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

(a) annually = 17.67 years

(b) monthly = 17.36 years

(c) daily = 17.34 years

(d) continuously = 17.328 years

Explanation:

given data

principal = $500

annual rate = 4%

solution

we know here amount formula that is

amount = principal ×
`(1+(r)/(n))^(n*t)` ..................1

put here value for compound annually

1000 = 500 ×
`(1+(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+(r)/(n))^(n*t)`

1000 = 500 ×
`(1+(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+(r)/(n))^(n*t)`

1000 = 500 ×
`(1+(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