To compare the two options, we'll use the formulas for calculating the final amount with different compounding intervals. Let's analyze each option step by step.
Option 1: Account pays 3.5% interest compounded monthly.
We'll use the formula for monthly compounding interest:
\(\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]\)
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 (decimal).
- \(\( n \)\) is the number of times that interest is compounded per year.
- \(\( t \)\) is the time the money is invested for, in years.
Let's plug in the values for the first option:
- \(\( P = $5500 \)\)
- \(\( r = 3.5\% = 0.035 \)\) (as a decimal)
- \(\( n = 12 \)\) (since the interest is compounded monthly)
- \(\( t = 7 \)\) (the time period in years)
\(\[ A_{\text{monthly}} = 5500 \cdot \left(1 + \frac{0.035}{12}\right)^{12 \cdot 7} \]\)
We calculate the amount for Option 1:
\(\[ A_{\text{monthly}} = 5500 \cdot \left(1 + \frac{0.035}{12}\right)^{84} \]\)
\(\[ A_{\text{monthly}} = 5500 \cdot \left(1 + 0.00291666667\right)^{84} \]\)
\(\[ A_{\text{monthly}} = 5500 \cdot \left(1.00291666667\right)^{84} \]\)
\(\[ A_{\text{monthly}} = 5500 \cdot (1.27374523406) \]\)
\(\[ A_{\text{monthly}} \approx $7005.60 \]\)
Option 2: Account pays 3.495% interest compounded continuously.
For continuous compounding, we use the formula:
\(\[ A = Pe^{rt} \]\)
Where:
- \(\( A \)\) is the amount of money accumulated after t years, including interest.
- \(\( P \)\) is the principal amount.
- \(\( e \)\) is Euler's number (approximately 2.71828).
- \(\( r \)\) is the annual interest rate (decimal).
- \(\( t \)\) is the time the money is invested for, in years.
Let's plug in the values for the second option:
- \(\( P = $5500 \)\)
- \(\( r = 3.495\% = 0.03495 \)\) (as a decimal)
- \(\( t = 7 \)\) (the time period in years)
\(\[ A_{\text{continuous}} = 5500 \cdot e^{0.03495 \cdot 7} \]\)
We calculate the amount using continuous compounding:
\(\[ A_{\text{continuous}} = 5500 \cdot e^{0.24465} \]\)
\(\[ A_{\text{continuous}} \approx 5500 \cdot 1.27736300585 \]\)
\(\[ A_{\text{continuous}} \approx $7024.50 \]\)
Conclusion:
Comparing the two amounts:
- Monthly Compounding: \(\( A_{\text{monthly}} \approx $7005.60 \)\)
- Continuous Compounding: \(\( A_{\text{continuous}} \approx $7024.50 \)\)
The account with 3.495% interest compounded continuously will give a higher amount after 7 years, so it would be the better option of the two for maximizing your returns.
