Occasionally a savings account may actually pay interest compounded continuously. For each deposit, find the interest earned if interest is compounded (a) semiannually, (b) quarterly, (c) monthly, (d) daily, and (e) continuously. Use 1 year=365 days.
Principal
Rate
Time
$1007
1.5%
4 years
To find the interest earned, we can use the formula:
A = P(1 + r/n)^(nt) - P,
where A is the amount after interest, P is the principal, r is the interest rate, n is the number of times compounded per year, and t is the time in years.
(a) Semiannually:
P = $1007
r = 1.5% = 0.015
n = 2 (compounded twice a year)
t = 4
A = $1007(1 + 0.015/2)^(2*4) - $1007
A = $1007(1 + 0.0075)^8 - $1007
A = $1007(1.0075)^8 - $1007
A ≈ $1007(1.0617) - $1007
A ≈ $1069.27 - $1007
A ≈ $62.27
The interest earned semiannually is $62.27.
(b) Quarterly:
P = $1007
r = 1.5% = 0.015
n = 4 (compounded four times a year)
t = 4
A = $1007(1 + 0.015/4)^(4*4) - $1007
A = $1007(1 + 0.00375)^16 - $1007
A ≈ $1007(1.00375)^16 - $1007
A ≈ $1007(1.0633) - $1007
A ≈ $1069.29 - $1007
A ≈ $62.29
The interest earned quarterly is $62.29.
(c) Monthly:
P = $1007
r = 1.5% = 0.015
n = 12 (compounded twelve times a year)
t = 4
A = $1007(1 + 0.015/12)^(12*4) - $1007
A = $1007(1 + 0.00125)^48 - $1007
A ≈ $1007(1.00125)^48 - $1007
A ≈ $1007(1.0662) - $1007
A ≈ $1069.80 - $1007
A ≈ $62.80
The interest earned monthly is $62.80.
(d) Daily:
P = $1007
r = 1.5% = 0.015
n = 365 (compounded daily)
t = 4
A = $1007(1 + 0.015/365)^(365*4) - $1007
A ≈ $1007(1.0000411)^1460 - $1007
A ≈ $1007(1.067) - $1007
A ≈ $1069.69 - $1007
A ≈ $62.69
The interest earned daily is $62.69.
(e) Continuously:
P = $1007
r = 1.5% = 0.015
t = 4
A = $1007e^(0.015*4) - $1007
A ≈ $1007e^(0.06) - $1007
A ≈ $1007(1.061837) - $1007
A ≈ $1069.36 - $1007
A ≈ $62.36
The interest earned continuously is $62.36.
Therefore, the interest earned for each deposit is approximately:
(a) Semiannually: $62.27
(b) Quarterly: $62.29
(c) Monthly: $62.80
(d) Daily: $62.69
(e) Continuously: $62.36
To find the interest earned for each compounding frequency, you can use the formula:
A = P(1 + r/n)^(nt) - P
Where:
A = Final amount
P = Principal (initial deposit)
r = Annual interest rate (in decimal form)
n = Number of times interest is compounded per year
t = Time in years
(a) Semiannually:
n = 2 (compounded twice a year)
t = 4 years
A = 1007(1 + 0.015/2)^(2*4) - 1007
A = 1007(1.0075)^(8) - 1007
A ≈ 1007(1.061717) - 1007
A ≈ 1069.29 - 1007
A ≈ $62.29
The interest earned if compounded semiannually is approximately $62.29.
(b) Quarterly:
n = 4 (compounded four times a year)
t = 4 years
A = 1007(1 + 0.015/4)^(4*4) - 1007
A = 1007(1.00375)^(16) - 1007
A ≈ 1007(1.062196) - 1007
A ≈ 1069.83 - 1007
A ≈ $62.83
The interest earned if compounded quarterly is approximately $62.83.
(c) Monthly:
n = 12 (compounded twelve times a year)
t = 4 years
A = 1007(1 + 0.015/12)^(12*4) - 1007
A = 1007(1.00125)^(48) - 1007
A ≈ 1007(1.061678) - 1007
A ≈ 1069.17 - 1007
A ≈ $62.17
The interest earned if compounded monthly is approximately $62.17.
(d) Daily:
n = 365 (compounded 365 times a year)
t = 4 years
A = 1007(1 + 0.015/365)^(365*4) - 1007
A = 1007(1.000041)^(1460) - 1007
A ≈ 1007(1.061836) - 1007
A ≈ 1069.52 - 1007
A ≈ $62.52
The interest earned if compounded daily is approximately $62.52.
(e) Continuously:
A = P*e^(rt)
Where:
e ≈ 2.71828 (Euler's number)
P = 1007
r = 0.015
t = 4 years
A = 1007*e^(0.015*4) - 1007
A ≈ 1007*e^(0.06) - 1007
A ≈ 1007*1.061836 - 1007
A ≈ 1069.52 - 1007
A ≈ $62.52
The interest earned if compounded continuously is approximately $62.52.
To find the interest earned using different compounding periods, we need to use the formula for compound interest:
A = P(1 + r/n)^(nt)
where:
A is the final amount
P is the principal amount
r is the annual interest rate (in decimal form)
n is the number of times interest is compounded per year
t is the time in years
Let's calculate the interest earned for each compounding period:
(a) Semiannually (compounded twice a year):
n = 2 (compounded semiannually)
t = 4 years
Using the formula, we have:
P = $1007
r = 1.5% = 0.015 (in decimal form)
A = P(1 + r/n)^(nt)
A = $1007(1 + 0.015/2)^(2*4)
Calculate A using the formula, then subtract the principal amount to find the interest earned.
(b) Quarterly (compounded four times a year):
n = 4 (compounded quarterly)
t = 4 years
Using the formula:
A = $1007(1 + 0.015/4)^(4*4)
Calculate A using the formula, then subtract the principal amount to find the interest earned.
(c) Monthly (compounded 12 times a year):
n = 12 (compounded monthly)
t = 4 years
Using the formula:
A = $1007(1 + 0.015/12)^(12*4)
Calculate A using the formula, then subtract the principal amount to find the interest earned.
(d) Daily (compounded 365 times a year, assuming 1 year = 365 days):
n = 365 (compounded daily)
t = 4 years
Using the formula:
A = $1007(1 + 0.015/365)^(365*4)
Calculate A using the formula, then subtract the principal amount to find the interest earned.
(e) Continuously:
For continuous compounding, we use the formula:
A = P*e^(rt)
where:
e is the mathematical constant approximately equal to 2.71828
Using the formula:
A = $1007*e^(0.015*4)
Calculate A using the formula, then subtract the principal amount to find the interest earned.
By following these steps, you can find the interest earned for each compounding period.