A projection estimates that a museum had 120,000 visitors in 2018. For the next 2 years, each quarter the number...
GMAT Advanced Math : (Adv_Math) Questions
A projection estimates that a museum had \(\mathrm{120,000}\) visitors in \(\mathrm{2018}\). For the next \(\mathrm{2}\) years, each quarter the number of visitors increases by \(\mathrm{3\%}\) compared to the previous quarter. Let \(\mathrm{x}\) be the number of years after \(\mathrm{2018}\), with \(\mathrm{0 \leq x \leq 2}\). Which equation best models the number of visitors \(\mathrm{V}\) as a function of \(\mathrm{x}\)?
1. TRANSLATE the problem information
- Given information:
- Initial visitors in 2018: 120,000
- Growth rate: 3% increase each quarter
- Time variable: \(\mathrm{x} = \text{years after 2018}\) (where \(0 \leq \mathrm{x} \leq 2\))
- What this tells us: We need an exponential growth model with quarterly compounding.
2. TRANSLATE the growth factor
- A 3% increase means we multiply by \((1 + 0.03) = 1.03\)
- This happens every quarter, not every year
3. INFER the time relationship
- The growth happens quarterly, but our variable x is in years
- In x years, there are 4x quarters (since 1 year = 4 quarters)
- So we need \((1.03)\) raised to the power of \(4\mathrm{x}\), not just x
4. INFER the complete model
- Start with the basic exponential form: \(\mathrm{V} = \mathrm{P}(\text{growth factor})^{(\text{number of periods})}\)
- \(\mathrm{P} = 120,000\) (initial amount)
- Growth factor = 1.03 (per quarter)
- Number of periods = \(4\mathrm{x}\) quarters
- Therefore: \(\mathrm{V} = 120,000(1.03)^{4\mathrm{x}}\)
Answer: C
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students read "3% increase each quarter" but don't properly connect this to the time variable x being in years. They might think since x is in years, they should use \((1.03)^{\mathrm{x}}\), not realizing they need to account for 4 quarters per year.
This leads them to select Choice A (\(\mathrm{V} = 120,000(1.03)^{\mathrm{x}}\)) - using the correct quarterly growth rate but with the wrong exponent.
Second Most Common Error:
Poor INFER reasoning about compound growth: Students might try to convert quarterly growth to annual growth by thinking "\(3\% \times 4 \text{ quarters} = 12\%\)" and then round up to 15% for "margin of error," creating a linear rather than compound relationship.
This may lead them to select Choice B (\(\mathrm{V} = 120,000(1.15)^{\mathrm{x}}\)) - using an incorrect annual rate instead of properly compounding quarterly growth.
The Bottom Line:
This problem tests whether students can distinguish between the time period of the growth rate (quarters) and the time variable in the function (years), requiring careful attention to unit consistency in exponential models.