A car is purchased for $25,000.At the end of each 6-month period, the car's value is 80% of its value...
GMAT Advanced Math : (Adv_Math) Questions
- A car is purchased for \(\$25,000\).
- At the end of each 6-month period, the car's value is \(80\%\) of its value at the end of the preceding 6-month period.
- Which of the following functions best models \(\mathrm{V}\), the value of the car in dollars, \(\mathrm{t}\) years after purchase?
- Assume the depreciation rate remains constant over time.
1. TRANSLATE the problem information
- Given information:
- Initial value: \(\$25,000\)
- Every 6 months: value becomes 80% of previous value
- Need: function V(t) where t = years after purchase
- What this tells us: We have exponential decay happening every 6 months, but need to express it in terms of years.
2. INFER the compounding pattern
- Key insight: The 80% reduction happens every 6 months, not every year
- In 1 year, there are 2 six-month periods
- In t years, there are 2t six-month periods
- Each period multiplies the value by 0.80
3. INFER the mathematical relationship
- After 1 six-month period: \(\$25,000 \times 0.80\)
- After 2 six-month periods (1 year): \(\$25,000 \times (0.80)^2\)
- After 2t six-month periods (t years): \(\$25,000 \times (0.80)^{2t}\)
4. Verify against answer choices
Looking at the choices, \(\mathrm{V} = 25,000(0.80)^{2t}\) matches choice (C) exactly.
Answer: C
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students recognize the 80% decay but fail to account for the frequency difference between the given 6-month periods and the requested yearly variable.
They think: "The value is 80% each period, and t represents time, so it should be \((0.80)^t\)."
This leads them to select Choice B [\(\mathrm{V} = 25,000(0.80)^t\)] because they treat the decay as if it happens once per year instead of twice per year.
Second Most Common Error:
Poor TRANSLATE reasoning: Students misinterpret "80% of previous value" as meaning the car loses 80% of its value (keeping only 20%).
They incorrectly use \(0.20\) as the multiplication factor instead of \(0.80\).
This may lead them to select Choice A [\(\mathrm{V} = 25,000(0.20)^t\)].
The Bottom Line:
This problem tests whether students can correctly handle exponential functions when the compounding period differs from the time variable units. The key is recognizing that 6-month periods must be converted to match the yearly time variable.