Today, Mark invested $5,000 into an account that guarantees 7.50% p.a., compounded monthly and Madonna invested $5,000 into account guaranteeing 8.125% p.a., compounded quarterly.

Question

How long will it take (in years) for the value of Madonna's investment to be three times as much as Mark's?

Answer 1

For mark:
interest rate
= 7.5% p.a.
= 7.5%/12 per period of one month
= .625%
Future value of $5000 in n years, FVK(n)
= 5000*(1.00625)^12*n

For Madonna,
interest rate
= 8.125% p.a.
= 8.125%/4 per quarter
= 2.03125% per quarter
Future value of $5000 in n years, FVD(n)
= 5000*(1.0203125)^4*n

We look for n where
FVD(n) = 3FVK(n)
5000*(1.0203125)^4*n
= 3*5000*(1.00625)^12*n
(1.0203125)^4*n / (1.00625)^12*n = 3

Take log on both sides
4*n*log(1.0203125)-12*n*log(1.00625) = 3
n = log(3)/(4log(1.0203125-12log(1.00625))
= 193.785 years

Check:
in 193.785 years,
Mark will have $9,801,861,882.04
Madonna will have $29,405,482,100.99