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

(a) annually = 28.07 years

(b) monthly = 27.75 years

(c) daily = 27.73 years

(d) continuously = 27.72 years

Explanation:

given data

principal = $400

annual rate = 2.5% = 0.025

solution

we know here amount formula that is

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

put here value for compound annually

800 = 400 ×
`(1+(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+(r)/(n))^(n*t)`

800 = 400 ×
`(1+(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+(r)/(n))^(n*t)`

800 = 400 ×
`(1+(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