Time (years)Total amount (dollars)0604.001606.422608.84Rosa opened a savings account at a bank. The table shows the exponential relationship between t...
GMAT Advanced Math : (Adv_Math) Questions
| Time (years) | Total amount (dollars) |
|---|---|
| 0 | 604.00 |
| 1 | 606.42 |
| 2 | 608.84 |
Rosa opened a savings account at a bank. The table shows the exponential relationship between the time \(\mathrm{t}\), in years, since Rosa opened the account and the total amount \(\mathrm{n}\), in dollars, in the account. If Rosa made no additional deposits or withdrawals, which of the following equations best represents the relationship between \(\mathrm{t}\) and \(\mathrm{n}\)?
\(\mathrm{n = (1+604)^t}\)
\(\mathrm{n = (1+ 0.004)^t}\)
\(\mathrm{n = 604(1+ 0.004)^t}\)
\(\mathrm{n = 0.004(1+ 604)^t}\)
Step-by-Step Solution
1. TRANSLATE the problem information
- Given information:
- Table showing exponential relationship between time t (years) and amount n (dollars)
- At \(\mathrm{t = 0}\): \(\mathrm{n = 604.00}\)
- At \(\mathrm{t = 1}\): \(\mathrm{n = 606.42}\)
- At \(\mathrm{t = 2}\): \(\mathrm{n = 608.84}\)
- Need to find: equation that represents this relationship
2. INFER the exponential form structure
- Since it's exponential growth, use the form: \(\mathrm{n = a(1 + r)^t}\)
- Where: \(\mathrm{a}\) = initial amount, \(\mathrm{r}\) = growth rate, \(\mathrm{t}\) = time
- Strategy: Use the data points to solve for \(\mathrm{a}\) and \(\mathrm{r}\)
3. SIMPLIFY to find the initial amount
- When \(\mathrm{t = 0}\): \(\mathrm{n = a(1 + r)^0}\)
- Since any number to the 0 power equals 1: \(\mathrm{n = a(1) = a}\)
- From the table when \(\mathrm{t = 0}\), \(\mathrm{n = 604}\)
- Therefore: \(\mathrm{a = 604}\)
4. SIMPLIFY to find the growth rate
- When \(\mathrm{t = 1}\): \(\mathrm{n = 604(1 + r)^1 = 604(1 + r)}\)
- From the table when \(\mathrm{t = 1}\), \(\mathrm{n = 606.42}\)
- Set up equation: \(\mathrm{606.42 = 604(1 + r)}\)
- Divide both sides by 604: \(\mathrm{606.42/604 = 1 + r}\)
- Calculate: \(\mathrm{1.004 = 1 + r}\) (use calculator)
- Subtract 1: \(\mathrm{r = 0.004}\)
5. INFER the final equation
- Substitute \(\mathrm{a = 604}\) and \(\mathrm{r = 0.004}\) into \(\mathrm{n = a(1 + r)^t}\)
- Final equation: \(\mathrm{n = 604(1 + 0.004)^t}\)
Answer: C. \(\mathrm{n = 604(1 + 0.004)^t}\)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students struggle to connect the table values to the components of the exponential formula. They might not realize that the value when \(\mathrm{t = 0}\) gives them the initial amount directly, or they might confuse which parameter represents what in the exponential equation.
This confusion often leads them to select Choice A (\(\mathrm{(1+604)^t}\)) because they mix up the initial amount with the growth factor, or Choice B (\(\mathrm{(1+ 0.004)^t}\)) because they identify the growth rate correctly but forget to include the initial amount coefficient.
Second Most Common Error:
Poor SIMPLIFY execution: Students make arithmetic errors when calculating 606.42 ÷ 604, getting an incorrect growth rate. They might calculate this as 0.004 directly instead of recognizing it equals 1.004, leading to \(\mathrm{r = 0.004}\).
This leads to confusion about which answer choice matches their work, causing them to second-guess their approach and potentially guess.
The Bottom Line:
This problem requires students to bridge the gap between tabular data and algebraic form. The key insight is recognizing that exponential growth problems give you direct access to parameters through strategic substitution - the \(\mathrm{t = 0}\) case immediately reveals the initial amount, and the \(\mathrm{t = 1}\) case allows direct calculation of the growth rate.
\(\mathrm{n = (1+604)^t}\)
\(\mathrm{n = (1+ 0.004)^t}\)
\(\mathrm{n = 604(1+ 0.004)^t}\)
\(\mathrm{n = 0.004(1+ 604)^t}\)