Answer :
Absolutely! Let's compare the final amounts for two investments of [tex]$2500 over 19 years, with one earning 6% interest compounded monthly and the other earning 8% interest compounded monthly.
To do this, we'll use the formula for compound interest:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
Where:
- \( A \) = the final amount
- \( P \) = the principal amount (initial investment), which is $[/tex]2500
- [tex]\( r \)[/tex] = the annual interest rate (expressed as a decimal)
- For 6%, [tex]\( r = 0.06 \)[/tex]
- For 8%, [tex]\( r = 0.08 \)[/tex]
- [tex]\( n \)[/tex] = the number of times interest is compounded per year (monthly compounding means [tex]\( n = 12 \)[/tex])
- [tex]\( t \)[/tex] = the number of years the money is invested, which is 19 years
Let's start by calculating the final amount for the investment at 6%.
### Investment at 6%
1. Substitute the values into the formula:
[tex]\[ A_{6\%} = 2500 \left(1 + \frac{0.06}{12}\right)^{12 \times 19} \][/tex]
2. Compute inside the parentheses first:
[tex]\[ A_{6\%} = 2500 \left(1 + 0.005\right)^{228} \][/tex]
3. [tex]\( 1 + 0.005 = 1.005 \)[/tex]:
[tex]\[ A_{6\%} = 2500 \left(1.005\right)^{228} \][/tex]
4. Now, compute [tex]\( (1.005)^{228} \)[/tex]:
[tex]\[ (1.005)^{228} \approx 3.06566313 \][/tex]
5. Finally, multiply by the principal [tex]\( 2500 \)[/tex]:
[tex]\[ A_{6\%} = 2500 \times 3.06566313 \approx 7664.16 \][/tex]
So, the final amount after investing [tex]$2500 at 6% compounded monthly for 19 years is approximately \$[/tex]7664.16.
### Investment at 8%
1. Substitute the values into the formula:
[tex]\[ A_{8\%} = 2500 \left(1 + \frac{0.08}{12}\right)^{12 \times 19} \][/tex]
2. Compute inside the parentheses first:
[tex]\[ A_{8\%} = 2500 \left(1 + 0.00667\right)^{228} \][/tex]
3. [tex]\( 1 + 0.00667 = 1.00667 \)[/tex]:
[tex]\[ A_{8\%} = 2500 \left(1.00667\right)^{228} \][/tex]
4. Now, compute [tex]\( (1.00667)^{228} \)[/tex]:
[tex]\[ (1.00667)^{228} \approx 4.27938787 \][/tex]
5. Finally, multiply by the principal [tex]\( 2500 \)[/tex]:
[tex]\[ A_{8\%} = 2500 \times 4.27938787 \approx 10698.47 \][/tex]
So, the final amount after investing [tex]$2500 at 8% compounded monthly for 19 years is approximately \$[/tex]10698.47.
In summary:
- The final amount after investing \[tex]$2500 at 6% compounded monthly for 19 years is approximately \$[/tex]7664.16.
- The final amount after investing \[tex]$2500 at 8% compounded monthly for 19 years is approximately \$[/tex]10698.47.
- [tex]\( r \)[/tex] = the annual interest rate (expressed as a decimal)
- For 6%, [tex]\( r = 0.06 \)[/tex]
- For 8%, [tex]\( r = 0.08 \)[/tex]
- [tex]\( n \)[/tex] = the number of times interest is compounded per year (monthly compounding means [tex]\( n = 12 \)[/tex])
- [tex]\( t \)[/tex] = the number of years the money is invested, which is 19 years
Let's start by calculating the final amount for the investment at 6%.
### Investment at 6%
1. Substitute the values into the formula:
[tex]\[ A_{6\%} = 2500 \left(1 + \frac{0.06}{12}\right)^{12 \times 19} \][/tex]
2. Compute inside the parentheses first:
[tex]\[ A_{6\%} = 2500 \left(1 + 0.005\right)^{228} \][/tex]
3. [tex]\( 1 + 0.005 = 1.005 \)[/tex]:
[tex]\[ A_{6\%} = 2500 \left(1.005\right)^{228} \][/tex]
4. Now, compute [tex]\( (1.005)^{228} \)[/tex]:
[tex]\[ (1.005)^{228} \approx 3.06566313 \][/tex]
5. Finally, multiply by the principal [tex]\( 2500 \)[/tex]:
[tex]\[ A_{6\%} = 2500 \times 3.06566313 \approx 7664.16 \][/tex]
So, the final amount after investing [tex]$2500 at 6% compounded monthly for 19 years is approximately \$[/tex]7664.16.
### Investment at 8%
1. Substitute the values into the formula:
[tex]\[ A_{8\%} = 2500 \left(1 + \frac{0.08}{12}\right)^{12 \times 19} \][/tex]
2. Compute inside the parentheses first:
[tex]\[ A_{8\%} = 2500 \left(1 + 0.00667\right)^{228} \][/tex]
3. [tex]\( 1 + 0.00667 = 1.00667 \)[/tex]:
[tex]\[ A_{8\%} = 2500 \left(1.00667\right)^{228} \][/tex]
4. Now, compute [tex]\( (1.00667)^{228} \)[/tex]:
[tex]\[ (1.00667)^{228} \approx 4.27938787 \][/tex]
5. Finally, multiply by the principal [tex]\( 2500 \)[/tex]:
[tex]\[ A_{8\%} = 2500 \times 4.27938787 \approx 10698.47 \][/tex]
So, the final amount after investing [tex]$2500 at 8% compounded monthly for 19 years is approximately \$[/tex]10698.47.
In summary:
- The final amount after investing \[tex]$2500 at 6% compounded monthly for 19 years is approximately \$[/tex]7664.16.
- The final amount after investing \[tex]$2500 at 8% compounded monthly for 19 years is approximately \$[/tex]10698.47.
