The equation \(\mathrm{V(t) = 12,500(0.86)^t}\) gives the estimated value, V, in dollars, of a certain asset t years after it...
GMAT Advanced Math : (Adv_Math) Questions
The equation \(\mathrm{V(t) = 12,500(0.86)^t}\) gives the estimated value, \(\mathrm{V}\), in dollars, of a certain asset \(\mathrm{t}\) years after it was purchased. Which of the following is the best interpretation of the number 12,500 in this context?
The estimated value of the asset when it was purchased
The amount the asset decreases in value each year
The number of years since the asset was purchased
The percent decrease in the asset's value each year
1. TRANSLATE the problem information
- Given: \(\mathrm{V(t) = 12,500(0.86)^t}\) represents the asset's value t years after purchase
- Question: What does 12,500 represent in this context?
2. INFER the approach
- This is an exponential function in the standard form \(\mathrm{a(b)^t}\)
- To understand what the coefficient (12,500) represents, I need to find the value when t = 0
- Since t represents years after purchase, t = 0 means "at the time of purchase"
3. SIMPLIFY by evaluating at t = 0
- \(\mathrm{V(0) = 12,500(0.86)^0}\)
- Since any number to the power of 0 equals 1: \(\mathrm{(0.86)^0 = 1}\)
- \(\mathrm{V(0) = 12,500(1) = 12,500}\)
4. INFER the meaning
- When t = 0 (at purchase time), the asset's value is $12,500
- Therefore, 12,500 represents the estimated value when the asset was purchased
Answer: A. The estimated value of the asset when it was purchased
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students don't recognize that they need to evaluate the function at t = 0 to interpret the coefficient. Instead, they might guess based on the wording of answer choices without doing the mathematical analysis. This leads to confusion and guessing among all the plausible-sounding options.
Second Most Common Error:
Conceptual confusion about exponential functions: Students might think that in exponential decay, the coefficient represents the amount lost each year (like in linear functions). They don't understand that exponential functions have a constant percentage rate, not a constant amount of change. This may lead them to select Choice B (The amount the asset decreases in value each year).
The Bottom Line:
This problem tests whether students understand the structure of exponential functions and can connect the mathematical model to its real-world meaning. The key insight is recognizing that the initial condition (t = 0) reveals what the coefficient represents.
The estimated value of the asset when it was purchased
The amount the asset decreases in value each year
The number of years since the asset was purchased
The percent decrease in the asset's value each year