Question:A textile manufacturer produces a fabric pattern that requires 450 square inches of material per unit. To accommodate a special...
GMAT Geometry & Trigonometry : (Geo_Trig) Questions
A textile manufacturer produces a fabric pattern that requires \(\mathrm{450}\) square inches of material per unit. To accommodate a special order, the manufacturer creates a scaled-down version where both the length and width of the pattern are reduced by \(\mathrm{20\%}\). How many square inches of material are needed for one unit of the scaled-down pattern?
1. TRANSLATE the scaling information
- Given information:
- Original pattern requires \(450\) square inches
- Both length and width reduced by \(20\%\)
- What this tells us: Each dimension becomes \(80\%\) of original, or \(0.8\) times the original dimension
2. INFER how area changes when dimensions are scaled
- Key insight: When you scale both length and width by the same factor, area scales by the square of that factor
- Since each dimension becomes \(0.8\) times original:
- New area = \((0.8 \times \text{length}) \times (0.8 \times \text{width})\)
- New area = \(0.8^2 \times \text{original area}\)
3. SIMPLIFY to find the scaling factor
- Calculate the area scaling factor: \(0.8^2 = 0.64\)
- This means the new area is \(64\%\) of the original area
4. Calculate the final answer
- New area = \(0.64 \times 450 = 288\) square inches (use calculator)
Answer: 288 square inches
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students incorrectly assume that if dimensions are reduced by \(20\%\), then area is also reduced by \(20\%\).
They think: "20% reduction in dimensions means 20% reduction in area, so new area = \(0.8 \times 450 = 360\) square inches."
This fundamental misunderstanding of how area scaling works leads them to an incorrect answer of \(360\).
Second Most Common Error:
Poor TRANSLATE reasoning: Students might misinterpret "reduced by 20%" as meaning the dimensions become \(20\%\) of original instead of \(80\%\) of original.
This leads them to calculate: \(0.2^2 \times 450 = 0.04 \times 450 = 18\) square inches, which is clearly unreasonable but could cause confusion and guessing.
The Bottom Line:
The key challenge is understanding that area scales quadratically, not linearly, when both dimensions are scaled by the same factor. This is a fundamental concept in similarity and scaling that many students struggle with initially.