most 8 hours" means the total time can be 8 hours or less
• This translates to: \(\mathrm{s + 2l \leq 8}\)

Answer: (A) \(\mathrm{s + 2l \leq 8}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Confusing "at most" with "at least"

Students sometimes interpret "at most 8 hours" as meaning they must work at least 8 hours to be efficient or profitable. This backwards reasoning leads them to use ≥ instead of ≤.

This may lead them to select Choice (B) (\(\mathrm{s + 2l \geq 8}\))

Second Most Common Error:

Poor TRANSLATE reasoning: Incorrectly assigning time coefficients

Students may forget which type of walk takes longer, or accidentally assign the coefficient 2 to small dogs instead of large dogs. They might think "small dogs are more work because there are more of them" and write 2s instead of 2l.

This may lead them to select Choice (D) (\(\mathrm{2s + l \leq 8}\))

The Bottom Line:

This problem tests your ability to carefully translate word relationships into mathematical symbols. The key is paying close attention to which quantity takes more time and what direction the inequality should face based on the constraint language.