A cleaning service that cleans both offices and homes can clean at most 14 places per day. Which inequality represents...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{f}\) = number of offices cleaned per day
  • \(\mathrm{h}\) = number of homes cleaned per day
  • Constraint: "can clean at most 14 places per day"

2. INFER what represents the total

• Since the service cleans both offices AND homes, the total number of places cleaned per day is \(\mathrm{f + h}\)
• The constraint applies to this total

3. TRANSLATE the constraint phrase

• "At most 14" means the total can be 14 or any number less than 14
• This translates to the inequality symbol \(\leq 14\)

4. Combine the pieces

• Total places: \(\mathrm{f + h}\)
• Constraint: \(\leq 14\)
• Complete inequality: \(\mathrm{f + h \leq 14}\)

Answer: A. \(\mathrm{f + h \leq 14}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

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

Students sometimes reverse the meaning of these constraint phrases. They might think "at most 14" means "14 or more" instead of "14 or fewer." This fundamental translation error leads them to use \(\geq\) instead of \(\leq\).

This leads them to select Choice B (\(\mathrm{f + h \geq 14}\))

Second Most Common Error:

Poor INFER reasoning: Focusing on the difference between offices and homes rather than the total

Some students get distracted by having two different types of places (offices vs homes) and think the problem is about comparing them rather than adding them. They might interpret the constraint as being about how many more offices than homes (or vice versa) can be cleaned.

This may lead them to select Choice C (\(\mathrm{f - h \leq 14}\)) or Choice D (\(\mathrm{f - h \geq 14}\))

The Bottom Line:

Success on this problem requires careful attention to what the constraint phrase "at most" actually means and recognizing that when dealing with a total limit, you add the individual quantities together.