A rectangular display is divided into 28 equal cells.Exactly 21 of the cells are illuminated.If the illuminated portion represents p...
GMAT Problem-Solving and Data Analysis : (PS_DA) Questions
- A rectangular display is divided into \(28\) equal cells.
- Exactly \(21\) of the cells are illuminated.
- If the illuminated portion represents \(\mathrm{p}\) percent of the display, what is the value of \(\mathrm{p}\)?
Answer Format: Enter an integer; do not include the percent sign.
1. TRANSLATE the problem information
- Given information:
- Total cells in display: 28
- Illuminated cells: 21
- Need to find: p percent that represents the illuminated portion
- What this tells us: We need to find what percent 21 is of 28
2. TRANSLATE this into a mathematical fraction
- Set up part-to-whole fraction: \(\frac{21}{28}\)
- This represents the fraction of the display that's illuminated
3. SIMPLIFY the fraction to make percentage conversion easier
- Find the greatest common divisor of 21 and 28
- Both numbers are divisible by 7: \(21 \div 7 = 3\), and \(28 \div 7 = 4\)
- Therefore: \(\frac{21}{28} = \frac{3}{4}\)
4. Convert fraction to percentage
- Multiply the simplified fraction by 100: \(\frac{3}{4} \times 100\)
- Calculate: \(\frac{3}{4} = 0.75\), so \(0.75 \times 100 = 75\)
- Since the problem asks for integer form without percent sign: 75
Answer: 75
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students may misinterpret 'what percent of the display' and set up the fraction backwards as \(\frac{28}{21}\) instead of \(\frac{21}{28}\).
This backwards setup leads to: \(\frac{28}{21} = \frac{4}{3} \approx 1.33\), and when converted to percentage: \(1.33 \times 100 = 133\). This creates an impossible result (over 100%) that should trigger reconsideration, but students often miss this logical constraint.
Second Most Common Error:
Conceptual gap about percentages: Students correctly set up \(\frac{21}{28}\) and may even simplify to \(\frac{3}{4}\), but then stop at the decimal 0.75 without converting to percentage form.
They might submit 0.75 or 0.75%, forgetting that percentages require multiplying the decimal by 100. This stems from not fully understanding that 'percent' means 'per hundred.'
The Bottom Line:
This problem tests whether students can correctly interpret percentage language and convert between fractions, decimals, and percentages. The key insight is recognizing that finding 'p percent' means determining what percentage the part (21) represents of the whole (28).