A rectangular garden has length l and width w. Both dimensions are multiplied by the same scale factor k to...

1. TRANSLATE the problem information

• Given information:
  • Original rectangular garden has dimensions \(\mathrm{l}\) and \(\mathrm{w}\)
  • Both dimensions multiplied by the same scale factor \(\mathrm{k}\)
  • New garden area is at least 69% larger than original area
• What this tells us: We need to find values of \(\mathrm{k}\) that satisfy an area constraint

2. INFER the scaling relationship

• Key insight: When you multiply both length and width by the same factor \(\mathrm{k}\), the area gets multiplied by \(\mathrm{k^2}\)
• This is because: New area = \(\mathrm{(kl) \times (kw) = k^2lw = k^2 \times (original\ area)}\)

3. TRANSLATE the constraint into mathematics

• 'At least 69% larger' means: \(\mathrm{New\ area \geq Original\ area + 69\%\ of\ original\ area}\)
• In mathematical terms: \(\mathrm{k^2 \times (original\ area) \geq (original\ area) + 0.69 \times (original\ area)}\)
• SIMPLIFY: \(\mathrm{k^2 \times (original\ area) \geq 1.69 \times (original\ area)}\)

4. SIMPLIFY to find the constraint on k

• Since original area > 0, we can divide both sides by original area:
  \(\mathrm{k^2 \geq 1.69}\)
• Taking the positive square root: \(\mathrm{k \geq \sqrt{1.69} = 1.3}\)

5. APPLY CONSTRAINTS to check answer choices

• We need \(\mathrm{k \geq 1.3}\):
  • (A) 1.4: Since \(\mathrm{1.4 \gt 1.3}\) ✓ This works
  • (B) 1.69: Since \(\mathrm{1.69 \gt 1.3}\) ✓ This works too
  • (C) 1.2: Since \(\mathrm{1.2 \lt 1.3}\) ✗ Too small
  • (D) 1.0: Since \(\mathrm{1.0 \lt 1.3}\) ✗ Too small

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that scaling both dimensions by \(\mathrm{k}\) means the area scales by \(\mathrm{k^2}\), not \(\mathrm{k}\).

Instead, they might think: 'If both dimensions increase by factor \(\mathrm{k}\), then area increases by factor \(\mathrm{k}\) too.' This leads to the wrong constraint \(\mathrm{k \geq 1.69}\), making them think only choice (B) 1.69 works, or causing confusion when multiple choices seem too small.

This may lead them to select Choice B (1.69) or abandon systematic solving and guess.

Second Most Common Error:

Poor TRANSLATE reasoning: Students misinterpret '69% larger' as meaning the new area equals 69% of the original area (instead of original + 69% of original).

This creates the wrong constraint \(\mathrm{k^2 \geq 0.69}\), making \(\mathrm{k \geq \sqrt{0.69} \approx 0.83}\). Since all answer choices satisfy this, they get confused about which to pick.

This leads to confusion and guessing.

The Bottom Line:

This problem tests whether students understand how area scales when linear dimensions are scaled - it's \(\mathrm{k^2}\), not \(\mathrm{k}\). The percentage language adds another layer of translation challenge, but the core insight is the quadratic relationship between linear scaling and area scaling.