A city employee will plant two types of bushes, azaleas and boxwoods, in a park. There will be no more...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{a}\) = number of azaleas that will be planted
  • \(\mathrm{b}\) = number of boxwoods that will be planted
  • "no more than 164 total bushes planted"
  • "number of azaleas planted will be at most three times the number of boxwoods planted"

2. TRANSLATE the first constraint

• "no more than 164 total bushes planted" means the total cannot exceed 164
• Total bushes = \(\mathrm{a + b}\)
• This gives us: \(\mathrm{a + b \leq 164}\)

3. TRANSLATE the second constraint

• "number of azaleas planted will be at most three times the number of boxwoods planted"
• "At most" means azaleas ≤ (three times boxwoods)
• This gives us: \(\mathrm{a \leq 3b}\)

4. INFER which answer choice matches

• Our system is:
  • \(\mathrm{a + b \leq 164}\)
  • \(\mathrm{a \leq 3b}\)
• Looking at the choices, this exactly matches Choice D

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Confusing "at most" with "at least" or mixing up inequality directions

Students might think "at most three times" means \(\mathrm{3a \geq b}\) (three times azaleas is at least the boxwoods), when it actually means \(\mathrm{a \leq 3b}\) (azaleas is at most three times the boxwoods). They get the relationship backwards.

This may lead them to select Choice A (\(\mathrm{a + b \geq 164}\), \(\mathrm{3a \geq b}\)) or Choice C (\(\mathrm{a + b \leq 164}\), \(\mathrm{3a \geq b}\))

Second Most Common Error:

Poor TRANSLATE reasoning: Misinterpreting "no more than" as "at least"

Students might read "no more than 164 total bushes" and think it means they need at least 164 bushes, translating it to \(\mathrm{a + b \geq 164}\) instead of \(\mathrm{a + b \leq 164}\).

This may lead them to select Choice A (\(\mathrm{a + b \geq 164}\), \(\mathrm{3a \geq b}\)) or Choice B (\(\mathrm{a + b \geq 164}\), \(\mathrm{a \leq 3b}\))

The Bottom Line:

This problem tests your ability to carefully translate constraint language into mathematical inequalities. The key is recognizing that "no more than" and "at most" both indicate upper limits, which correspond to ≤ symbols, not ≥ symbols.