The function \(\mathrm{P(t) = 45,000(0.88)^{(t/24)}}\) models a city's population after t months. Due to economic factors, the city's population decre...
GMAT Advanced Math : (Adv_Math) Questions
The function \(\mathrm{P(t) = 45,000(0.88)^{(t/24)}}\) models a city's population after \(\mathrm{t}\) months. Due to economic factors, the city's population decreases by a fixed percentage every two years. What is the percentage by which the population decreases every two years?
8
12
24
88
1. TRANSLATE the function at key time points
- Given: \(\mathrm{P(t) = 45,000(0.88)^{(t/24)}}\) models population after t months
- We need the percentage decrease every 2 years (24 months)
- Let's find population values at the start and after 24 months:
- Initial: \(\mathrm{P(0) = 45,000(0.88)^{0} = 45,000}\)
- After 2 years: \(\mathrm{P(24) = 45,000(0.88)^{1} = 45,000(0.88)}\)
2. INFER what the mathematical result means
- After 24 months, population = \(\mathrm{0.88 \times original\ population}\)
- This tells us that 0.88 (or 88%) of the original population remains
- The key insight: if 88% remains, then the decrease is the complement
3. SIMPLIFY to find the percentage decrease
- Percentage remaining = 88%
- Percentage decrease = \(\mathrm{100\% - 88\% = 12\%}\)
Answer: B (12)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students correctly evaluate the function but misinterpret what 0.88 represents in the context of percentage decrease.
They see that the population becomes 0.88 times the original and incorrectly conclude that the population "decreases by 88%" rather than understanding that 88% remains after the decrease. This leads them to select Choice D (88).
Second Most Common Error:
Poor TRANSLATE execution: Students struggle with the exponent structure (t/24) and don't recognize that for a 2-year period, they need t = 24 months.
They might substitute t = 2 instead of t = 24, getting \(\mathrm{P(2) = 45,000(0.88)^{(2/24)} = 45,000(0.88)^{(1/12)}}\), leading to a much smaller decrease percentage. This causes confusion and may lead them to select Choice A (8).
The Bottom Line:
This problem tests whether students can distinguish between "what fraction remains" versus "what fraction decreased" when interpreting exponential decay models. The mathematical calculation is straightforward, but the conceptual interpretation requires careful attention to what the multiplier represents.
8
12
24
88