At the beginning of 2020, a startup has 8,000 users. Each month thereafter, the number of users increases by 4%...
GMAT Advanced Math : (Adv_Math) Questions
At the beginning of 2020, a startup has 8,000 users. Each month thereafter, the number of users increases by 4% of the number of users in the previous month, with growth compounding monthly. Which equation defines U(y), the number of users y years after the beginning of 2020?
- \(\mathrm{U(y) = 8000(0.96)^{12y}}\)
- \(\mathrm{U(y) = 8000(1.004)^{12y}}\)
- \(\mathrm{U(y) = 8000(1.04)^{y}}\)
- \(\mathrm{U(y) = 8000(1.04)^{12y}}\)
1. TRANSLATE the problem information
- Given information:
- Initial users: 8,000 (at beginning of 2020)
- Monthly growth rate: 4% increase each month
- Growth compounds monthly
- Find: U(y) where y = years after beginning of 2020
- What this tells us: We have exponential growth with monthly compounding over a time period measured in years.
2. INFER the approach
- Since we have monthly growth but want a function of years, we need to account for 12 months per year
- The exponential growth formula is: \(\mathrm{Final\ Amount = Initial \times (Growth\ Factor)^{Number\ of\ periods}}\)
- We need to determine the monthly growth factor and express the number of periods in terms of y
3. TRANSLATE the growth rate to a growth factor
- A 4% increase means the new amount is 100% + 4% = 104% of the previous amount
- As a decimal: \(104\% = 1.04\)
- So the monthly growth factor is 1.04
4. INFER the time component
- The function input y represents years
- Since growth happens monthly, the number of growth periods in y years is 12y months
- Therefore, we raise the growth factor to the power of 12y
5. Combine all components
- \(\mathrm{U(y) = 8000 \times (1.04)^{12y}}\)
- This matches choice (D)
Answer: D
Why Students Usually Falter on This Problem
Most Common Error Path:
TRANSLATE reasoning error: Students may convert 4% incorrectly as 0.04 instead of recognizing it as a growth factor of 1.04.
When they see "4% increase," they might think the growth factor is 0.04, leading to an equation like \(\mathrm{U(y) = 8000(0.04)^{12y}}\). Since this doesn't match any answer choice exactly, they might look for the closest option or get confused. Alternatively, they might think 4% means 0.4% and select 1.004 as the growth factor.
This may lead them to select Choice B (\(\mathrm{8000(1.004)^{12y}}\)) if they interpret 4% as 0.4%.
Second Most Common Error:
INFER reasoning error: Students correctly identify the monthly growth factor as 1.04 but fail to account for the time unit conversion from years to months.
They might think since the growth factor is monthly, they can directly use y as the exponent, creating \(\mathrm{U(y) = 8000(1.04)^y}\). This treats y as if it represents months instead of years.
This may lead them to select Choice C (\(\mathrm{8000(1.04)^y}\)).
The Bottom Line:
This problem tests both language interpretation skills (converting percentage increases to growth factors) and time unit reasoning (converting years to months for the exponent). Success requires carefully translating the English description into the correct mathematical components of the exponential growth formula.