For a certain rectangular region, the ratio of its length to its width is 35 to 10. If the width...

1. TRANSLATE the problem information

• Given information:
  • Original ratio: length to width = 35 to 10
  • Width increases by 7 units
  • Need to maintain the same ratio

• What this tells us: We need to find how much the length changes to keep the proportional relationship intact.

2. INFER the mathematical approach

• Key insight: If the ratio stays the same, then \(\mathrm{length/width}\) must equal \(\mathrm{35/10}\) before AND after the changes
• Strategy: Set up equations for both the original and new dimensions, then solve for the length change

3. TRANSLATE the ratio into an equation

• The ratio \(\mathrm{35:10}\) means \(\mathrm{length/width = 35/10 = 3.5}\)
• This gives us: \(\mathrm{length = 3.5 \times width}\)
• Using variables: \(\mathrm{ℓ = 3.5w}\) (where \(\mathrm{ℓ}\) = original length, \(\mathrm{w}\) = original width)

4. INFER the new dimension relationship

• After changes: new width = \(\mathrm{w + 7}\), new length = \(\mathrm{ℓ + x}\) (where \(\mathrm{x}\) = change in length)
• To maintain ratio: \(\mathrm{(ℓ + x)/(w + 7) = 3.5}\)

5. SIMPLIFY by solving the equation

• From the ratio constraint: \(\mathrm{ℓ + x = 3.5(w + 7)}\)
• Expand: \(\mathrm{ℓ + x = 3.5w + 24.5}\)
• Since we know \(\mathrm{ℓ = 3.5w}\), substitute: \(\mathrm{3.5w + x = 3.5w + 24.5}\)
• Solve: \(\mathrm{x = 24.5}\)

Answer: B (It must increase by 24.5 units)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students think that if the width increases by 7, the length should also increase by 7 to "keep things equal."

This represents a fundamental misunderstanding of proportional relationships. They confuse maintaining a ratio with maintaining equal changes. Since the original ratio is \(\mathrm{35:10}\) (or \(\mathrm{3.5:1}\)), when width increases by 7, length must increase by \(\mathrm{3.5 \times 7 = 24.5}\) to maintain the same proportional relationship.

This may lead them to select Choice D (It must increase by 7 units)

Second Most Common Error:

Poor TRANSLATE reasoning: Students incorrectly set up the proportion or misunderstand what "maintain the ratio" means algebraically.

Some students might think that increasing the width means the length should decrease to somehow "balance out" the rectangle, not recognizing that maintaining a ratio means both dimensions scale together in the same direction.

This may lead them to select Choice A or C (decreasing options)

The Bottom Line:

The key challenge is understanding that proportional relationships require scaled changes, not equal changes. When one dimension changes, the other must change by a factor determined by the original ratio.