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:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
Where:
- [tex]\( A \)[/tex] is the amount of money accumulated after n years, including interest.
- [tex]\( P \)[/tex] is the principal amount (the initial amount of money).
- [tex]\( r \)[/tex] is the annual interest rate (decimal).
- [tex]\( n \)[/tex] is the number of times that interest is compounded per year.
- [tex]\( t \)[/tex] is the time the money is invested for, in years.
Let's plug in the values for the first option:
- [tex]\( P = $5500 \)[/tex]
- [tex]\( r = 3.5\% = 0.035 \)[/tex] (as a decimal)
- [tex]\( n = 12 \)[/tex] (since the interest is compounded monthly)
- [tex]\( t = 7 \)[/tex] (the time period in years)
[tex]\[ A_{\text{monthly}} = 5500 \cdot \left(1 + \frac{0.035}{12}\right)^{12 \cdot 7} \][/tex]
We calculate the amount for Option 1:
[tex]\[ A_{\text{monthly}} = 5500 \cdot \left(1 + \frac{0.035}{12}\right)^{84} \][/tex]
[tex]\[ A_{\text{monthly}} = 5500 \cdot \left(1 + 0.00291666667\right)^{84} \][/tex]
[tex]\[ A_{\text{monthly}} = 5500 \cdot \left(1.00291666667\right)^{84} \][/tex]
[tex]\[ A_{\text{monthly}} = 5500 \cdot (1.27374523406) \][/tex]
[tex]\[ A_{\text{monthly}} \approx $7005.60 \][/tex]
Option 2: Account pays 3.495% interest compounded continuously.
For continuous compounding, we use the formula:
[tex]\[ A = Pe^{rt} \][/tex]
Where:
- [tex]\( A \)[/tex] is the amount of money accumulated after t years, including interest.
- [tex]\( P \)[/tex] is the principal amount.
- [tex]\( e \)[/tex] is Euler's number (approximately 2.71828).
- [tex]\( r \)[/tex] is the annual interest rate (decimal).
- [tex]\( t \)[/tex] is the time the money is invested for, in years.
Let's plug in the values for the second option:
- [tex]\( P = $5500 \)[/tex]
- [tex]\( r = 3.495\% = 0.03495 \)[/tex] (as a decimal)
- [tex]\( t = 7 \)[/tex] (the time period in years)
[tex]\[ A_{\text{continuous}} = 5500 \cdot e^{0.03495 \cdot 7} \][/tex]
We calculate the amount using continuous compounding:
[tex]\[ A_{\text{continuous}} = 5500 \cdot e^{0.24465} \][/tex]
[tex]\[ A_{\text{continuous}} \approx 5500 \cdot 1.27736300585 \][/tex]
[tex]\[ A_{\text{continuous}} \approx $7024.50 \][/tex]
Conclusion:
Comparing the two amounts:
- Monthly Compounding: [tex]\( A_{\text{monthly}} \approx $7005.60 \)[/tex]
- Continuous Compounding: [tex]\( A_{\text{continuous}} \approx $7024.50 \)[/tex]
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.
