The function \(\mathrm{S(n) = 38,000a^n}\) above models the annual salary, in dollars, of an employee n years after starting a...
GMAT Advanced Math : (Adv_Math) Questions
The function \(\mathrm{S(n) = 38,000a^n}\) above models the annual salary, in dollars, of an employee \(\mathrm{n}\) years after starting a job, where \(\mathrm{a}\) is a constant. If the employee's salary increases by 4% each year, what is the value of \(\mathrm{a}\)?
\(\mathrm{0.04}\)
\(\mathrm{0.4}\)
\(\mathrm{1.04}\)
\(\mathrm{1.4}\)
1. TRANSLATE the growth information
- Given information:
- Function: \(\mathrm{S(n) = 38,000a^n}\)
- Salary increases by 4% each year
- Need to find: value of a
- What "increases by 4% each year" means:
- Each year, salary becomes \(\mathrm{100\% + 4\% = 104\%}\) of previous year
- This equals \(\mathrm{1.04}\) times the previous year's salary
2. INFER the pattern for exponential growth
- Since salary grows by the same percentage each year, we can trace the pattern:
- Year 0: \(\mathrm{S(0) = 38,000}\) (starting salary)
- Year 1: \(\mathrm{S(1) = 38,000 \times 1.04}\)
- Year 2: \(\mathrm{S(2) = 38,000 \times (1.04)^2}\)
- Year n: \(\mathrm{S(n) = 38,000 \times (1.04)^n}\)
3. INFER the value of constant a
- Comparing our derived formula \(\mathrm{S(n) = 38,000 \times (1.04)^n}\) with the given form \(\mathrm{S(n) = 38,000a^n}\):
- The constant a must equal \(\mathrm{1.04}\)
Answer: C. 1.04
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students misinterpret what "4% increase" means mathematically.
Many students think that a 4% increase means using 0.04 as the multiplier, reasoning that "\(\mathrm{4\% = 4/100 = 0.04}\), so the salary should be multiplied by 0.04 each year." This fundamental misunderstanding of percentage increase leads them to think \(\mathrm{a = 0.04}\).
This may lead them to select Choice A (0.04).
Second Most Common Error:
Incomplete TRANSLATE reasoning: Students partially understand percentage increase but make calculation errors.
Some students correctly recognize that 4% increase means adding 4% to the original, but then calculate \(\mathrm{4/100 = 0.04}\) and think the multiplier is \(\mathrm{1 + 0.04 = 1.4}\) instead of \(\mathrm{1 + 0.04 = 1.04}\). They skip the decimal place conversion.
This may lead them to select Choice D (1.4).
The Bottom Line:
The key challenge is correctly translating percentage language into mathematical operations. Students must understand that "r% increase" means multiplying by \(\mathrm{(1 + r/100)}\), not just \(\mathrm{r/100}\).
\(\mathrm{0.04}\)
\(\mathrm{0.4}\)
\(\mathrm{1.04}\)
\(\mathrm{1.4}\)