The value V, in dollars, of a piece of industrial equipment is modeled by the equation V = 45,000 -...
GMAT Advanced Math : (Adv_Math) Questions
The value V, in dollars, of a piece of industrial equipment is modeled by the equation \(\mathrm{V = 45,000 - 2,500y}\). In the equation, y is the number of years since the equipment was purchased. Which of the following is the best interpretation of the number 45,000 in this context?
The value of the equipment, in dollars, when it was purchased.
The decrease in the value of the equipment, in dollars, each year.
The percent decrease in the value of the equipment each year.
The number of years it will take for the equipment's value to decrease to \(\$0\).
1. TRANSLATE the problem information
- Given: \(\mathrm{V = 45,000 - 2,500y}\) where V is value in dollars and y is years since purchase
- Need to interpret: What does 45,000 represent in this context?
2. INFER the key insight
- To understand what 45,000 means, I need to think about when this value would be "isolated" in the equation
- The constant term in a linear equation represents the initial value (when the variable equals zero)
- In this context: \(\mathrm{y = 0}\) means "zero years since purchase" = "when purchased"
3. SIMPLIFY by substitution
- Substitute \(\mathrm{y = 0}\): \(\mathrm{V = 45,000 - 2,500(0) = 45,000}\)
- This gives us the equipment value at the time of purchase
4. INFER why other choices are incorrect
- (B) The yearly decrease would be 2,500 (the coefficient of y)
- (C) This is linear (constant dollar) depreciation, not percentage-based
- (D) Time to reach $0: solve \(\mathrm{0 = 45,000 - 2,500y}\) → \(\mathrm{y = 18}\) years
Answer: A
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students don't recognize that the constant term represents the y-intercept, which is the initial value when \(\mathrm{y = 0}\).
Instead, they might confuse 45,000 with other aspects of the model, such as thinking it represents the total depreciation or confusing it with the rate of change. Without understanding that \(\mathrm{y = 0}\) corresponds to "when purchased," they may randomly guess among the choices.
This leads to confusion and guessing among the remaining choices.
Second Most Common Error:
Poor TRANSLATE reasoning: Students correctly identify that 45,000 is important but misunderstand what "years since purchase" means mathematically.
They might think larger y-values represent earlier times, or not connect \(\mathrm{y = 0}\) to the purchase date. This prevents them from making the crucial substitution that reveals 45,000 as the initial value.
This may lead them to select Choice D (18) by incorrectly calculating \(\mathrm{45,000/2,500}\) as some kind of interpretation.
The Bottom Line:
This problem tests whether students can connect the mathematical structure of linear equations to real-world context. The key insight is recognizing that the constant term represents the starting point when the independent variable equals zero.
The value of the equipment, in dollars, when it was purchased.
The decrease in the value of the equipment, in dollars, each year.
The percent decrease in the value of the equipment each year.
The number of years it will take for the equipment's value to decrease to \(\$0\).