A financial model estimates that at the end of each year from 2020 to 2025, the value of an investment...
GMAT Advanced Math : (Adv_Math) Questions
A financial model estimates that at the end of each year from 2020 to 2025, the value of an investment account was \(200\%\) more than its value at the end of the previous year. The model estimates that at the end of 2022, the account had a value of \($2,700\). Which of the following equations represents this model, where \(\mathrm{V}\) is the estimated account value in dollars \(\mathrm{t}\) years after the end of 2020, and \(\mathrm{t} \leq 5\)?
1. TRANSLATE the percentage increase language
- Given information:
- Each year the value is "200% more" than the previous year
- At end of 2022, the value was $2,700
- Need equation where V is value and t is years after end of 2020
- What "200% more" means: If last year's value was x, this year's value is \(\mathrm{x + 200\% \text{ of } x = x + 2x = 3x}\)
So the account triples each year.
2. INFER the exponential model structure
- Since the account triples each year, we have exponential growth with factor 3
- The general form is: \(\mathrm{V = V_0(3)^t}\)
- Here, \(\mathrm{V_0}\) = initial value at \(\mathrm{t = 0}\) (end of 2020)
- \(\mathrm{t}\) = years after the end of 2020
3. INFER what t = 2 represents and use the given condition
- End of 2022 is 2 years after end of 2020, so \(\mathrm{t = 2}\)
- At \(\mathrm{t = 2}\), we know \(\mathrm{V = 2700}\)
- Substitute: \(\mathrm{2700 = V_0(3)^2}\)
4. SIMPLIFY to find the initial value
\(\mathrm{2700 = V_0(9)}\)
\(\mathrm{V_0 = 2700 \div 9 = 300}\)
5. Write the complete equation
\(\mathrm{V = 300(3)^t}\)
Answer: A
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Many students misinterpret "200% more" as meaning the new value is 200% of the original (multiply by 2) rather than 200% more than the original (multiply by 3).
With this misunderstanding, they would set up \(\mathrm{V = V_0(2)^t}\). Using the condition at \(\mathrm{t = 2}\):
\(\mathrm{2700 = V_0(2)^2 = 4V_0}\)
\(\mathrm{V_0 = 675}\)
This leads them to think the equation is \(\mathrm{V = 675(2)^t}\), but this doesn't match any answer choice exactly. They might then select Choice B (\(\mathrm{V = 1350(2)^t}\)) thinking they made a small arithmetic error, or Choice D (\(\mathrm{V = 2700(2)^t}\)) if they think 2700 should be the coefficient.
The Bottom Line:
This problem hinges entirely on correctly interpreting percentage language. "200% more" is fundamentally different from "200% of" - it means you add 200% to what you already had, resulting in 300% of the original value.