On a scale drawing of a city park, the main pavilion measures 7.5 centimeters in length. The actual pavilion measures...

1. TRANSLATE the problem information

• Given information:
  • Pavilion on drawing: 7.5 cm
  • Pavilion in reality: 90 m
  • Gazebo on drawing: 2.0 cm
  • Gazebo in reality: unknown (what we need to find)

2. INFER the key relationship

• Since both objects are on the same scale drawing, they must have the same scale ratio
• This means: \(\frac{\text{drawing length}}{\text{actual length}}\) is constant for both objects
• We can set up a proportion: \(\frac{7.5}{90} = \frac{2.0}{x}\)

3. SIMPLIFY using cross multiplication

• Cross multiply: \(7.5 \times x = 90 \times 2.0\)
• Calculate: \(7.5x = 180\)
• Solve for x: \(x = \frac{180}{7.5} = 24\) (use calculator)

Answer: C (24)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students may set up the proportion incorrectly, such as putting drawing measurements in different positions relative to actual measurements.

For example, they might write: \(\frac{7.5}{2.0} = \frac{90}{x}\) instead of \(\frac{7.5}{90} = \frac{2.0}{x}\)

This flipped proportion leads to: \(7.5x = 180\), so \(x = \frac{180}{7.5} = 24\)... wait, this actually gives the same answer! Let me reconsider.

Actually, a more likely error: \(\frac{7.5}{2.0} = \frac{x}{90}\), leading to \(x = \frac{7.5 \times 90}{2.0} = \frac{675}{2.0} = 337.5\), which is way off from any answer choice.

This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the proportion but make arithmetic errors in cross multiplication or division.

For instance, miscalculating \(90 \times 2.0\) as 160 instead of 180, leading to \(x = \frac{160}{7.5} \approx 21.3\), which might cause them to select Choice (B) (20) as the closest option.

The Bottom Line:

Scale drawing problems require recognizing that the ratio between drawing and reality stays constant for all objects. The key insight is maintaining consistent positioning in your proportion setup.