Certainly! Let's work through the problem step-by-step to compare the two different compounding methods over a 5-year period.
### Initial Information
- Initial investment (principal): [tex]$10,000
- Time period: 5 years
### Scenario 1: Compounding Monthly at 9%
- Annual interest rate: 9% (0.09 as a decimal)
- Compounding frequency: monthly (12 times per year)
The formula for compound interest when compounded monthly is:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
where:
- \( P \) is the principal amount ($[/tex]10,000)
- [tex]\( r \)[/tex] is the annual interest rate (0.09)
- [tex]\( n \)[/tex] is the number of times interest applied per time period (12)
- [tex]\( t \)[/tex] is the time in years (5)
Plugging in the values, we get:
[tex]\[ A = 10{,}000 \left(1 + \frac{0.09}{12}\right)^{12 \times 5} \][/tex]
After calculating, we find:
[tex]\[ A \approx 15{,}656.81 \][/tex]
### Scenario 2: Compounding Continuously at 8.87%
- Annual interest rate: 8.87% (0.0887 as a decimal)
The formula for continuous compounding is:
[tex]\[ A = Pe^{rt} \][/tex]
where:
- [tex]\( P \)[/tex] is the principal amount ([tex]$10,000)
- \( r \) is the annual interest rate (0.0887)
- \( t \) is the time in years (5)
- \( e \) is the base of the natural logarithm, approximately equal to 2.71828
Plugging in the values, we get:
\[ A = 10{,}000 \cdot e^{0.0887 \times 5} \]
After calculating, we find:
\[ A \approx 15{,}581.51 \]
### Conclusion
- With 9% compounded monthly, the investment grows to approximately $[/tex]15,656.81.
- With 8.87% compounded continuously, the investment grows to approximately $15,581.51.
Therefore, the 9% compounded monthly rate yields a larger amount in 5 years compared to the 8.87% compounded continuously rate. The difference in the final amounts is:
[tex]\[ 15{,}656.81 - 15{,}581.51 \approx 75.30 \][/tex]
So, the 9% compounded monthly investment is the better option for yielding a larger amount after 5 years.
