If $400 is invested at an interest rate of 5.5% per year, find the amount of the investment at the end of 12 years for the following compounding methods. (Round your answers to the nearest cent.)
" ....for the following compounding methods"
none included
To find the amount of the investment at the end of 12 years for different compounding methods, we need to use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment
P = the principal amount (initial investment)
r = the interest rate (as a decimal)
n = the number of times interest is compounded per year
t = the number of years
Let's calculate the amount for each compounding method:
1. Compounded annually (n = 1):
A = 400(1 + 0.055/1)^(1*12)
A = 400(1.055)^12
A ≈ $784.33
2. Compounded semi-annually (n = 2):
A = 400(1 + 0.055/2)^(2*12)
A = 400(1.0275)^24
A ≈ $787.13
3. Compounded quarterly (n = 4):
A = 400(1 + 0.055/4)^(4*12)
A = 400(1.01375)^48
A ≈ $788.26
4. Compounded monthly (n = 12):
A = 400(1 + 0.055/12)^(12*12)
A = 400(1.00458)^144
A ≈ $788.75
5. Compounded daily (n = 365):
A = 400(1 + 0.055/365)^(365*12)
A = 400(1.00015)^4380
A ≈ $789.49
Therefore, the amount of the investment at the end of 12 years for different compounding methods are approximately:
1. Compounded annually: $784.33
2. Compounded semi-annually: $787.13
3. Compounded quarterly: $788.26
4. Compounded monthly: $788.75
5. Compounded daily: $789.49
To calculate the amount of the investment at the end of 12 years for different compounding methods, we will use the formula for compound interest: A = P(1 + r/n)^(nt), where:
A = the amount of the investment at the end of the time period
P = the initial principal (the amount invested)
r = the annual interest rate (as a decimal)
n = the number of times interest is compounded per year
t = the number of years
Let's calculate the amount of the investment for the given values:
1. Annual Compounding:
In this case, the interest is compounded only once a year. Therefore, n = 1.
A = 400(1 + 0.055/1)^(1*12)
A = 400(1 + 0.055)^12
A = 400(1.055)^12
A ≈ $804.20 (rounded to the nearest cent)
2. Semi-annual Compounding:
In this case, the interest is compounded twice a year. Therefore, n = 2.
A = 400(1 + 0.055/2)^(2*12)
A = 400(1 + 0.0275)^24
A ≈ $814.44 (rounded to the nearest cent)
3. Quarterly Compounding:
In this case, the interest is compounded four times a year. Therefore, n = 4.
A = 400(1 + 0.055/4)^(4*12)
A = 400(1 + 0.01375)^48
A ≈ $818.66 (rounded to the nearest cent)
4. Monthly Compounding:
In this case, the interest is compounded twelve times a year. Therefore, n = 12.
A = 400(1 + 0.055/12)^(12*12)
A = 400(1 + 0.004583)^144
A ≈ $822.99 (rounded to the nearest cent)
So, the amount of the investment at the end of 12 years for each compounding method is approximately:
- Annual Compounding: $804.20
- Semi-annual Compounding: $814.44
- Quarterly Compounding: $818.66
- Monthly Compounding: $822.99