Rectangle A has length 15 and width w. Rectangle B has length 20 and the same length-to-width ratio as rectangle...

1. TRANSLATE the problem information

• Given information:
  • Rectangle A: length = 15, width = w
  • Rectangle B: length = 20, width = unknown
  • Rectangle B has the same length-to-width ratio as Rectangle A

• What this tells us: We need to find Rectangle B's width using the fact that both rectangles have equal length-to-width ratios.

2. INFER the mathematical relationship

• Since both rectangles have the same length-to-width ratio, we can set up an equation
• Rectangle A's ratio: \(\frac{15}{w}\)
• Rectangle B's ratio: \(\frac{20}{x}\) (where x is Rectangle B's width)
• These ratios must be equal: \(\frac{15}{w} = \frac{20}{x}\)

3. SIMPLIFY to solve for the unknown width

• Cross multiply: \(15 \times x = 20 \times w\)
• This gives us: \(15x = 20w\)
• Solve for x: \(x = \frac{20w}{15}\)
• Reduce the fraction: \(x = \frac{4w}{3}\)

Answer: A. \(\frac{4}{3}w\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret "same length-to-width ratio" as meaning the rectangles have the same dimensions or that the difference between length and width is the same.

Instead of setting up the ratio relationship \(\frac{15}{w} = \frac{20}{x}\), they might think: "Rectangle A's length is 5 more than Rectangle B's length \((20-15=5)\), so Rectangle B's width should be 5 more than Rectangle A's width."

This may lead them to select Choice B \((w + 5)\).

Second Most Common Error:

Poor TRANSLATE reasoning: Students set up the ratio backwards, thinking Rectangle A's ratio is \(\frac{w}{15}\) instead of \(\frac{15}{w}\).

This leads to the equation \(\frac{w}{15} = \frac{20}{x}\), which gives \(x = \frac{300w}{15} = 20w\). When they try to match this to the answer choices, they might incorrectly simplify or select Choice C \((\frac{3}{4}w)\) by confusing the relationship.

The Bottom Line:

The key challenge is correctly translating "same length-to-width ratio" into a mathematical proportion. Students must resist the temptation to work with differences instead of ratios and carefully maintain the correct order when setting up their proportion.