Marcus purchased a car for personal use. The table shows the exponential relationship between the time t, in years, since...
GMAT Advanced Math : (Adv_Math) Questions
Marcus purchased a car for personal use. The table shows the exponential relationship between the time \(\mathrm{t}\), in years, since Marcus purchased the car and the car's value \(\mathrm{v}\), in dollars. Which of the following equations best represents the relationship between \(\mathrm{t}\) and \(\mathrm{v}\)?
| Time (years) | Value (dollars) |
|---|---|
| 0 | 18,000.00 |
| 1 | 17,100.00 |
| 2 | 16,245.00 |
- \(\mathrm{v = 0.05(1 + 18000)^t}\)
- \(\mathrm{v = (1 - 0.05)^t}\)
- \(\mathrm{v = 18000(1 - 0.05)^t}\)
- \(\mathrm{v = (1 + 18000)^t}\)
1. TRANSLATE the problem information
- Given information:
- Table showing car values at different times
- \(\mathrm{t = 0: v = \$18,000}\)
- \(\mathrm{t = 1: v = \$17,100}\)
- \(\mathrm{t = 2: v = \$16,245}\)
- Need to find equation relating t and v
2. INFER the type of exponential relationship
- Values are decreasing over time → exponential decay
- Need to find the decay factor (how much value remains each year)
3. INFER the decay factor from the pattern
- Calculate ratios between consecutive years:
- Year 1 to Year 0: \(\mathrm{17,100 \div 18,000 = 0.95}\)
- Year 2 to Year 1: \(\mathrm{16,245 \div 17,100 = 0.95}\) (use calculator)
- The decay factor is consistently 0.95
4. TRANSLATE decay factor to mathematical form
- \(\mathrm{0.95 = 1 - 0.05}\)
- This means 5% decay rate per year
5. INFER the complete exponential decay formula
- General form: \(\mathrm{v = initial\_value \times (decay\_factor)^t}\)
- Substitute values: \(\mathrm{v = 18,000(1 - 0.05)^t}\)
- This matches choice C
Answer: C
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students don't recognize how to find the decay factor from the data table.
Instead of calculating ratios between consecutive terms, they might try to find differences (\(\mathrm{17,100 - 18,000 = -900}\)) or look for additive patterns. This leads them away from understanding the multiplicative nature of exponential relationships. Without the decay factor, they cannot identify the correct exponential form and may guess randomly among the choices.
Second Most Common Error:
Poor TRANSLATE reasoning: Students confuse the decay rate with the decay factor.
They correctly identify that the car loses value each year but incorrectly think the 5% loss means the factor should be 0.05 rather than \(\mathrm{(1 - 0.05) = 0.95}\). This misconception might lead them to select Choice A (\(\mathrm{v = 0.05(1 + 18000)^t}\)) because it contains 0.05, even though the structure is completely wrong.
The Bottom Line:
This problem requires understanding that exponential relationships show up as consistent ratios in data tables, not as consistent differences. The key insight is translating between real-world decay rates and mathematical decay factors.