Question:A digital library had 2,800 items at the beginning of the first quarter of 2022. For the next 8 quarters...
GMAT Advanced Math : (Adv_Math) Questions
A digital library had 2,800 items at the beginning of the first quarter of 2022. For the next 8 quarters after the first quarter of 2022, the total number of items increased by 12% at the end of each quarter compared to the number at the end of the previous quarter. Which equation gives the total number of items, I, at the end of m months after the beginning of the first quarter of 2022, where \(\mathrm{m} \leq 24\)?
1. TRANSLATE the problem information
- Given information:
- Starting amount: 2,800 items
- Growth rate: 12% increase each quarter
- Time variable: m months (where m ≤ 24)
- Need: Equation for total items I
2. INFER the growth pattern
- 12% increase each quarter means the growth factor is \(1 + 0.12 = 1.12\)
- This follows exponential growth: \(\mathrm{I = Initial × (growth\: factor)^{(number\: of\: periods)}}\)
- But there's a timing mismatch: growth happens quarterly, but we want months
3. TRANSLATE the time period relationship
- Growth occurs every quarter (every 3 months)
- After m months, the number of quarters that have passed is \(\mathrm{m/3}\)
- So the number of growth periods is \(\mathrm{m/3}\), not m
4. INFER the final equation
- Combining everything: \(\mathrm{I = 2,800(1.12)^{m/3}}\)
- This matches the exponential growth pattern with the correct time conversion
Answer: C
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students correctly identify 12% quarterly growth as factor 1.12, but miss the time period conversion between months and quarters.
They think: "The equation asks for m months, so the exponent should be m." This leads them to write \(\mathrm{I = 2,800(1.12)^m}\), ignoring that growth happens quarterly, not monthly.
This may lead them to select Choice D (\(\mathrm{I = 2,800(1.12)^m}\))
Second Most Common Error:
Poor TRANSLATE reasoning: Students try to convert quarterly growth to monthly growth by dividing: "\(12\% ÷ 3\: \mathrm{months} = 4\%\: \mathrm{per\: month}\)."
They create \(\mathrm{I = 2,800(1.04)^m}\), thinking this represents 4% monthly growth. This misses the compounding nature of quarterly growth.
This may lead them to select Choice B (\(\mathrm{I = 2,800(1.04)^m}\))
The Bottom Line:
The challenge lies in coordinating two different time scales - quarterly growth periods with a monthly time variable. Students must recognize that the exponent represents the number of growth periods, not simply the time variable given in the problem.