When a photograph is enlarged, each linear dimension is increased by a factor of 1.2. The area of the original...
GMAT Problem-Solving and Data Analysis : (PS_DA) Questions
When a photograph is enlarged, each linear dimension is increased by a factor of \(1.2\). The area of the original photograph is what percent of the area of the enlarged photograph, to the nearest whole number?
1. TRANSLATE the problem information
- Given information:
- Each linear dimension increases by factor 1.2
- Need: original area as percent of enlarged area
2. INFER the area scaling relationship
- Key insight: When linear dimensions scale by factor k, area scales by \(\mathrm{k}^2\)
- Since linear factor = \(1.2\), area factor = \(1.2^2 = 1.44\)
- This means: \(\mathrm{enlarged\ area} = 1.44 \times \mathrm{original\ area}\)
3. TRANSLATE the percentage question
- "Original is what percent of enlarged" means:
\(\frac{\mathrm{original\ area}}{\mathrm{enlarged\ area}} \times 100\%\)
4. SIMPLIFY the calculation
- Substitute: \(\frac{\mathrm{original\ area}}{1.44 \times \mathrm{original\ area}} \times 100\%\)
- Simplify: \(\frac{1}{1.44} \times 100\%\)
- Calculate: \(1 \div 1.44 \approx 0.6944\) (use calculator)
- Convert to percentage: \(0.6944 \times 100\% = 69.44\%\)
- Round to nearest whole number: \(69\%\)
Answer: B) 69
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students don't recognize the \(\mathrm{k}^2\) scaling rule for area. They think that if linear dimensions increase by \(1.2\), then area also increases by \(1.2\) (instead of \(1.2^2\)).
Following this incorrect reasoning: original area would be \(\frac{1}{1.2} \times 100\% = 83.33\% \approx 83\%\) of the enlarged area.
This may lead them to select Choice D (83).
Second Most Common Error:
Poor TRANSLATE reasoning: Students misinterpret the percentage question and calculate what percent larger the enlarged photo is compared to the original, rather than what percent the original is of the enlarged.
They calculate: \((1.44 - 1) \times 100\% = 44\%\) increase, then mistakenly think this means the answer is related to 144% or 120%.
This may lead them to select Choice E (120).
The Bottom Line:
This problem requires recognizing that scaling affects area quadratically, not linearly. Students who miss this fundamental relationship about similar figures will struggle with the entire setup.