A landscaping company has an average monthly water bill of $410. The company considers installing a high-efficiency irrigation system that...

1. TRANSLATE the problem information

• Given information:
  • Current monthly bill: \(\$410\)
  • Installation cost: \(\$3,900\)
  • New monthly bill: \(\$265\)
  • Monthly maintenance: \(\$15\)
  • Find when total savings > installation cost

2. INFER what "total savings exceed installation cost" means

• We need monthly savings × number of months > \(\$3,900\)
• Monthly savings = what we save each month = old bill - (new bill + maintenance)

3. SIMPLIFY to find monthly savings

• Monthly savings = \(\mathrm{410 - (265 + 15)}\)
• Monthly savings = \(\mathrm{410 - 280 = 130}\) dollars per month

4. TRANSLATE "exceed" into inequality notation

• Total savings after m months = \(\mathrm{130m}\)
• For savings to exceed installation cost: \(\mathrm{130m \gt 3,900}\)
• This can be rewritten as: \(\mathrm{3,900 \lt 130m}\)

5. SIMPLIFY to match answer format

• Looking at the choices, we need the form \(\mathrm{3,900 \lt (expression)m}\)
• Our expression should equal 130
• Check: \(\mathrm{(410 - 265 - 15) = 145 - 15 = 130}\) ✓

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students misinterpret what costs to include in the monthly savings calculation. They might calculate savings as just \(\mathrm{(410 - 265) = 145}\), forgetting that the new system adds \(\$15\)/month in maintenance costs.

This leads them to think the monthly savings is 145 instead of 130, making them select Choice D \(\mathrm{(3,900 \lt (410 - 265)m)}\).

Second Most Common Error:

Poor INFER reasoning about inequality direction: Students correctly calculate that monthly savings is 130 and set up \(\mathrm{130m \gt 3,900}\), but when rewriting this inequality, they flip it incorrectly to \(\mathrm{3,900 \gt 130m}\) instead of \(\mathrm{3,900 \lt 130m}\).

This may lead them to select Choice A \(\mathrm{(3,900 \gt (410 - 265 - 15)m)}\).

The Bottom Line:

This problem tests whether students can carefully track all costs (including maintenance) and properly handle inequality direction when rewriting expressions. The key insight is that "exceed" creates a "greater than" relationship, and maintenance costs reduce the actual monthly savings.