Question

A deposit of $500 is made in an account that earns interest at an annual rate of 4%. 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 = 17.67 years

(b) monthly = 17.36 years

(c) daily =  17.34 years

(d) continuously = 17.328 years

Step-by-step explanation:

given data

principal =  $500

annual rate = 4%

solution

we know here amount formula that is

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

put here value for compound annually

1000 = 500 × $(1+\frac{0.04}{1})^{t}$

take ln both side

ln 2 = ln ${1.04}^{t}$

t = 17.67 years

and

put value now in equation 1 for monthly

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

1000 = 500 × $(1+\frac{0.04}{12})^{12*t}$

take ln both side

ln 2 = 12t × ln(1.003333)

t = 17.36 years

and

put value now in equation 1 for daily

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

1000 = 500 × $(1+\frac{0.04}{365})^{365*t}$

take ln both side

ln 2 = 365 t × ln (1.0001095)

t = 17.34 years

and

for compound continuously

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

put here value

1000 = 500 × $e^{0.04*t}$

t = 17.328 years