If $500 is invested at an interest rate of 4.5% per year, find the amount of the investment at the end of 11 years for the following compounding methods. (Round your answers to the nearest cent.)
Find a, Anually
Find b, SemiAnually
Find c, Quarterly
Find d, Continuosly
d. P = Po*e^rt.
P/Po = e^rt.
r*t = 0.045/yr. * 11yrs. = 0.495,
e^0.495 = P/Po,
1.6305 = P/500, P = $820.25.
To find the amount of the investment at the end of 11 years using different compounding methods, we can use the formula for compound interest:
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
where:
- A is the amount of the investment at the end of the specified time period
- P is the principal amount (initial investment)
- r is the interest rate
- n is the number of times the interest is compounded per year
- t is the number of years
Given:
P = $500
r = 4.5% = 0.045
t = 11 years
a) Annually (n = 1):
Using the formula, we have:
\[
A = 500 \left(1 + \frac{0.045}{1}\right)^{1 \times 11}
\]
Calculating,
\[
A = 500 \times 1.045^{11} \approx $817.56
\]
b) SemiAnnually (n = 2):
Using the formula, we have:
\[
A = 500 \left(1 + \frac{0.045}{2}\right)^{2 \times 11}
\]
Calculating,
\[
A = 500 \times 1.0225^{22} \approx $819.97
\]
c) Quarterly (n = 4):
Using the formula, we have:
\[
A = 500 \left(1 + \frac{0.045}{4}\right)^{4 \times 11}
\]
Calculating,
\[
A = 500 \times 1.01125^{44} \approx $820.74
\]
d) Continuously (n approximating to infinity):
Using the formula, we have:
\[
A = 500 \times e^{0.045 \times 11}
\]
Calculating,
\[
A \approx 500 \times e^{0.495} \approx $821.99
\]
Therefore, the amount of the investment at the end of 11 years, rounded to the nearest cent, for the different compounding methods are as follows:
a) Annually: $817.56
b) Semiannually: $819.97
c) Quarterly: $820.74
d) Continuously: $821.99
To find the amount of the investment at the end of 11 years for each compounding method, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the amount at the end of the investment period
P = principal amount (initial investment)
r = annual interest rate (as a decimal)
n = number of times the interest is compounded per year
t = number of years
Now, let's calculate each amount using the given information:
a) Annually compounded interest:
In this case, the interest is compounded once a year.
P = $500
r = 4.5% = 0.045 (as a decimal)
n = 1 (compounded annually)
t = 11 years
Plugging in these values into the formula:
A = 500(1 + 0.045/1)^(1*11)
A ≈ $861.40
Therefore, the amount at the end of 11 years, with annual compounding, will be approximately $861.40.
b) Semiannually compounded interest:
In this case, the interest is compounded twice a year.
P = $500
r = 4.5% = 0.045 (as a decimal)
n = 2 (compounded semiannually)
t = 11 years
Plugging in these values into the formula:
A = 500(1 + 0.045/2)^(2*11)
A ≈ $865.63
Therefore, the amount at the end of 11 years, with semiannual compounding, will be approximately $865.63.
c) Quarterly compounded interest:
In this case, the interest is compounded four times a year.
P = $500
r = 4.5% = 0.045 (as a decimal)
n = 4 (compounded quarterly)
t = 11 years
Plugging in these values into the formula:
A = 500(1 + 0.045/4)^(4*11)
A ≈ $867.68
Therefore, the amount at the end of 11 years, with quarterly compounding, will be approximately $867.68.
d) Continuously compounded interest:
For continuous compounding, we use the formula:
A = Pe^(rt)
Where:
A = the amount at the end of the investment period
P = principal amount (initial investment)
r = annual interest rate (as a decimal)
t = number of years
e = Euler's number, approximately 2.71828
P = $500
r = 4.5% = 0.045 (as a decimal)
t = 11 years
Plugging in these values into the formula:
A = 500 * e^(0.045 * 11)
A ≈ $868.97
Therefore, the amount at the end of 11 years, with continuous compounding, will be approximately $868.97.
In summary:
a) Annually compounded interest: $861.40
b) Semiannually compounded interest: $865.63
c) Quarterly compounded interest: $867.68
d) Continuously compounded interest: $868.97
a. P = Po(1+r)^n.
Po = $500.
r = 0.045/yr. = Annual % rate.
n = 1Comp/yr. * 11yrs. = 11Compounding periods.
b. r = 0.045/yr. * 0.5yr. = 0.0225 = Semi-annual % rate.
n = 2Comp./yr. * 11yrs. = 22 Compounding periods.
c. r = 0.045/yr. * 0.25yr. = 0.01125 = Quarterly % rate.
n = 4Comp./yr. * 11yrs. = 44 Compounding periods.