A developer plans a subdivision with 55 residential lots, each with a planned area of 0.75 acres. A county ordinance...

1. TRANSLATE the problem information

• Given information:
  • 55 residential lots planned
  • Each lot = 0.75 acres
  • 18% of total land must be open space/infrastructure
  • Remaining land will be subdivided into lots

2. TRANSLATE what we need to find

• We need the minimum total area the developer must purchase
• The residential lots will occupy only part of this total area

3. Calculate the total area needed for all residential lots

• Total lot area = \(\mathrm{55\,lots \times 0.75\,acres/lot = 41.25\,acres}\)

4. INFER the key relationship

• If 18% goes to open space/infrastructure, then 82% can be used for residential lots
• This means: \(\mathrm{82\%\,of\,total\,land = 41.25\,acres}\)
• As a decimal: \(\mathrm{0.82 \times (total\,area) = 41.25\,acres}\)

5. SIMPLIFY to find the total area

• Set up the equation: \(\mathrm{0.82 \times T = 41.25}\)
• Solve for T: \(\mathrm{T = 41.25 \div 0.82}\) (use calculator)
• \(\mathrm{T = 50.30487... \approx 50.3\,acres}\)

Answer: D. 50.3

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that the lot area represents only a portion of the total land purchase, not the entire parcel.

Instead of understanding that 41.25 acres = 82% of total, they incorrectly treat 41.25 acres as the base amount and try to add 18% to it: \(\mathrm{41.25 + (0.18 \times 41.25) = 48.675 \approx 48.7}\).

This leads them to select Choice C (48.7).

Second Most Common Error:

Poor TRANSLATE reasoning: Students misunderstand the relationship and think they should multiply the lot area by 0.82 instead of dividing by it.

They calculate: \(\mathrm{41.25 \times 0.82 = 33.825 \approx 33.8}\), thinking this accounts for the restriction somehow.

This leads them to select Choice A (33.8).

The Bottom Line:

The key insight is recognizing that when a restriction removes a percentage of total area, the usable area represents the remaining percentage of the whole. Students must set up the equation correctly: \(\mathrm{(remaining\,percentage) \times (total) = (usable\,area)}\).