To compare these two investments, we need to calculate the future value of each investment after 18 years using the compound interest formula:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
where:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount (the initial amount of money).
- \( r \) is the annual interest rate (in decimal).
- \( n \) is the number of times that interest is compounded per year.
- \( t \) is the number of years the money is invested or borrowed.
Ella's Account:
- Principal (\( P \)): $4,800
- Annual interest rate (\( r \)): \( 2 \frac{1}{8}% = 2.125% = 0.02125 \)
- Compounding frequency (\( n \)): quarterly = 4 times per year
- Time (\( t \)): 18 years
Using the formula for Ella:
\[ A_E = 4800 \left(1 + \frac{0.02125}{4}\right)^{4 \cdot 18} \]
Calculating \( \frac{0.02125}{4} \):
\[ \frac{0.02125}{4} = 0.0053125 \]
Now substitute into the formula:
\[ A_E = 4800 \left(1 + 0.0053125\right)^{72} \] \[ A_E = 4800 \left(1.0053125\right)^{72} \]
Calculating \( (1.0053125)^{72} \):
\[ (1.0053125)^{72} \approx 1.485947 \]
Now calculate \( A_E \):
\[ A_E \approx 4800 \times 1.485947 \approx 7137.54 \]
Santiago's Account:
- Principal (\( P \)): $4,800
- Annual interest rate (\( r \)): \( 2 \frac{3}{8}% = 2.375% = 0.02375 \)
- Compounding frequency (\( n \)): daily = 365 times per year
- Time (\( t \)): 18 years
Using the formula for Santiago:
\[ A_S = 4800 \left(1 + \frac{0.02375}{365}\right)^{365 \cdot 18} \]
Calculating \( \frac{0.02375}{365} \):
\[ \frac{0.02375}{365} \approx 0.0000650685 \]
Now substitute into the formula:
\[ A_S = 4800 \left(1 + 0.0000650685\right)^{6570} \] \[ A_S = 4800 \left(1.0000650685\right)^{6570} \]
Calculating \( (1.0000650685)^{6570} \):
\[ (1.0000650685)^{6570} \approx 1.490592 \]
Now calculate \( A_S \):
\[ A_S \approx 4800 \times 1.490592 \approx 7150.84 \]
Difference between Santiago's and Ella's Account:
\[ \text{Difference} = A_S - A_E \approx 7150.84 - 7137.54 \approx 13.30 \]
Rounding to the nearest dollar:
\[ \text{Difference} \approx 13 \]
Thus, the answer is Santiago would have approximately $13 more than Ella after 18 years.
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