\(\mathrm{V(m) = 32,000 - 250m}\) The function V models the value, in dollars, of a certain car m months after...
GMAT Algebra : (Alg) Questions
\(\mathrm{V(m) = 32,000 - 250m}\)
The function V models the value, in dollars, of a certain car \(\mathrm{m}\) months after it was purchased. According to the model, what is the predicted decrease in the car's value each year?
- 250
- 2,500
- 3,000
- 3,750
250
2,500
3,000
3,750
1. TRANSLATE the function information
- Given: \(\mathrm{V(m) = 32,000 - 250m}\) models car value after m months
- Question asks: "decrease in car's value each year"
- What this tells us: We need to find how much the car loses in value annually
2. TRANSLATE the coefficient meaning
- In the linear function \(\mathrm{V(m) = 32,000 - 250m}\):
- \(\mathrm{32,000}\) = initial car value
- \(\mathrm{-250}\) = the coefficient of m (number of months)
- This coefficient tells us the car decreases by \(\mathrm{\$250}\) each month
3. INFER the conversion strategy
- We have: monthly decrease = \(\mathrm{\$250}\)
- We need: annual decrease
- Key insight: Must convert monthly rate to yearly rate
- Strategy: Multiply monthly decrease by 12 (months per year)
4. SIMPLIFY the calculation
- Annual decrease = \(\mathrm{Monthly\ decrease \times 12}\)
- Annual decrease = \(\mathrm{250 \times 12 = 3,000}\)
Answer: C ($3,000)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE reasoning: Students misinterpret what the coefficient \(\mathrm{-250}\) represents or confuse the question's timeframe.
Some students see the coefficient \(\mathrm{-250}\) and think this is already the annual decrease, missing that it represents the monthly rate. This may lead them to select Choice A (250) by just taking the absolute value of the coefficient.
Second Most Common Error:
Poor unit conversion in INFER: Students understand that \(\mathrm{-250}\) is monthly decrease but make errors in the time conversion.
They might think there are 10 months in a year or make other conversion errors, leading to calculations like \(\mathrm{250 \times 10 = 2,500}\). This may lead them to select Choice B (2,500).
The Bottom Line:
This problem tests whether students can correctly interpret linear function coefficients and perform proper unit conversions. The key is recognizing that the coefficient represents a monthly rate that must be scaled up to find the annual rate.
250
2,500
3,000
3,750