A rectangular photograph has an area of 900 square centimeters. A photo lab creates a reduced copy of this photograph...

1. TRANSLATE the problem information

• Given information:
  • Original photograph area: \(\mathrm{900}\) square centimeters
  • Each side of reduced copy is \(\mathrm{\frac{1}{5}}\) times the corresponding side of original
  • Need to find the area of the reduced copy

2. INFER the scaling relationship

• The key insight: When a 2D figure is scaled, the area doesn't scale the same way as the sides
• If linear dimensions scale by factor \(\mathrm{k}\), then area scales by \(\mathrm{k^2}\)
• This happens because area involves \(\mathrm{length \times width}\), so both dimensions get multiplied by \(\mathrm{k}\)

3. SIMPLIFY to find the area scale factor

• Linear scale factor: \(\mathrm{k = \frac{1}{5}}\)
• Area scale factor: \(\mathrm{k^2 = (\frac{1}{5})^2 = \frac{1}{25}}\)

4. SIMPLIFY to calculate the final area

• Area of reduced copy = Original area × Area scale factor
• Area of reduced copy = \(\mathrm{900 \times \frac{1}{25} = 900 \div 25 = 36}\) square centimeters

Answer: A (36)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students assume that area scales linearly with the sides

Many students think: "If each side is \(\mathrm{\frac{1}{5}}\) the original, then the area should also be \(\mathrm{\frac{1}{5}}\) the original." This linear thinking ignores the fact that area is a 2-dimensional measurement.

Following this incorrect reasoning: \(\mathrm{900 \times \frac{1}{5} = 180}\)

This may lead them to select Choice D (180)

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly identify that area scales by \(\mathrm{(\frac{1}{5})^2}\), but make calculation errors

They might incorrectly calculate \(\mathrm{(\frac{1}{5})^2}\) or make arithmetic errors in the final multiplication, leading to confusion and potentially guessing among the remaining choices.

The Bottom Line:

This problem tests whether students understand the fundamental difference between linear scaling and area scaling. The critical insight is recognizing that area is a 2-dimensional quantity, so when you scale both length and width by the same factor, the area changes by that factor squared.