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.

Question

Principal
Rate
Time
​$1007
1.5​%
4 years

Answer 1

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