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Math  /  Algebra

QuestionCompare investing $1250\$ 1250 at 9\% compounded monthly for 25 years with investing $1250\$ 1250 at 14%14 \% compounded monthly for 25 years.
The final amount after investing at 9%9 \% is $\$ □\square. □\square (Type an integer or a decimal. Round to the nearest cent as needed.)

Studdy Solution

STEP 1

1. The principal amount for both investments is $1250.
2. The interest rates are 9% and 14%, compounded monthly.
3. The investment period is 25 years for both cases.
4. We need to calculate the final amount for both investments and compare them.
5. The formula for compound interest is \( A = P \left(1 + \frac{r}{n}\right)^{nt} \), 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.

STEP 2

1. Calculate the final amount for the investment at 9% compounded monthly.
2. Calculate the final amount for the investment at 14% compounded monthly.
3. Compare the final amounts from both investments.

STEP 3

Calculate the final amount for the investment at 9% compounded monthly.
- Convert the annual interest rate to a decimal: r=9100=0.09 r = \frac{9}{100} = 0.09 . - The number of times interest is compounded per year is n=12 n = 12 . - The time in years is t=25 t = 25 .
Use the compound interest formula: A=1250(1+0.0912)12×25 A = 1250 \left(1 + \frac{0.09}{12}\right)^{12 \times 25}
Calculate the expression inside the parentheses: 1+0.0912=1.0075 1 + \frac{0.09}{12} = 1.0075
Calculate the exponent: 12×25=300 12 \times 25 = 300
Calculate the final amount: A=1250×(1.0075)300 A = 1250 \times (1.0075)^{300}
Using a calculator, compute (1.0075)300≈9.816 (1.0075)^{300} \approx 9.816 .
Now, calculate: A=1250×9.816≈12270.00 A = 1250 \times 9.816 \approx 12270.00
The final amount after investing at 9% is 12270.00 \boxed{12270.00} .

STEP 4

Calculate the final amount for the investment at 14% compounded monthly.
- Convert the annual interest rate to a decimal: r=14100=0.14 r = \frac{14}{100} = 0.14 . - The number of times interest is compounded per year is n=12 n = 12 . - The time in years is t=25 t = 25 .
Use the compound interest formula: A=1250(1+0.1412)12×25 A = 1250 \left(1 + \frac{0.14}{12}\right)^{12 \times 25}
Calculate the expression inside the parentheses: 1+0.1412=1.0116667 1 + \frac{0.14}{12} = 1.0116667
Calculate the exponent: 12×25=300 12 \times 25 = 300
Calculate the final amount: A=1250×(1.0116667)300 A = 1250 \times (1.0116667)^{300}
Using a calculator, compute (1.0116667)300≈29.96 (1.0116667)^{300} \approx 29.96 .
Now, calculate: A=1250×29.96≈37450.00 A = 1250 \times 29.96 \approx 37450.00
The final amount after investing at 14% is 37450.00 \boxed{37450.00} .

STEP 5

Compare the final amounts from both investments.
- Final amount at 9%: 12270.00 12270.00 - Final amount at 14%: 37450.00 37450.00
The investment at 14% yields a significantly higher final amount compared to the investment at 9%.

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