Question

Compound Interest A deposit of $400 is made in an account that earns interest at an annual rate of 2.5%. How long will it take for the balance to double when the interest is compounded (a) annually, (b) monthly, (c) daily, and (d) continuously?

Answer 1

Answer:

(a) annually = 28.07 years

(b) monthly = 27.75 years

(c) daily =  27.73 years

(d) continuously = 27.72 years

Step-by-step explanation:

given data

principal =  $400

annual rate = 2.5%  = 0.025

solution

we know here amount formula that is

amount = principal × $(1+\frac{r}{n})^{n*t}$    ..................1

put here value for compound annually

800 = 400 × $(1+\frac{0.025}{1})^{t}$

take ln both side

ln 2 = ln ${1.025}^{t}$

t = 28.07 years

and

put value now in equation 1 for monthly

amount = principal × $(1+\frac{r}{n})^{n*t}$

800 = 400 × $(1+\frac{0.025}{12})^{12*t}$

take ln both side

ln 2 = 12t × ln(1.00208333)

t = 27.75 years

and

put value now in equation 1 for daily

amount = principal × $(1+\frac{r}{n})^{n*t}$

800 = 400 × $(1+\frac{0.025}{365})^{365*t}$

take ln both side

ln 2 = 365 t × ln (1.0000684932)

t = 27.73 years

and

for compound continuously

amount = principal × $e^{r*t}$   .................2

put here value

800 = 400 × $e^{0.025*t}$

t = 27.72 years